Finite Group Databasebeta

← #36356381

Subgroups — #36356381

Order 22144.

1252 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×D173C211yesmaximal, characteristic
2C2×C4×C22⋊D173C211yesmaximal, characteristic, complemented
3C2×#19149639C211yesmaximal, characteristic, complemented
4C22×#14583803C211yesmaximal, characteristic, complemented
5C2×C22⋊C4×D173C211yesmaximal, characteristic, complemented
6#26011391C211yesmaximal, characteristic, complemented
7#26011391C211yesmaximal, characteristic, complemented
8#26011371C211yesmaximal, characteristic, complemented
9#26011369C211yesmaximal, characteristic, complemented
10#26011369C211yesmaximal, characteristic, complemented
11#26011350C211yesmaximal, characteristic, complemented
12#26011346C211yesmaximal, characteristic, complemented
13#26011346C211yesmaximal, characteristic, complemented
14#26011338C211yesmaximal, characteristic, complemented
15#386×C173C211yesmaximal, characteristic, complemented
16C24×D173C2211yescharacteristic
17C24×D173C2211yescharacteristic
18C22×D692C2211yescharacteristic
19C22×C22⋊D173C2211yescharacteristic
20C22×C22⋊D173C2211yescharacteristic
21C2×D4×D173—24no—
22C2×D4×D173—24no—
23C2×D4×D173C418yescomplemented
24C23×Dic173C2211yescharacteristic, complemented
25C22×C4×D173C2211yescharacteristic
26C22×C4×D173C2211yescharacteristic, complemented
27C22×C4×D173C2211yescharacteristic, complemented
28C4×C22⋊D173—24no—
29#19149639—24no—
30C2×#14583805C2211yescharacteristic, complemented
31C2×#14583805C2211yescharacteristic, complemented
32C2×C22⋊Dic173C2211yescharacteristic, complemented
33C2×C22⋊Dic173C2211yescharacteristic, complemented
34C2×C22⋊Dic173C2211yescharacteristic, complemented
35C2×#14583803—24no—
36C2×#14583803C2211yescharacteristic
37C2×#14583803C2211yescharacteristic
38C2×#14583803C2211yescharacteristic
39C2×#14583803C2211yescharacteristic
40C2×#14583803C2211yescharacteristic, complemented
41C2×#14583803C2211yescharacteristic, complemented
42C2×#14583803C2211yescharacteristic, complemented
43C2×#14583803C2211yescharacteristic, complemented
44C2×#14583803C2211yescharacteristic, complemented
45C2×#14583803C2211yescharacteristic, complemented
46C22⋊C4×D173—24no—
47C2×C4×Dic173C2211yescharacteristic, complemented
48C2×C4×Dic173C2211yescharacteristic, complemented
49#19149586C2211yescharacteristic, complemented
50#19149586C2211yescharacteristic, complemented
51C22×D4×C173C2211yescharacteristic
52C23×C4×C173C2211yescharacteristic, complemented
53C2×C22⋊C4×C173C2211yescharacteristic, complemented
54C2×C22⋊C4×C173C2211yescharacteristic, complemented
55C2×C22⋊C4×C173C2211yescharacteristic, complemented
56#95×C173C2211yescharacteristic, complemented
57#95×C173C2211yescharacteristic, complemented
58C23×D173—22no—
59C23×D173—22no—
60C23×D173—22no—
61C23×D173—22no—
62C23×D173—22no—
63C23×D173—22no—
64C23×D173—22no—
65C23×D173—22no—
66C23×D173—41no—
67C23×D173—41no—
68C23×D173—41no—
69C23×D173—41no—
70C23×D173C2×C414yescomplemented
71C23×D173C2×C414yescomplemented
72C23×D173C2311yescharacteristic
73C23×D173C2311yescharacteristic
74C23×D173C2311yescharacteristic
75C2×D692—22no—
76C2×D692—22no—
77C2×D692C2×C414yescomplemented
78C2×C22⋊D173—22no—
79C2×C22⋊D173—22no—
80C2×C22⋊D173C2×C414yes—
81C2×C22⋊D173C2×C414yescomplemented
82C2×C22⋊D173D414yes—
83C2×C22⋊D173D414yes—
84D4×D173—232no—
85C22×Dic173—22no—
86C22×Dic173—22no—
87C22×Dic173—41no—
88C22×Dic173C2311yescharacteristic
89C22×Dic173C2311yescharacteristic
90C22×Dic173C2311yescharacteristic
91C22×Dic173C2311yescharacteristic
92C22×Dic173C2311yescharacteristic
93C2×C4×D173—22no—
94C2×C4×D173—22no—
95C2×C4×D173—22no—
96C2×C4×D173—22no—
97C2×C4×D173—22no—
98C2×C4×D173—22no—
99C2×C4×D173—41no—
100C2×C4×D173—41no—
101C2×C4×D173C2×C414yescomplemented
102#14583805—22no—
103#14583805—22no—
104C22⋊Dic173—22no—
105C22⋊Dic173—22no—
106C22⋊Dic173—41no—
107#14583803—22no—
108#14583803—22no—
109#14583803—22no—
110#14583803—22no—
111#14583803—22no—
112#14583803—22no—
113#14583803—22no—
114#14583803—41no—
115#14583803—41no—
116#14583803D414yescomplemented
117C4×Dic173—22no—
118C4×Dic173—22no—
119C24×C173C2311yescharacteristic
120C24×C173C2311yescharacteristic
121C2×D4×C173—22no—
122C2×D4×C173—22no—
123C2×D4×C173C2×C414yescomplemented
124C22×C4×C173—22no—
125C22×C4×C173—22no—
126C22×C4×C173—41no—
127C22×C4×C173C2311yescharacteristic
128C22×C4×C173C2311yescharacteristic
129C22×C4×C173C2311yescharacteristic
130C22×C4×C173C2311yescharacteristic
131C22×C4×C173C2311yescharacteristic
132C22⋊C4×C173—22no—
133C22⋊C4×C173—22no—
134C22⋊C4×C173—41no—
135C22×D173—21no—
136C22×D173—21no—
137C22×D173—21no—
138C22×D173—21no—
139C22×D173—21no—
140C22×D173—21no—
141C22×D173—21no—
142C22×D173—21no—
143C22×D173—28no—
144C22×D173—21no—
145C22×D173—21no—
146C22×D173—21no—
147C22×D173—21no—
148C22×D173—21no—
149C22×D173—21no—
150C22×D173—28no—
151C22×D173—28no—
152C22×D173—28no—
153C22×D173—44no—
154C22×D173—44no—
155C22×D173—44no—
156C22×D173—44no—
157C22×D173—44no—
158C22×D173—44no—
159C22×D173—44no—
160C22×D173—44no—
161C22×D173—44no—
162C22×D173—44no—
163C22×D173C22×C412yes—
164C22×D173C22×C412yes—
165C22×D173C22×C412yes—
166C22×D173C2×D412yes—
167C22×D173C2×D412yes—
168C22×D173C2×D412yes—
169C22×D173#4112yes—
170D692—28no—
171C22⋊D173—28no—
172C22⋊D173C22⋊C4116yescomplemented
173C2×Dic173—21no—
174C2×Dic173—21no—
175C2×Dic173—21no—
176C2×Dic173—21no—
177C2×Dic173—21no—
178C2×Dic173—21no—
179C2×Dic173—21no—
180C2×Dic173—21no—
181C2×Dic173—21no—
182C2×Dic173—21no—
183C2×Dic173—21no—
184C2×Dic173—44no—
185C2×Dic173C22×C412yes—
186C2×Dic173C2×D412yes—
187C2×Dic173C2×D412yes—
188C2×Dic173C2×D412yescomplemented
189C4×D173—28no—
190C4×D173—44no—
191C4×D173—44no—
192C23×C173—21no—
193C23×C173—21no—
194C23×C173—21no—
195C23×C173—21no—
196C23×C173—21no—
197C23×C173—21no—
198C23×C173—21no—
199C23×C173—21no—
200C23×C173—21no—
201C23×C173—21no—
202C23×C173C22×C412yes—
203C23×C173C22×C412yes—
204C23×C173C2×D412yes—
205C23×C173C2×D412yes—
206C23×C173C2411yescharacteristic
207D4×C173—28no—
208C2×C4×C173—21no—
209C2×C4×C173—21no—
210C2×C4×C173—21no—
211C2×C4×C173—21no—
212C2×C4×C173—21no—
213C2×C4×C173—21no—
214C2×C4×C173—21no—
215C2×C4×C173—21no—
216C2×C4×C173—21no—
217C2×C4×C173—21no—
218C2×C4×C173—21no—
219C2×C4×C173—21no—
220C2×C4×C173—21no—
221C2×C4×C173—44no—
222C2×C4×C173C22×C412yes—
223C2×C4×C173C2×D412yescomplemented
224C22×C173—22no—
225C22×C173—22no—
226C22×C173—22no—
227C22×C173—22no—
228C22×C173—22no—
229C22×C173—22no—
230C22×C173—22no—
231C22×C173—22no—
232C22×C173—22no—
233C22×C173—41no—
234C22×C173—41no—
235C22×C173—41no—
236C22×C173—41no—
237C22×C173C2×C22⋊C414yescomplemented
238C22×C173C23×C411yescharacteristic
239C22×C173C22×D411yescharacteristic
240C22×C173C22×D411yescharacteristic
241C22×C173C22×D411yescharacteristic
242C22×C173C2×#4111yescharacteristic
243C22×C173H(5,𝔽2)11yescharacteristic
244C22×C173H(5,𝔽2)11yescharacteristic
245C2×D173—22no—
246C2×D173—22no—
247C2×D173—22no—
248C2×D173—22no—
249C2×D173—22no—
250C2×D173—22no—
251C2×D173—22no—
252C2×D173—22no—
253C2×D173—22no—
254C2×D173—22no—
255C2×D173—22no—
256C2×D173—22no—
257C2×D173—22no—
258C2×D173—22no—
259C2×D173—22no—
260C2×D173—416no—
261C2×D173—416no—
262C2×D173—41no—
263C2×D173—41no—
264C2×D173—41no—
265C2×D173—41no—
266C2×D173C2×C22⋊C414yescomplemented
267C2×D173C4×D414yescomplemented
268C4×C173—22no—
269C4×C173—22no—
270C4×C173—41no—
271C4×C173—41no—
272C4×C173—41no—
273Dic173—22no—
274Dic173—41no—
275Dic173—41no—
276Dic173—41no—
277Dic173C2×C22⋊C414yescomplemented
278C2×C173—24no—
279C2×C173—42no—
280C2×C173C22×C22⋊C411yescharacteristic
281C2×C173C2×C4×D411yescharacteristic
282C2×C173#51811yescharacteristic
283C2×C173D4211yescharacteristic
284C2×C173D4211yescharacteristic
285C2×C173#54611yescharacteristic
286C2×C173#54611yescharacteristic
287D173—24no—
288D173—24no—
289D173—42no—
290C173D4×C22⋊C411yescharacteristic, complemented, Sylow
291D4×C22⋊C4—1731nomaximal, Sylow
292C23×D4—1731no—
293C2×C4×D4—1731no—
294C2×C4×D4—1731no—
295C22×C22⋊C4—1731no—
296C22×C22⋊C4—1731no—
297#390—1731no—
298#390—1731no—
299#386—1731no—
300#386—1731no—
301#386—1731no—
302#386D17311yescharacteristic, complemented
303#380—1731no—
304#379—1731no—
305#379—1731no—
306C4×C22⋊C4—1731no—
307C25—1731no—
308C25—1731no—
309C22×D4—3464no—
310C22×D4—3464no—
311C22×D4—1731no—
312C22×D4—1738no—
313C22×D4—1731no—
314C22×D4—1731no—
315C22×D4C2×D17311yescharacteristic
316C23×C4—1731no—
317C23×C4—1731no—
318C23×C4—1731no—
319C23×C4—1731no—
320C23×C4C2×D17311yescharacteristic, complemented
321C4×D4—3464no—
322C4×D4—3464no—
323C2×C4⋊C4—1731no—
324C2×C4⋊C4—1731no—
325C2×C22⋊C4—3464no—
326C2×C22⋊C4—3464no—
327C2×C22⋊C4—1731no—
328C2×C22⋊C4—1731no—
329C2×C22⋊C4—1731no—
330C2×C22⋊C4—1731no—
331C2×C22⋊C4—1731no—
332C2×C22⋊C4—1731no—
333C2×C22⋊C4—1731no—
334C2×C22⋊C4—1731no—
335C2×C22⋊C4—1731no—
336C2×C22⋊C4—1731no—
337C2×C22⋊C4—1731no—
338C2×C22⋊C4—1731no—
339C2×C22⋊C4—1731no—
340C2×C22⋊C4C2×D17311yescharacteristic, complemented
341C2×C22⋊C4C2×D17311yescharacteristic, complemented
342C2×C22⋊C4C2×D17311yescharacteristic, complemented
343C2×C42—1731no—
344C2×C42—1731no—
345#95—1731no—
346#95—1731no—
347#95C2×D17311yescharacteristic, complemented
348#95C2×D17311yescharacteristic, complemented
349C24—3462no—
350C24—3462no—
351C24—3462no—
352C24—3462no—
353C24—3462no—
354C24—3462no—
355C24—3462no—
356C24—3462no—
357C24—1731no—
358C24—1731no—
359C24—1731no—
360C24—1734no—
361C24—1734no—
362C24—6921no—
363C24—6921no—
364C24—6921no—
365C24—6921no—
366C24C22×D17311yescharacteristic
367C24C22×D17311yescharacteristic
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—3462no—
371C2×D4—3462no—
372C2×D4—34632no—
373C2×D4—3462no—
374C2×D4—3462no—
375C2×D4—1734no—
376C2×D4—1734no—
377C2×D4—1734no—
378C2×D4—1734no—
379C2×D4—1734no—
380C2×D4C4×D17314yescomplemented
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—41no—
384C22×C4—3462no—
385C22×C4—3462no—
386C22×C4—3462no—
387C22×C4—3462no—
388C22×C4—3462no—
389C22×C4—3462no—
390C22×C4—3462no—
391C22×C4—3462no—
392C22×C4—1731no—
393C22×C4—1731no—
394C22×C4—1731no—
395C22×C4—1731no—
396C22×C4—1734no—
397C22×C4—1731no—
398C22×C4—6921no—
399C22×C4—6921no—
400C22×C4—6921no—
401C22×C4C22×D17311yescharacteristic
402C22×C4C22×D17311yescharacteristic
403C22×C4C22×D17311yescharacteristic
404C22×C4C22×D17311yescharacteristic
405C22×C4C22×D17311yescharacteristic
406C4⋊C4—3462no—
407C4⋊C4—3462no—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—41no—
411C22⋊C4—3462no—
412C22⋊C4—3462no—
413C22⋊C4—3462no—
414C22⋊C4—3462no—
415C22⋊C4—3462no—
416C22⋊C4—3462no—
417C22⋊C4—3462no—
418C22⋊C4—3462no—
419C22⋊C4—3462no—
420C22⋊C4—1734no—
421C22⋊C4—6921no—
422C22⋊C4—6921no—
423C22⋊C4—6921no—
424C42—3462no—
425C42—3462no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—3468no—
437C23—3468no—
438C23—3468no—
439C23—3461no—
440C23—3461no—
441C23—3461no—
442C23—3461no—
443C23—3461no—
444C23—3461no—
445C23—3461no—
446C23—3461no—
447C23—3468no—
448C23—3461no—
449C23—3461no—
450C23—3461no—
451C23—3461no—
452C23—3461no—
453C23—3461no—
454C23—1732no—
455C23—1732no—
456C23—1732no—
457C23—1732no—
458C23—1732no—
459C23—1732no—
460C23—1732no—
461C23—6924no—
462C23—6924no—
463C23—6924no—
464C23—6924no—
465C23—6924no—
466C23—6924no—
467C23—6924no—
468C23—6924no—
469C23—6924no—
470C23—6924no—
471C23C2×C4×D17312yes—
472C23C2×C4×D17312yes—
473C23D4×D17312yes—
474C23D4×D17312yes—
475C23C23×D17311yescharacteristic
476D4—28no—
477D4—3468no—
478D4—3468no—
479D4—17316no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—44no—
494C2×C4—3468no—
495C2×C4—3461no—
496C2×C4—3461no—
497C2×C4—3461no—
498C2×C4—3461no—
499C2×C4—3461no—
500C2×C4—3461no—
501C2×C4—3461no—
502C2×C4—3461no—
503C2×C4—3461no—
504C2×C4—3461no—
505C2×C4—3461no—
506C2×C4—1732no—
507C2×C4—1732no—
508C2×C4—1732no—
509C2×C4—1732no—
510C2×C4—6924no—
511C2×C4—6924no—
512C2×C4—6924no—
513C2×C4C2×C4×D17312yes—
514C2×C4D4×D17312yescomplemented
515C22—22no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—3462no—
529C22—3462no—
530C22—3462no—
531C22—3462no—
532C22—3462no—
533C22—3462no—
534C22—3462no—
535C22—3462no—
536C22—3462no—
537C22—3462no—
538C22—3462no—
539C22—3462no—
540C22—3462no—
541C22—3462no—
542C22—3462no—
543C22—1734no—
544C22—1734no—
545C22—69216no—
546C22—69216no—
547C22—6921no—
548C22—6921no—
549C22—6921no—
550C22—6921no—
551C22C22⋊C4×D17314yescomplemented
552C22C22×C4×D17311yescharacteristic
553C22C2×D4×D17311yescharacteristic
554C22C2×D4×D17311yescharacteristic
555C22C2×D4×D17311yescharacteristic
556C22#41×D17311yescharacteristic
557C22#1914972411yescharacteristic
558C22#1914972511yescharacteristic
559C4—22no—
560C4—22no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—3462no—
565C4—1734no—
566C4—6921no—
567C4—6921no—
568C4—6921no—
569C2—24no—
570C2—42no—
571C2—3464no—
572C2—3464no—
573C2—6922no—
574C2C2×C22⋊C4×D17311yescharacteristic
575C2C4×D4×D17311yescharacteristic
576C2#2601196311yescharacteristic
577C2#2601212311yescharacteristic
578C2#2601212511yescharacteristic
579C2#2601213011yescharacteristic
580C2#2601213311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutatorC22×C173C23×C4
Fitting#386×C173C2
socle, cosocleC23×C173C24
prime radicalC22×D4×D173C2
hypercenter#386D173
hypocenterC173D4×C22⋊C4
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ #386

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C4×C22⋊D173C2semidirect
3. C2×#19149639C2semidirect
4. C22×#14583803C2semidirect
5. C2×C22⋊C4×D173C2semidirect
6. #26011391C2semidirect
7. #26011391C2semidirect
8. #26011371C2semidirect
9. #26011369C2semidirect
10. #26011369C2semidirect
11. #26011350C2semidirect
12. #26011346C2semidirect
13. #26011346C2semidirect
14. #26011338C2semidirect
15. #386×C173C2semidirect
23. C2×D4×D173C4semidirect
24. C23×Dic173C22semidirect
26. C22×C4×D173C22semidirect
27. C22×C4×D173C22semidirect
30. C2×#14583805C22semidirect
31. C2×#14583805C22semidirect
32. C2×C22⋊Dic173C22semidirect
33. C2×C22⋊Dic173C22semidirect
34. C2×C22⋊Dic173C22semidirect
40. C2×#14583803C22semidirect
41. C2×#14583803C22semidirect
42. C2×#14583803C22semidirect
43. C2×#14583803C22semidirect
44. C2×#14583803C22semidirect
45. C2×#14583803C22semidirect
47. C2×C4×Dic173C22semidirect
48. C2×C4×Dic173C22semidirect
49. #19149586C22semidirect
50. #19149586C22semidirect
52. C23×C4×C173C22semidirect
53. C2×C22⋊C4×C173C22semidirect
54. C2×C22⋊C4×C173C22semidirect
55. C2×C22⋊C4×C173C22semidirect
56. #95×C173C22semidirect
57. #95×C173C22semidirect
70. C23×D173C2×C4semidirect
71. C23×D173C2×C4semidirect
77. C2×D692C2×C4semidirect
81. C2×C22⋊D173C2×C4semidirect
101. C2×C4×D173C2×C4semidirect
116. #14583803D4semidirect
123. C2×D4×C173C2×C4semidirect
172. C22⋊D173C22⋊C4semidirect
188. C2×Dic173C2×D4semidirect
223. C2×C4×C173C2×D4semidirect
237. C22×C173C2×C22⋊C4semidirect
266. C2×D173C2×C22⋊C4semidirect
267. C2×D173C4×D4semidirect
277. Dic173C2×C22⋊C4semidirect
290. C173D4×C22⋊C4semidirect
302. #386D173semidirect
320. C23×C4C2×D173semidirect
340. C2×C22⋊C4C2×D173semidirect
341. C2×C22⋊C4C2×D173semidirect
342. C2×C22⋊C4C2×D173semidirect
347. #95C2×D173semidirect
348. #95C2×D173semidirect
380. C2×D4C4×D173semidirect
514. C2×C4D4×D173semidirect
551. C22C22⋊C4×D173semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×D4×D1732
C2×C4×C22⋊D1732
C2×#191496392
C22×#145838032
C2×C22⋊C4×D1732
#260113912
#260113712
#260113692
#260113502
#260113462
#260113382
#386×C1732
D4×C22⋊C4173