Finite Group Databasebeta

← #33104400

Subgroups — #33104400

Order 18688.

986 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×Dic73C211yesmaximal, characteristic
2C2×SD16×Dic73C211yesmaximal, characteristic
3#22612849C211yesmaximal, characteristic, complemented
4#22612836C211yesmaximal, characteristic
5#22610659C211yesmaximal, characteristic, complemented
6#22610657C211yesmaximal, characteristic, complemented
7#22610185C211yesmaximal, characteristic, complemented
8#22610183C211yesmaximal, characteristic, complemented
9#22610178C211yesmaximal, characteristic
10#22610176C211yesmaximal, characteristic
11#22610092C211yesmaximal, characteristic, complemented
12#22609944C211yesmaximal, characteristic, complemented
13#22609939C211yesmaximal, characteristic
14C8×#11924007C211yesmaximal, characteristic, complemented
15C2×C4×SD16×C73C211yesmaximal, characteristic
16C2×Q8×Dic73C2211yescharacteristic
17C2×Q8×Dic73C2211yescharacteristic
18C2×D4×Dic73C2211yescharacteristic
19C2×D4×Dic73C2211yescharacteristic
20SD16×Dic73—24no—
21SD16×Dic73C418yescomplemented
22C2×#14556524C2211yescharacteristic
23C2×#14556524C2211yescharacteristic
24C2×#14556523C2211yescharacteristic
25C2×#14556523C2211yescharacteristic
26C2×#14556520C2211yescharacteristic
27C2×#14556515C2211yescharacteristic
28C2×#14556515C2211yescharacteristic
29C2×#14556512C2211yescharacteristic
30C2×#14556512C2211yescharacteristic
31C2×C8×Dic73C2211yescharacteristic
32C2×C8×Dic73C2211yescharacteristic
33#17988485—21no—
34#17988485C2211yescharacteristic
35#17988485C2211yescharacteristic
36#17988483—21no—
37#17988483C2211yescharacteristic
38#17988483C2211yescharacteristic
39#17988460—21no—
40#17988459—21no—
41#17988452—21no—
42#17988452C2211yescharacteristic
43#17988452C2211yescharacteristic
44#17988445—21no—
45C4×#11924007—21no—
46C4×#11924007C2211yescharacteristic
47C2×#14556498—21no—
48#17988350C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988342C2211yescharacteristic
53#17988341C2211yescharacteristic
54C22×SD16×C73C2211yescharacteristic
55C2×C4×Q8×C73C2211yescharacteristic
56C2×C4×D4×C73C2211yescharacteristic
57C4×SD16×C73—24no—
58C2×#106×C73C2211yescharacteristic
59C2×#103×C73C2211yescharacteristic
60C2×#102×C73C2211yescharacteristic
61C2×C4×C8×C73C2211yescharacteristic
62C23×Dic73—21no—
63C23×Dic73—21no—
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66Q8×Dic73—22no—
67Q8×Dic73—22no—
68Q8×Dic73—41no—
69Q8×Dic73C2×C414yescomplemented
70D4×Dic73—22no—
71D4×Dic73—22no—
72D4×Dic73—41no—
73D4×Dic73C2×C414yes—
74C2×C4⋊Dic73—21no—
75C2×C4⋊Dic73—21no—
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×#11924007—21no—
81C2×#11924007—21no—
82C2×#11924007—21no—
83C2×#11924007—21no—
84C2×#11924007D411yescharacteristic
85C2×#11924007D411yescharacteristic
86C2×C22⋊Dic73—21no—
87C2×C22⋊Dic73—21no—
88C2×C4×Dic73—21no—
89C2×C4×Dic73—21no—
90C2×C4×Dic73C2311yescharacteristic
91C2×C4×Dic73C2311yescharacteristic
92#14556524—41no—
93#14556524—41no—
94#14556523—22no—
95#14556523C2×C414yescomplemented
96#14556520—22no—
97#14556515—22no—
98#14556515C2×C414yescomplemented
99#14556512—22no—
100#14556512C2×C414yes—
101C8×Dic73—22no—
102C8×Dic73C2×C414yescomplemented
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108#14556498—21no—
109#14556498—21no—
110#14556498—21no—
111#14556498D411yescharacteristic
112#14556498D411yescharacteristic
113C22×Q8×C73C2311yescharacteristic
114C22×D4×C73C2311yescharacteristic
115C23×C4×C73—21no—
116C2×SD16×C73C2×C414yes—
117C2×SD16×C73D414yes—
118C2×SD16×C73Q814yes—
119C22×C8×C73C2311yescharacteristic
120C22×C8×C73C2311yescharacteristic
121C4×Q8×C73—22no—
122C4×Q8×C73—41no—
123C4×D4×C73—22no—
124C4×D4×C73—41no—
125C2×C4⋊C4×C73—21no—
126C2×C4⋊C4×C73C2311yescharacteristic
127C2×C4⋊C4×C73C2311yescharacteristic
128C2×C22⋊C4×C73—21no—
129C2×C42×C73—21no—
130C2×C42×C73C2311yescharacteristic
131#106×C73—22no—
132#103×C73—22no—
133#102×C73—22no—
134C4×C8×C73—22no—
135C22×Dic73—21no—
136C22×Dic73—21no—
137C22×Dic73—24no—
138C22×Dic73—21no—
139C22×Dic73—21no—
140C22×Dic73—42no—
141C22×Dic73—42no—
142C22×Dic73—81no—
143C22×Dic73—81no—
144C22×Dic73—81no—
145C22×Dic73C2×D411yescharacteristic
146C22×Dic73C2×D411yescharacteristic
147C22×Dic73C2×D411yescharacteristic
148C22×Dic73C2×D411yescharacteristic
149C2×#10596831—21no—
150C2×#10596831—21no—
151C2×#10596831—21no—
152C2×#10596831—21no—
153C2×#10596831—21no—
154C2×#10596831C22×C412yes—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73—24no—
158C4⋊Dic73—21no—
159C4⋊Dic73—21no—
160C4⋊Dic73—21no—
161C4⋊Dic73—21no—
162C4⋊Dic73—42no—
163C4⋊Dic73C22×C412yes—
164C4⋊Dic73C22×C412yes—
165#11924007—22no—
166#11924007—22no—
167#11924007—42no—
168#11924007—42no—
169#11924007—81no—
170#11924007—81no—
171C22⋊Dic73—24no—
172C22⋊Dic73—42no—
173C4×Dic73—21no—
174C4×Dic73—21no—
175C4×Dic73—24no—
176C4×Dic73—42no—
177C4×Dic73C22×C412yes—
178C4×Dic73#4112yes—
179C24×C73—21no—
180C2×Q8×C73—21no—
181C2×Q8×C73—21no—
182C2×Q8×C73—21no—
183C2×Q8×C73C22×C412yes—
184C2×Q8×C73C2×D412yes—
185C2×Q8×C73C2×Q812yes—
186C2×D4×C73—21no—
187C2×D4×C73—21no—
188C2×D4×C73—21no—
189C2×D4×C73C22×C412yes—
190C2×D4×C73C2×D412yes—
191C2×D4×C73C2×Q812yes—
192C22×C4×C73—21no—
193C22×C4×C73—21no—
194C22×C4×C73—21no—
195C22×C4×C73—42no—
196C22×C4×C73—42no—
197C22×C4×C73—81no—
198C22×C4×C73C2×D411yescharacteristic
199C22×C4×C73C2×D411yescharacteristic
200C22×C4×C73C2411yescharacteristic
201SD16×C73C4⋊C4116yescomplemented
202C2×C8×C73—21no—
203C2×C8×C73—21no—
204C2×C8×C73—21no—
205C2×C8×C73C22×C412yes—
206C2×C8×C73C2×D412yes—
207C2×C8×C73C2×Q812yes—
208C4⋊C4×C73—21no—
209C4⋊C4×C73—21no—
210C4⋊C4×C73—21no—
211C4⋊C4×C73—21no—
212C4⋊C4×C73—42no—
213C22⋊C4×C73—42no—
214C42×C73—21no—
215C42×C73—21no—
216C42×C73—42no—
217C23×C73—22no—
218C23×C73—22no—
219C23×C73—22no—
220C23×C73—41no—
221C23×C73—41no—
222C23×C73—41no—
223C23×C73—41no—
224C23×C73C22×D411yescharacteristic
225Q8×C73—22no—
226Q8×C73—22no—
227Q8×C73—22no—
228Q8×C73C2×C4⋊C414yes—
229D4×C73—22no—
230D4×C73—22no—
231D4×C73—22no—
232D4×C73C2×C4⋊C414yes—
233C2×C4×C73—22no—
234C2×C4×C73—21no—
235C2×C4×C73—22no—
236C2×C4×C73—22no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73—41no—
242C2×C4×C73—41no—
243C2×C4×C73—41no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73—84no—
248C2×C4×C73#13512yes—
249C2×C4×C73C23×C411yescharacteristic
250C2×C4×C73C22×D411yescharacteristic
251C2×C4×C73C22×Q811yescharacteristic
252C2×C4×C73C2×#4111yescharacteristic
253C2×C4×C73H(5,𝔽2)11yescharacteristic
254C2×C4×C732−1+411yescharacteristic
255C2×Dic73—21no—
256C2×Dic73—21no—
257C2×Dic73—21no—
258C2×Dic73—21no—
259C2×Dic73—21no—
260C2×Dic73—21no—
261C2×Dic73—22no—
262C2×Dic73—22no—
263C2×Dic73—21no—
264C2×Dic73—21no—
265C2×Dic73—21no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272C2×Dic73—41no—
273C2×Dic73—41no—
274C2×Dic73—41no—
275C2×Dic73—41no—
276C2×Dic73—84no—
277C2×Dic73—84no—
278C2×Dic73C4×D412yes—
279C2×Dic73C4×D412yes—
280C2×Dic73#13512yes—
281C8×C73—41no—
282C8×C73C2×C4⋊C414yes—
283#10596831—22no—
284#10596831—41no—
285C22×C73—24no—
286C22×C73—42no—
287C22×C73—42no—
288C22×C73—42no—
289C22×C73—42no—
290C22×C73—42no—
291C22×C73—42no—
292C22×C73C2×C4×D411yescharacteristic
293C22×C73D4211yescharacteristic
294C22×C73D4×Q811yescharacteristic
295C22×C73C2×#13511yescharacteristic
296C22×C73C2×#13511yescharacteristic
297C22×C73#57711yescharacteristic
298C22×C73#57711yescharacteristic
299C4×C73—24no—
300C4×C73—22no—
301C4×C73—41no—
302C4×C73—81no—
303C4×C73—81no—
304C4×C73C22×C4⋊C411yescharacteristic
305C4×C73#52011yescharacteristic
306C4×C73#54711yescharacteristic
307C4×C73#55411yescharacteristic
308Dic73—22no—
309Dic73—22no—
310Dic73—22no—
311Dic73—42no—
312Dic73—42no—
313Dic73—41no—
314Dic73—81no—
315Dic73—81no—
316#22683—731nomaximal, Sylow
317C2×C73—44no—
318C2×C73D4×C4⋊C411yescharacteristic
319C2×C73C4×#13511yescharacteristic
320C2×C73#294911yescharacteristic
321C2×C73#328211yescharacteristic
322C2×C73#328211yescharacteristic
323C2×C73#338111yescharacteristic
324C2×C73#338111yescharacteristic
325C2×C4×SD16—731no—
326C2×C4×SD16—731no—
327C2×C4×SD16D7311yescharacteristic
328#2351—731no—
329#2349—731no—
330#1941—731no—
331#1941—731no—
332#1863—731no—
333#1863—731no—
334#1862—731no—
335#1862—731no—
336#1846—731no—
337#1805—731no—
338#1804—731no—
339C8×C4⋊C4—731no—
340C73#2268311yescharacteristic, complemented, Sylow
341C22×SD16C2×D7311yescharacteristic
342C2×C4×Q8—731no—
343C2×C4×Q8—731no—
344C2×C4×Q8C2×D7311yescharacteristic
345C2×C4×D4—731no—
346C2×C4×D4—731no—
347C2×C4×D4C2×D7311yescharacteristic
348C4×SD16—24no—
349C4×SD16—738no—
350C4×SD16—1464no—
351C2×#107—731no—
352C2×#107—731no—
353C2×#106—731no—
354C2×#106—731no—
355C2×#106C2×D7311yescharacteristic
356C2×C4⋊C8—731no—
357C2×#103—731no—
358C2×#103—731no—
359C2×#103C2×D7311yescharacteristic
360C2×#102—731no—
361C2×#102—731no—
362C2×#102C2×D7311yescharacteristic
363C2×C4×C8—731no—
364C2×C4×C8—731no—
365C2×C4×C8C2×D7311yescharacteristic
366#389—731no—
367#389—731no—
368#389—731no—
369#389—731no—
370#389—1461no—
371#389—1461no—
372#385—1461no—
373#385—1461no—
374#382—731no—
375#382—731no—
376#382—1461no—
377#380—1461no—
378C4×C4⋊C4—731no—
379C4×C4⋊C4—1461no—
380C2×#95—1461no—
381#340—731no—
382#340—731no—
383#340—731no—
384#340—731no—
385#336—731no—
386#336—731no—
387C22×Q8C22×D7311yescharacteristic
388C22×D4C22×D7311yescharacteristic
389C23×C4—21no—
390C23×C4—1461no—
391C23×C4—1461no—
392C2×SD16Dic14614yes—
393C2×SD16C4×D7314yes—
394C2×SD16C22⋊D7314yes—
395C22×C8—731no—
396C22×C8—731no—
397C22×C8C22×D7311yescharacteristic
398C22×C8C22×D7311yescharacteristic
399C4×Q8—22no—
400C4×Q8—41no—
401C4×Q8—2921no—
402C4×Q8—734no—
403C4×Q8—1462no—
404C4×Q8—1462no—
405C4×D4—22no—
406C4×D4—41no—
407C4×D4—2921no—
408C4×D4—734no—
409C4×D4—1462no—
410C4×D4—1462no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—731no—
413C2×C4⋊C4—731no—
414C2×C4⋊C4—731no—
415C2×C4⋊C4—731no—
416C2×C4⋊C4—731no—
417C2×C4⋊C4—731no—
418C2×C4⋊C4—1461no—
419C2×C4⋊C4—1461no—
420C2×C4⋊C4—1461no—
421C2×C4⋊C4—1461no—
422C2×C4⋊C4—1461no—
423C2×C4⋊C4—1461no—
424C2×C4⋊C4C22×D7311yescharacteristic
425C2×C4⋊C4C22×D7311yescharacteristic
426C2×C22⋊C4—21no—
427C2×C22⋊C4—1461no—
428C2×C22⋊C4—1461no—
429C2×C42—21no—
430C2×C42—731no—
431C2×C42—731no—
432C2×C42—1461no—
433C2×C42—1461no—
434C2×C42C22×D7311yescharacteristic
435#107—2921no—
436#107—2921no—
437#106—22no—
438#106—734no—
439#106—1462no—
440C4⋊C8—1462no—
441#103—22no—
442#103—734no—
443#103—1462no—
444#102—22no—
445#102—734no—
446#102—1462no—
447C4×C8—22no—
448C4×C8—734no—
449C4×C8—1462no—
450#95—731no—
451#95—731no—
452#95—1461no—
453#95—1461no—
454#95—1461no—
455#95—1461no—
456#95—1461no—
457#95—1461no—
458#95—1461no—
459#95—1461no—
460C24—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8C2×C4×D7312yes—
465C2×Q8C2×C22⋊D7312yes—
466C2×Q8C2×Dic14612yes—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4C2×C4×D7312yes—
471C2×D4C2×C22⋊D7312yes—
472C2×D4C2×Dic14612yes—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—81no—
479C22×C4—2922no—
480C22×C4—2922no—
481C22×C4—5841no—
482C22×C4—5841no—
483C22×C4—5841no—
484C22×C4—731no—
485C22×C4—731no—
486C22×C4—731no—
487C22×C4—731no—
488C22×C4—1461no—
489C22×C4—1461no—
490C22×C4—1461no—
491C22×C4—1461no—
492C22×C4—1464no—
493C22×C4D4×D7311yescharacteristic
494C22×C4D4×D7311yescharacteristic
495C22×C4C23×D7311yescharacteristic
496SD16#11924007116yescomplemented
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—732no—
501C2×C8—1461no—
502C2×C8—1461no—
503C2×C8—1461no—
504C2×C8—1461no—
505C2×C8—1461no—
506C2×C8C2×C4×D7312yes—
507C2×C8C2×C22⋊D7312yes—
508C2×C8C2×Dic14612yes—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—42no—
514C4⋊C4—2922no—
515C4⋊C4—2922no—
516C4⋊C4—2922no—
517C4⋊C4—5841no—
518C4⋊C4—5841no—
519C4⋊C4—732no—
520C4⋊C4—732no—
521C4⋊C4—1462no—
522C4⋊C4—1464no—
523C4⋊C4—1461no—
524C4⋊C4—1461no—
525C4⋊C4—1461no—
526C4⋊C4—1461no—
527C4⋊C4—1462no—
528C4⋊C4—1461no—
529C4⋊C4—1461no—
530C22⋊C4—42no—
531C22⋊C4—2922no—
532C22⋊C4—1464no—
533C42—21no—
534C42—21no—
535C42—42no—
536C42—2922no—
537C42—732no—
538C42—732no—
539C42—1464no—
540C42—1461no—
541C42—1461no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23C2×D4×D7311yescharacteristic
550Q8—22no—
551Q8—22no—
552Q8—22no—
553Q8C2×#1192400714yes—
554D4—22no—
555D4—22no—
556D4—22no—
557D4C2×#1192400714yes—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—84no—
573C2×C4—2921no—
574C2×C4—2921no—
575C2×C4—2921no—
576C2×C4—2921no—
577C2×C4—2921no—
578C2×C4—2921no—
579C2×C4—2921no—
580C2×C4—2921no—
581C2×C4—2921no—
582C2×C4—2921no—
583C2×C4—5844no—
584C2×C4—5844no—
585C2×C4—732no—
586C2×C4—732no—
587C2×C4—732no—
588C2×C4—1461no—
589C2×C4—1461no—
590C2×C4—1462no—
591C2×C4—1462no—
592C2×C4—1461no—
593C2×C4—1461no—
594C2×C4—1461no—
595C2×C4—1461no—
596C2×C4—1461no—
597C2×C4—1461no—
598C2×C4—1461no—
599C2×C4#1455661212yes—
600C2×C4C22×C4×D7311yescharacteristic
601C2×C4C22×C22⋊D7311yescharacteristic
602C2×C4C22×Dic14611yescharacteristic
603C2×C4#41×D7311yescharacteristic
604C2×C4#1455663611yescharacteristic
605C2×C4#1455663811yescharacteristic
606C8—41no—
607C8—2921no—
608C8—1462no—
609C8C2×#1192400714yes—
610C22—24no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22C4×D4×D7311yescharacteristic
618C22D4×C22⋊D7311yescharacteristic
619C22D4×Dic14611yescharacteristic
620C22#135×D7311yescharacteristic
621C22C2×#1455661211yescharacteristic
622C22#1798939311yescharacteristic
623C22#1798939411yescharacteristic
624C4—24no—
625C4—22no—
626C4—41no—
627C4—81no—
628C4—81no—
629C4—2922no—
630C4—2922no—
631C4—2921no—
632C4—5841no—
633C4—5841no—
634C4—1462no—
635C4—1462no—
636C4—1462no—
637C4C22×#1192400711yescharacteristic
638C4#1798906611yescharacteristic
639C4#1798923611yescharacteristic
640C4#1798929611yescharacteristic
641C2—44no—
642C2D4×#1192400711yescharacteristic
643C2C4×#1455661211yescharacteristic
644C2#2261878511yescharacteristic
645C2#2262163211yescharacteristic
646C2#2262163811yescharacteristic
647C2#2262280411yescharacteristic
648C2#2262280711yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×C4×SD16×C73C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
cosocleC22×C4×C73C24
hypercenterC2×C4×SD16D73
hypocenterC73#22683
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
3. #22612849C2semidirect
5. #22610659C2semidirect
6. #22610657C2semidirect
7. #22610185C2semidirect
8. #22610183C2semidirect
11. #22610092C2semidirect
12. #22609944C2semidirect
14. C8×#11924007C2semidirect
21. SD16×Dic73C4semidirect
69. Q8×Dic73C2×C4semidirect
95. #14556523C2×C4semidirect
98. #14556515C2×C4semidirect
102. C8×Dic73C2×C4semidirect
201. SD16×C73C4⋊C4semidirect
340. C73#22683semidirect
496. SD16#11924007semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×Dic732
#226128492
#226128362
#226106592
#226106572
#226101852
#226101832
#226101782
#226101762
#226100922
#226099442
#226099392
C8×#119240072
C2×C4×SD16×C732
#2268373