Finite Group Databasebeta

← #46839897

Subgroups — #46839897

Order 34688.

922 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576131C211yesmaximal, characteristic, complemented
2C2×#22576128C211yesmaximal, characteristic, complemented
3C2×#22576128C211yesmaximal, characteristic, complemented
4C2×C4×C22⋊D271C211yesmaximal, characteristic, complemented
5C2×C22⋊C4×D271C211yesmaximal, characteristic, complemented
6#31773819C211yesmaximal, characteristic, complemented
7#31773811C211yesmaximal, characteristic, complemented
8#31773811C211yesmaximal, characteristic, complemented
9#31773805C211yesmaximal, characteristic, complemented
10#31773797C211yesmaximal, characteristic, complemented
11#31773790C211yesmaximal, characteristic, complemented
12#31773776C211yesmaximal, characteristic, complemented
13#31773773C211yesmaximal, characteristic, complemented
14#31773768C211yesmaximal, characteristic, complemented
15C2×#121×C271C211yesmaximal, characteristic, complemented
16C24×D271C2211yescharacteristic
17C22×C22⋊D271C2211yescharacteristic
18C22×C22⋊D271C2211yescharacteristic
19C22×C22⋊D271C2211yescharacteristic
20C22×C22⋊D271C2211yescharacteristic
21C22×C4×D271C2211yescharacteristic
22C22×C4×D271C2211yescharacteristic
23#22576131—24no—
24#22576128—24no—
25#22576128—24no—
26C4×C22⋊D271—24no—
27C2×C4⋊Dic271C2211yescharacteristic, complemented
28C2×Dic271⋊C4C2211yescharacteristic, complemented
29C2×Dic271⋊C4C2211yescharacteristic, complemented
30C2×C22⋊Dic271C2211yescharacteristic
31C2×C22⋊Dic271C2211yescharacteristic
32C2×C22⋊Dic271C2211yescharacteristic
33C2×C22⋊Dic271C2211yescharacteristic, complemented
34C2×C22⋊Dic271C2211yescharacteristic, complemented
35C2×C22⋊Dic271C2211yescharacteristic, complemented
36C2×C22⋊Dic271C2211yescharacteristic, complemented
37C2×(C2×D271)⋊C4C2211yescharacteristic
38C2×(C2×D271)⋊C4C2211yescharacteristic
39C2×(C2×D271)⋊C4C2211yescharacteristic
40C2×(C2×D271)⋊C4C2211yescharacteristic
41C2×(C2×D271)⋊C4C2211yescharacteristic
42C2×(C2×D271)⋊C4C2211yescharacteristic
43C22⋊C4×D271—24no—
44C2×C4×Dic271C2211yescharacteristic, complemented
45C271⋊#95C2211yescharacteristic, complemented
46C271⋊#95C2211yescharacteristic, complemented
47C271⋊#95C2211yescharacteristic, complemented
48C271⋊#95C2211yescharacteristic, complemented
49C22×D4×C271C2211yescharacteristic
50C22×D4×C271C2211yescharacteristic
51C22×D4×C271C2211yescharacteristic, complemented
52C23×C4×C271C2211yescharacteristic, complemented
53#121×C271—24no—
54C2×C4⋊C4×C271C2211yescharacteristic, complemented
55C2×C22⋊C4×C271C2211yescharacteristic
56C2×C22⋊C4×C271C2211yescharacteristic, complemented
57C23×D271—22no—
58C23×D271—22no—
59C23×D271—22no—
60C23×D271—22no—
61C23×D271—22no—
62C23×D271—41no—
63C23×D271—41no—
64C23×D271C2311yescharacteristic
65C23×D271C2311yescharacteristic
66C2×C22⋊D271—22no—
67C2×C22⋊D271—22no—
68C2×C22⋊D271—22no—
69C2×C22⋊D271—22no—
70C2×C22⋊D271—22no—
71C2×C22⋊D271—22no—
72C2×C22⋊D271—22no—
73C2×C22⋊D271—22no—
74C2×C22⋊D271—22no—
75C2×C22⋊D271—41no—
76C2×C22⋊D271—41no—
77C2×C22⋊D271D414yescomplemented
78C22×Dic271C2311yescharacteristic
79C22×Dic271C2311yescharacteristic
80C22×Dic271C2311yescharacteristic
81C22×Dic271C2311yescharacteristic
82C22×Dic271C2311yescharacteristic
83C22×Dic271C2311yescharacteristic
84C2×C4×D271—22no—
85C2×C4×D271—22no—
86C2×C4×D271—22no—
87C2×C4×D271—22no—
88C2×C4×D271—41no—
89C2×C4×D271—41no—
90C4⋊Dic271—22no—
91Dic271⋊C4—22no—
92Dic271⋊C4—41no—
93C22⋊Dic271—22no—
94C22⋊Dic271—22no—
95C22⋊Dic271—22no—
96C22⋊Dic271—22no—
97C22⋊Dic271—22no—
98C22⋊Dic271—22no—
99C22⋊Dic271—41no—
100(C2×D271)⋊C4—22no—
101(C2×D271)⋊C4—22no—
102(C2×D271)⋊C4—22no—
103(C2×D271)⋊C4—22no—
104(C2×D271)⋊C4—22no—
105(C2×D271)⋊C4D414yescomplemented
106C4×Dic271—22no—
107C24×C271C2311yescharacteristic
108C24×C271C2311yescharacteristic
109C24×C271C2311yescharacteristic
110C2×D4×C271—22no—
111C2×D4×C271—22no—
112C2×D4×C271—22no—
113C2×D4×C271—22no—
114C2×D4×C271—22no—
115C2×D4×C271—22no—
116C2×D4×C271—41no—
117C2×D4×C271—41no—
118C2×D4×C271—41no—
119C22×C4×C271—22no—
120C22×C4×C271—22no—
121C22×C4×C271—41no—
122C22×C4×C271C2311yescharacteristic
123C22×C4×C271C2311yescharacteristic
124C22×C4×C271C2311yescharacteristic
125C22×C4×C271C2311yescharacteristic
126C4⋊C4×C271—22no—
127C22⋊C4×C271—22no—
128C22⋊C4×C271D414yescomplemented
129C22×D271—21no—
130C22×D271—21no—
131C22×D271—21no—
132C22×D271—21no—
133C22×D271—28no—
134C22×D271—21no—
135C22×D271—21no—
136C22×D271—21no—
137C22×D271—21no—
138C22×D271—21no—
139C22×D271—21no—
140C22×D271—44no—
141C22×D271—44no—
142C22×D271—44no—
143C22×D271—44no—
144C22×D271—44no—
145C22×D271—44no—
146C22×D271C2×D412yes—
147C22×D271C2×D412yes—
148C22×D271#4112yes—
149C22×D271#4112yes—
150C22⋊D271—28no—
151C22⋊D271—44no—
152C22⋊D271—44no—
153C22⋊D271—44no—
154C2×Dic271—21no—
155C2×Dic271—21no—
156C2×Dic271—21no—
157C2×Dic271—21no—
158C2×Dic271—21no—
159C2×Dic271—21no—
160C2×Dic271—21no—
161C2×Dic271—21no—
162C2×Dic271—21no—
163C2×Dic271—21no—
164C2×Dic271—21no—
165C2×Dic271—21no—
166C2×Dic271—21no—
167C2×Dic271—21no—
168C2×Dic271—21no—
169C2×Dic271—21no—
170C2×Dic271C2×D412yes—
171C2×Dic271C2×D412yes—
172C4×D271—44no—
173C4×D271—44no—
174C23×C271—21no—
175C23×C271—21no—
176C23×C271—21no—
177C23×C271—21no—
178C23×C271—21no—
179C23×C271—21no—
180C23×C271—21no—
181C23×C271—21no—
182C23×C271—21no—
183C23×C271—21no—
184C23×C271—21no—
185C23×C271—21no—
186C23×C271—21no—
187C23×C271—21no—
188C23×C271—21no—
189C23×C271—21no—
190C23×C271—21no—
191C23×C271—21no—
192C23×C271—21no—
193C23×C271C2×D412yes—
194C23×C271C2×D412yes—
195C23×C271C2411yescharacteristic
196D4×C271—44no—
197D4×C271—44no—
198D4×C271—44no—
199C2×C4×C271—21no—
200C2×C4×C271—21no—
201C2×C4×C271—21no—
202C2×C4×C271—21no—
203C2×C4×C271—21no—
204C2×C4×C271—21no—
205C2×C4×C271—21no—
206C2×C4×C271—21no—
207C2×C4×C271—21no—
208C2×C4×C271—44no—
209C2×C4×C271C2×D412yes—
210C2×C4×C271C2×D412yes—
211C2×C4×C271C2×D412yes—
212C22×C271—22no—
213C22×C271—22no—
214C22×C271—22no—
215C22×C271—22no—
216C22×C271—22no—
217C22×C271—22no—
218C22×C271—41no—
219C22×C271—41no—
220C22×C271—41no—
221C22×C271—41no—
222C22×C271—41no—
223C22×C271—41no—
224C22×C271—41no—
225C22×C271—41no—
226C22×C271—41no—
227C22×C271—41no—
228C22×C271—41no—
229C22×C271—41no—
230C22×C271—41no—
231C22×C271—41no—
232C22×C271—41no—
233C22×C271C22×D411yescharacteristic
234C22×C271C22×D411yescharacteristic
235C22×C271C22×D411yescharacteristic
236C22×C271C2×#4111yescharacteristic
237C22×C271C2×#4111yescharacteristic
238C22×C271H(5,𝔽2)11yescharacteristic
239C22×C271H(5,𝔽2)11yescharacteristic
240C2×D271—22no—
241C2×D271—22no—
242C2×D271—22no—
243C2×D271—22no—
244C2×D271—22no—
245C2×D271—22no—
246C2×D271—22no—
247C2×D271—22no—
248C2×D271—22no—
249C2×D271—416no—
250C2×D271—41no—
251C2×D271—41no—
252C2×D271—41no—
253C2×D271—41no—
254C2×D271#12114yescomplemented
255C4×C271—22no—
256C4×C271—22no—
257C4×C271—22no—
258C4×C271—41no—
259Dic271—22no—
260Dic271—22no—
261Dic271—41no—
262Dic271—41no—
263Dic271—41no—
264Dic271—41no—
265C2×C271—42no—
266C2×C271—42no—
267C2×C271—81no—
268C2×C271C2×#12111yescharacteristic
269C2×C271#52511yescharacteristic
270C2×C271#53811yescharacteristic
271C2×C271D4211yescharacteristic
272C2×C271D4211yescharacteristic
273C2×C271#54611yescharacteristic
274C2×C271#54611yescharacteristic
275D271—24no—
276D271—42no—
277C271#242411yescharacteristic, complemented, Sylow
278#2424—2711nomaximal, Sylow
279C2×#121—2711no—
280C2×#121D27111yescharacteristic, complemented
281C2×#120—2711no—
282C2×#120—2711no—
283C2×C4×D4—2711no—
284C22×C22⋊C4—2711no—
285#396—2711no—
286#394—2711no—
287#394—2711no—
288#392—2711no—
289#390—2711no—
290#388—2711no—
291#386—2711no—
292#386—2711no—
293#385—2711no—
294C25—2711no—
295C22×D4—2711no—
296C22×D4—2711no—
297C22×D4—2711no—
298C22×D4—2711no—
299C22×D4C2×D27111yescharacteristic
300C22×D4C2×D27111yescharacteristic
301C22×D4C2×D27111yescharacteristic, complemented
302C23×C4—2711no—
303C23×C4—2711no—
304C23×C4C2×D27111yescharacteristic, complemented
305#121—24no—
306#121—5424no—
307#120—5424no—
308#120—5424no—
309C4×D4—5424no—
310C2×C4⋊C4—2711no—
311C2×C4⋊C4—2711no—
312C2×C4⋊C4—2711no—
313C2×C4⋊C4C2×D27111yescharacteristic, complemented
314C2×C22⋊C4—2711no—
315C2×C22⋊C4—2711no—
316C2×C22⋊C4—2711no—
317C2×C22⋊C4—2711no—
318C2×C22⋊C4—2711no—
319C2×C22⋊C4—2711no—
320C2×C22⋊C4—2711no—
321C2×C22⋊C4—2711no—
322C2×C22⋊C4—2711no—
323C2×C22⋊C4—2711no—
324C2×C22⋊C4—2711no—
325C2×C22⋊C4—2711no—
326C2×C22⋊C4—2711no—
327C2×C22⋊C4—5424no—
328C2×C22⋊C4C2×D27111yescharacteristic
329C2×C22⋊C4C2×D27111yescharacteristic, complemented
330C2×C42—2711no—
331#95—2711no—
332#95—2711no—
333#95—2711no—
334#95—2711no—
335C24—2711no—
336C24—2711no—
337C24—5422no—
338C24—5422no—
339C24—5422no—
340C24—5422no—
341C24—5422no—
342C24—10841no—
343C24—10841no—
344C24C22×D27111yescharacteristic
345C24C22×D27111yescharacteristic
346C24C22×D27111yescharacteristic
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—2714no—
357C2×D4—5422no—
358C2×D4—5422no—
359C2×D4—5422no—
360C2×D4—5422no—
361C2×D4—5422no—
362C2×D4—5422no—
363C2×D4—5422no—
364C2×D4—5422no—
365C2×D4—5422no—
366C2×D4—10841no—
367C2×D4—10841no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—41no—
371C22×C4—2711no—
372C22×C4—2711no—
373C22×C4—2711no—
374C22×C4—2711no—
375C22×C4—2711no—
376C22×C4—2711no—
377C22×C4—5422no—
378C22×C4—5422no—
379C22×C4—5422no—
380C22×C4—5422no—
381C22×C4—10841no—
382C22×C4—10841no—
383C22×C4C22×D27111yescharacteristic
384C22×C4C22×D27111yescharacteristic
385C22×C4C22×D27111yescharacteristic
386C22×C4C22×D27111yescharacteristic
387C4⋊C4—22no—
388C4⋊C4—5422no—
389C4⋊C4—5422no—
390C4⋊C4—10841no—
391C22⋊C4—22no—
392C22⋊C4—2714no—
393C22⋊C4—5422no—
394C22⋊C4—5422no—
395C22⋊C4—5422no—
396C22⋊C4—5422no—
397C22⋊C4—5422no—
398C22⋊C4—5422no—
399C22⋊C4—5422no—
400C22⋊C4—5422no—
401C22⋊C4—5422no—
402C22⋊C4—5422no—
403C22⋊C4—5422no—
404C22⋊C4—10841no—
405C22⋊C4C22⋊D27114yescomplemented
406C42—5422no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—2712no—
427C23—2712no—
428C23—2712no—
429C23—2712no—
430C23—5428no—
431C23—5421no—
432C23—5421no—
433C23—5421no—
434C23—5421no—
435C23—5421no—
436C23—5421no—
437C23—5421no—
438C23—5421no—
439C23—5421no—
440C23—5421no—
441C23—10844no—
442C23—10844no—
443C23—10844no—
444C23—10844no—
445C23—10844no—
446C23—10844no—
447C23D4×D27112yes—
448C23C2×C22⋊D27112yes—
449C23C23×D27111yescharacteristic
450D4—44no—
451D4—44no—
452D4—44no—
453D4—5428no—
454D4—10844no—
455D4—10844no—
456D4—10844no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—44no—
467C2×C4—2712no—
468C2×C4—2712no—
469C2×C4—5421no—
470C2×C4—5421no—
471C2×C4—5421no—
472C2×C4—5421no—
473C2×C4—5421no—
474C2×C4—5421no—
475C2×C4—5421no—
476C2×C4—5421no—
477C2×C4—5421no—
478C2×C4—5421no—
479C2×C4—5421no—
480C2×C4—5421no—
481C2×C4—5421no—
482C2×C4—5421no—
483C2×C4—5421no—
484C2×C4—5421no—
485C2×C4—10844no—
486C2×C4—10844no—
487C2×C4D4×D27112yes—
488C2×C4C2×C22⋊D27112yes—
489C2×C4C2×C22⋊D27112yes—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—2714no—
512C22—5422no—
513C22—5422no—
514C22—5422no—
515C22—5422no—
516C22—5422no—
517C22—5422no—
518C22—5422no—
519C22—5422no—
520C22—5422no—
521C22—108416no—
522C22—10841no—
523C22—10841no—
524C22—10841no—
525C22—10841no—
526C22C2×D4×D27111yescharacteristic
527C22C2×D4×D27111yescharacteristic
528C22C22×C22⋊D27111yescharacteristic
529C22#41×D27111yescharacteristic
530C22#41×D27111yescharacteristic
531C22#2257620511yescharacteristic
532C22#2257620511yescharacteristic
533C4—22no—
534C4—22no—
535C4—22no—
536C4—41no—
537C4—5422no—
538C4—5422no—
539C4—10841no—
540C4—10841no—
541C4—10841no—
542C4—10841no—
543C2—42no—
544C2—42no—
545C2—81no—
546C2—5424no—
547C2—10842no—
548C2#121×D27111yescharacteristic
549C2#3177440411yescharacteristic
550C2#3177447911yescharacteristic
551C2D4×C22⋊D27111yescharacteristic
552C2D4×C22⋊D27111yescharacteristic
553C2#3177453611yescharacteristic
554C2#3177454411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
FittingC2×#121×C271C2
hypercenterC2×#121D271
hypocenterC271#2424
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#22576131C2semidirect
2. C2×#22576128C2semidirect
3. C2×#22576128C2semidirect
4. C2×C4×C22⋊D271C2semidirect
5. C2×C22⋊C4×D271C2semidirect
6. #31773819C2semidirect
7. #31773811C2semidirect
8. #31773811C2semidirect
9. #31773805C2semidirect
10. #31773797C2semidirect
11. #31773790C2semidirect
12. #31773776C2semidirect
13. #31773773C2semidirect
14. #31773768C2semidirect
15. C2×#121×C271C2semidirect
27. C2×C4⋊Dic271C22semidirect
28. C2×Dic271⋊C4C22semidirect
29. C2×Dic271⋊C4C22semidirect
33. C2×C22⋊Dic271C22semidirect
34. C2×C22⋊Dic271C22semidirect
35. C2×C22⋊Dic271C22semidirect
36. C2×C22⋊Dic271C22semidirect
44. C2×C4×Dic271C22semidirect
45. C271⋊#95C22semidirect
46. C271⋊#95C22semidirect
47. C271⋊#95C22semidirect
48. C271⋊#95C22semidirect
51. C22×D4×C271C22semidirect
52. C23×C4×C271C22semidirect
54. C2×C4⋊C4×C271C22semidirect
56. C2×C22⋊C4×C271C22semidirect
77. C2×C22⋊D271D4semidirect
105. (C2×D271)⋊C4D4semidirect
128. C22⋊C4×C271D4semidirect
254. C2×D271#121semidirect
277. C271#2424semidirect
280. C2×#121D271semidirect
301. C22×D4C2×D271semidirect
304. C23×C4C2×D271semidirect
313. C2×C4⋊C4C2×D271semidirect
329. C2×C22⋊C4C2×D271semidirect
405. C22⋊C4C22⋊D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761312
C2×#225761282
C2×C4×C22⋊D2712
C2×C22⋊C4×D2712
#317738192
#317738112
#317738052
#317737972
#317737902
#317737762
#317737732
#317737682
C2×#121×C2712
#2424271