Finite Group Databasebeta

← #46839970

Subgroups — #46839970

Order 34688.

914 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#16864653C211yesmaximal, characteristic, complemented
2C2×#22576141C211yesmaximal, characteristic, complemented
3C2×#22576132C211yesmaximal, characteristic, complemented
4#31773817C211yesmaximal, characteristic, complemented
5#31773812C211yesmaximal, characteristic, complemented
6#31773809C211yesmaximal, characteristic, complemented
7#31773805C211yesmaximal, characteristic, complemented
8#31773777C211yesmaximal, characteristic, complemented
9#31773777C211yesmaximal, characteristic, complemented
10#31773776C211yesmaximal, characteristic, complemented
11#31773773C211yesmaximal, characteristic, complemented
12#31773769C211yesmaximal, characteristic, complemented
13#31773768C211yesmaximal, characteristic, complemented
14C2×C271⋊#95C211yesmaximal, characteristic, complemented
15C2×#121×C271C211yesmaximal, characteristic, complemented
16C2×#16864653—24no—
17C2×#16864653—24no—
18C2×#16864653—24no—
19C22×Dic542C2211yescharacteristic
20C22×C22⋊D271C2211yescharacteristic
21C22×C22⋊D271C2211yescharacteristic
22C22×C22⋊D271C2211yescharacteristic
23C23×Dic271C2211yescharacteristic
24C23×Dic271C2211yescharacteristic
25C23×Dic271C2211yescharacteristic
26C22×C4×D271C2211yescharacteristic
27#22576141—24no—
28#22576132—24no—
29C2×Dic271⋊C4C2211yescharacteristic
30C2×Dic271⋊C4C2211yescharacteristic
31C2×Dic271⋊C4C2211yescharacteristic
32C2×C22⋊Dic271C2211yescharacteristic
33C2×C22⋊Dic271C2211yescharacteristic
34C2×C22⋊Dic271C2211yescharacteristic
35C2×C22⋊Dic271C2211yescharacteristic
36C2×C22⋊Dic271C2211yescharacteristic
37C2×C22⋊Dic271C2211yescharacteristic
38C2×(C2×D271)⋊C4C2211yescharacteristic
39C2×(C2×D271)⋊C4C2211yescharacteristic
40C2×(C2×D271)⋊C4C2211yescharacteristic
41C271⋊#95C2211yescharacteristic
42C271⋊#95C2211yescharacteristic
43C271⋊#95C2211yescharacteristic
44C271⋊#95C2211yescharacteristic
45C271⋊#95C2211yescharacteristic
46C271⋊#95C2211yescharacteristic
47C271⋊#95C2211yescharacteristic
48C271⋊#95C2211yescharacteristic
49C22×D4×C271C2211yescharacteristic
50C22×D4×C271C2211yescharacteristic
51C22×D4×C271C2211yescharacteristic
52C23×C4×C271C2211yescharacteristic
53#121×C271—24no—
54C2×C4⋊C4×C271C2211yescharacteristic
55C2×C22⋊C4×C271C2211yescharacteristic
56C2×C22⋊C4×C271C2211yescharacteristic
57C23×D271C2311yescharacteristic
58#16864653—232no—
59C2×Dic542—22no—
60C2×Dic542—22no—
61C2×Dic542—22no—
62C2×C22⋊D271—22no—
63C2×C22⋊D271—22no—
64C2×C22⋊D271—22no—
65C2×C22⋊D271—22no—
66C2×C22⋊D271—22no—
67C2×C22⋊D271—22no—
68C2×C22⋊D271—22no—
69C2×C22⋊D271—41no—
70C2×C22⋊D271D414yescomplemented
71C22×Dic271—22no—
72C22×Dic271—22no—
73C22×Dic271—22no—
74C22×Dic271—22no—
75C22×Dic271—22no—
76C22×Dic271—22no—
77C22×Dic271—22no—
78C22×Dic271—41no—
79C22×Dic271D414yescomplemented
80C22×Dic271C2311yescharacteristic
81C22×Dic271C2311yescharacteristic
82C22×Dic271C2311yescharacteristic
83C22×Dic271C2311yescharacteristic
84C22×Dic271C2311yescharacteristic
85C22×Dic271C2311yescharacteristic
86C22×Dic271C2311yescharacteristic
87C2×C4×D271—22no—
88C2×C4×D271—22no—
89C2×C4×D271—22no—
90Dic271⋊C4—22no—
91Dic271⋊C4—22no—
92Dic271⋊C4—22no—
93C22⋊Dic271—22no—
94C22⋊Dic271—22no—
95C22⋊Dic271—22no—
96C22⋊Dic271—22no—
97C22⋊Dic271—22no—
98C22⋊Dic271—41no—
99(C2×D271)⋊C4—22no—
100(C2×D271)⋊C4—22no—
101(C2×D271)⋊C4—41no—
102C24×C271C2311yescharacteristic
103C24×C271C2311yescharacteristic
104C24×C271C2311yescharacteristic
105C2×D4×C271—22no—
106C2×D4×C271—22no—
107C2×D4×C271—22no—
108C2×D4×C271—22no—
109C2×D4×C271—22no—
110C2×D4×C271—22no—
111C2×D4×C271—22no—
112C2×D4×C271—41no—
113C2×D4×C271D414yescomplemented
114C22×C4×C271—22no—
115C22×C4×C271—22no—
116C22×C4×C271—41no—
117C22×C4×C271C2311yescharacteristic
118C22×C4×C271C2311yescharacteristic
119C22×C4×C271C2311yescharacteristic
120C22×C4×C271C2311yescharacteristic
121C4⋊C4×C271—22no—
122C22⋊C4×C271—22no—
123C22⋊C4×C271—22no—
124C22×D271—21no—
125C22×D271—21no—
126C22×D271—21no—
127C22×D271—21no—
128C22×D271—21no—
129C22×D271—21no—
130C22×D271C2×D412yes—
131Dic542—28no—
132C22⋊D271—28no—
133C22⋊D271—28no—
134C22⋊D271—44no—
135C2×Dic271—28no—
136C2×Dic271—28no—
137C2×Dic271—21no—
138C2×Dic271—21no—
139C2×Dic271—21no—
140C2×Dic271—21no—
141C2×Dic271—21no—
142C2×Dic271—21no—
143C2×Dic271—21no—
144C2×Dic271—21no—
145C2×Dic271—21no—
146C2×Dic271—21no—
147C2×Dic271—21no—
148C2×Dic271—21no—
149C2×Dic271—21no—
150C2×Dic271—21no—
151C2×Dic271—21no—
152C2×Dic271—21no—
153C2×Dic271—21no—
154C2×Dic271—44no—
155C2×Dic271C2×D412yes—
156C2×Dic271C2×D412yes—
157C2×Dic271C2×D412yes—
158C2×Dic271#4112yescomplemented
159C4×D271—28no—
160C23×C271—21no—
161C23×C271—21no—
162C23×C271—21no—
163C23×C271—21no—
164C23×C271—21no—
165C23×C271—21no—
166C23×C271—21no—
167C23×C271—21no—
168C23×C271—21no—
169C23×C271—21no—
170C23×C271—21no—
171C23×C271—21no—
172C23×C271—21no—
173C23×C271—21no—
174C23×C271—21no—
175C23×C271—21no—
176C23×C271—21no—
177C23×C271C2×D412yes—
178C23×C271C2×D412yes—
179C23×C271C2×D412yes—
180C23×C271C2×D412yes—
181C23×C271C2411yescharacteristic
182D4×C271—28no—
183D4×C271—44no—
184D4×C271—44no—
185C2×C4×C271—21no—
186C2×C4×C271—21no—
187C2×C4×C271—21no—
188C2×C4×C271—21no—
189C2×C4×C271—21no—
190C2×C4×C271—21no—
191C2×C4×C271—21no—
192C2×C4×C271—21no—
193C2×C4×C271—21no—
194C2×C4×C271—21no—
195C2×C4×C271—44no—
196C2×C4×C271C2×D412yes—
197C2×C4×C271#4112yescomplemented
198C22×C271—22no—
199C22×C271—22no—
200C22×C271—22no—
201C22×C271—22no—
202C22×C271—22no—
203C22×C271—22no—
204C22×C271—22no—
205C22×C271—22no—
206C22×C271—22no—
207C22×C271—41no—
208C22×C271—41no—
209C22×C271—41no—
210C22×C271—41no—
211C22×C271—41no—
212C22×C271—41no—
213C22×C271—41no—
214C22×C271—41no—
215C22×C271—41no—
216C22×C271—41no—
217C22×C271—41no—
218C22×C271#12014yes—
219C22×C271C22×D411yescharacteristic
220C22×C271C22×D411yescharacteristic
221C22×C271C22×D411yescharacteristic
222C22×C271C2×#4111yescharacteristic
223C22×C271C2×#4111yescharacteristic
224C22×C271H(5,𝔽2)11yescharacteristic
225C22×C2712−1+411yescharacteristic
226C2×D271—22no—
227C2×D271—22no—
228C2×D271—22no—
229C2×D271—41no—
230C2×D271—41no—
231C2×D271—41no—
232C2×D271—41no—
233C4×C271—22no—
234C4×C271—22no—
235C4×C271—41no—
236C4×C271—41no—
237Dic271—22no—
238Dic271—22no—
239Dic271—22no—
240Dic271—22no—
241Dic271—41no—
242Dic271—41no—
243Dic271—41no—
244C2×C271—24no—
245C2×C271—42no—
246C2×C271—81no—
247C2×C271C2×#12011yescharacteristic
248C2×C271#53211yescharacteristic
249C2×C271#53711yescharacteristic
250C2×C271#54611yescharacteristic
251C2×C271#54611yescharacteristic
252C2×C271#54711yescharacteristic
253C2×C271#54711yescharacteristic
254D271—42no—
255C271#243011yescharacteristic, complemented, Sylow
256#2430—2711nomaximal, Sylow
257C22×#41—2711no—
258C2×#123—2711no—
259C2×#121—2711no—
260C2×#121D27111yescharacteristic, complemented
261#395—2711no—
262#394—2711no—
263#393—2711no—
264#392—2711no—
265#386—2711no—
266#386—2711no—
267#386—2711no—
268#386—2711no—
269#385—2711no—
270#385—2711no—
271C2×#95—2711no—
272C2×#41—5424no—
273C2×#41—5424no—
274C2×#41—5424no—
275C22×Q8—2711no—
276C22×D4—2711no—
277C22×D4—2711no—
278C22×D4—2711no—
279C22×D4C2×D27111yescharacteristic
280C22×D4C2×D27111yescharacteristic
281C22×D4C2×D27111yescharacteristic
282C23×C4—2711no—
283C23×C4—2711no—
284C23×C4—2711no—
285C23×C4—2711no—
286C23×C4C2×D27111yescharacteristic
287#123—5424no—
288#121—24no—
289#121—5424no—
290C2×C4⋊C4—2711no—
291C2×C4⋊C4—2711no—
292C2×C4⋊C4—2711no—
293C2×C4⋊C4C2×D27111yescharacteristic
294C2×C22⋊C4—2711no—
295C2×C22⋊C4—2711no—
296C2×C22⋊C4—2711no—
297C2×C22⋊C4—2711no—
298C2×C22⋊C4—2711no—
299C2×C22⋊C4—2711no—
300C2×C22⋊C4—2711no—
301C2×C22⋊C4—2711no—
302C2×C22⋊C4—2711no—
303C2×C22⋊C4C2×D27111yescharacteristic
304C2×C22⋊C4C2×D27111yescharacteristic
305#95—2711no—
306#95—2711no—
307#95—2711no—
308#95—2711no—
309#95—2711no—
310#95—2711no—
311#95—2711no—
312#95—2711no—
313C24—2711no—
314C24C22×D27111yescharacteristic
315C24C22×D27111yescharacteristic
316C24C22×D27111yescharacteristic
317#41—54232no—
318C2×Q8—5422no—
319C2×Q8—5422no—
320C2×Q8—5422no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—41no—
329C2×D4—2714no—
330C2×D4—5422no—
331C2×D4—5422no—
332C2×D4—5422no—
333C2×D4—5422no—
334C2×D4—5422no—
335C2×D4—5422no—
336C2×D4—5422no—
337C2×D4—10841no—
338C2×D4C22⋊D27114yescomplemented
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—41no—
342C22×C4—2714no—
343C22×C4—2711no—
344C22×C4—2711no—
345C22×C4—2711no—
346C22×C4—2711no—
347C22×C4—2711no—
348C22×C4—2711no—
349C22×C4—2711no—
350C22×C4—5422no—
351C22×C4—5422no—
352C22×C4—5422no—
353C22×C4—5422no—
354C22×C4—5422no—
355C22×C4—5422no—
356C22×C4—5422no—
357C22×C4—5422no—
358C22×C4—5422no—
359C22×C4—5422no—
360C22×C4—10841no—
361C22×C4C22×D27111yescharacteristic
362C22×C4C22×D27111yescharacteristic
363C22×C4C22×D27111yescharacteristic
364C22×C4C22×D27111yescharacteristic
365C4⋊C4—22no—
366C4⋊C4—5422no—
367C4⋊C4—5422no—
368C4⋊C4—5422no—
369C22⋊C4—22no—
370C22⋊C4—22no—
371C22⋊C4—5422no—
372C22⋊C4—5422no—
373C22⋊C4—5422no—
374C22⋊C4—5422no—
375C22⋊C4—5422no—
376C22⋊C4—5422no—
377C22⋊C4—5422no—
378C22⋊C4—10841no—
379C22⋊C4—10841no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—2712no—
398C23—5421no—
399C23—5421no—
400C23—5421no—
401C23—5421no—
402C23—5421no—
403C23—5421no—
404C23D4×D27112yes—
405C23D4×D27112yes—
406C23C2×C22⋊D27112yes—
407C23C2×C22⋊D27112yes—
408C23C23×D27111yescharacteristic
409Q8—5428no—
410D4—28no—
411D4—44no—
412D4—44no—
413D4—5428no—
414D4—5428no—
415D4—10844no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—44no—
427C2×C4—2712no—
428C2×C4—2712no—
429C2×C4—2712no—
430C2×C4—2712no—
431C2×C4—5428no—
432C2×C4—5428no—
433C2×C4—5428no—
434C2×C4—5421no—
435C2×C4—5421no—
436C2×C4—5421no—
437C2×C4—5421no—
438C2×C4—5421no—
439C2×C4—5421no—
440C2×C4—5421no—
441C2×C4—5421no—
442C2×C4—5421no—
443C2×C4—5421no—
444C2×C4—5421no—
445C2×C4—5421no—
446C2×C4—5421no—
447C2×C4—5421no—
448C2×C4—5421no—
449C2×C4—5421no—
450C2×C4—5421no—
451C2×C4—10844no—
452C2×C4C2×C22⋊D27112yes—
453C2×C4#1686465312yescomplemented
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22—5422no—
475C22—5422no—
476C22—5422no—
477C22—10841no—
478C22—10841no—
479C22—10841no—
480C22—10841no—
481C22#2257612814yes—
482C22C2×D4×D27111yescharacteristic
483C22C2×D4×D27111yescharacteristic
484C22C22×C22⋊D27111yescharacteristic
485C22#41×D27111yescharacteristic
486C22C2×#1686465311yescharacteristic
487C22#2257620511yescharacteristic
488C22#2257620711yescharacteristic
489C4—22no—
490C4—22no—
491C4—41no—
492C4—41no—
493C4—5422no—
494C4—5422no—
495C4—5422no—
496C4—5422no—
497C4—10841no—
498C4—10841no—
499C4—10841no—
500C2—24no—
501C2—42no—
502C2—81no—
503C2—10842no—
504C2C2×#2257612811yescharacteristic
505C2#3177444211yescharacteristic
506C2#3177447411yescharacteristic
507C2#3177453611yescharacteristic
508C2#3177454111yescharacteristic
509C2#3177454611yescharacteristic
510C2#3177454711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
FittingC2×#121×C271C2
hypercenterC2×#121D271
hypocenterC271#2430
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#16864653C2semidirect
2. C2×#22576141C2semidirect
3. C2×#22576132C2semidirect
4. #31773817C2semidirect
5. #31773812C2semidirect
6. #31773809C2semidirect
7. #31773805C2semidirect
8. #31773777C2semidirect
9. #31773777C2semidirect
10. #31773776C2semidirect
11. #31773773C2semidirect
12. #31773769C2semidirect
13. #31773768C2semidirect
14. C2×C271⋊#95C2semidirect
15. C2×#121×C271C2semidirect
70. C2×C22⋊D271D4semidirect
79. C22×Dic271D4semidirect
113. C2×D4×C271D4semidirect
158. C2×Dic271#41semidirect
197. C2×C4×C271#41semidirect
255. C271#2430semidirect
260. C2×#121D271semidirect
338. C2×D4C22⋊D271semidirect
453. C2×C4#16864653semidirect

Maximal subgroups

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

subgroupindex
C22×#168646532
C2×#225761412
C2×#225761322
#317738172
#317738122
#317738092
#317738052
#317737772
#317737762
#317737732
#317737692
#317737682
C2×C271⋊#952
C2×#121×C2712
#2430271