Finite Group Databasebeta

← #46839981

Subgroups — #46839981

Order 34688.

952 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#16864654C211yesmaximal, characteristic, complemented
2C2×#22576156C211yesmaximal, characteristic, complemented
3C2×#22576146C211yesmaximal, characteristic, complemented
4C2×#22576133C211yesmaximal, characteristic, complemented
5#31773807C211yesmaximal, characteristic, complemented
6#31773806C211yesmaximal, characteristic, complemented
7#31773806C211yesmaximal, characteristic, complemented
8#31773805C211yesmaximal, characteristic, complemented
9#31773801C211yesmaximal, characteristic, complemented
10#31773797C211yesmaximal, characteristic, complemented
11#31773776C211yesmaximal, characteristic, complemented
12#31773775C211yesmaximal, characteristic, complemented
13#31773773C211yesmaximal, characteristic, complemented
14#31773744C211yesmaximal, characteristic, complemented
15#386×C271C211yesmaximal, characteristic, complemented
16C2×#16864654—24no—
17C2×#16864654—24no—
18C2×#16864654—24no—
19C22×Dic542C2211yescharacteristic
20C22×D1084C2211yescharacteristic
21C22×D1084C2211yescharacteristic
22C22×D1084C2211yescharacteristic
23C22×C22⋊D271C2211yescharacteristic
24C22×C22⋊D271C2211yescharacteristic
25C22×C22⋊D271C2211yescharacteristic
26C22×C22⋊D271C2211yescharacteristic
27C22×C22⋊D271C2211yescharacteristic
28C22×C22⋊D271C2211yescharacteristic
29C22×C4×D271C2211yescharacteristic
30C22×C4×D271C2211yescharacteristic
31#22576156—24no—
32#22576146—24no—
33#22576133—24no—
34C2×Dic271⋊C4C2211yescharacteristic
35C2×C22⋊Dic271C2211yescharacteristic
36C2×C22⋊Dic271C2211yescharacteristic
37C2×(C2×D271)⋊C4C2211yescharacteristic
38C2×(C2×D271)⋊C4C2211yescharacteristic
39C2×(C2×D271)⋊C4C2211yescharacteristic
40C2×(C2×D271)⋊C4C2211yescharacteristic
41C2×(C2×D271)⋊C4C2211yescharacteristic
42C2×(C2×D271)⋊C4C2211yescharacteristic
43C2×(C2×D271)⋊C4C2211yescharacteristic
44C2×C4×Dic271C2211yescharacteristic
45C2×C4×Dic271C2211yescharacteristic
46C271⋊#95C2211yescharacteristic
47C271⋊#95C2211yescharacteristic
48C271⋊#95C2211yescharacteristic
49C271⋊#95C2211yescharacteristic
50C22×D4×C271C2211yescharacteristic
51C23×C4×C271C2211yescharacteristic
52C2×C22⋊C4×C271C2211yescharacteristic
53C2×C22⋊C4×C271C2211yescharacteristic
54C2×C22⋊C4×C271C2211yescharacteristic
55#95×C271C2211yescharacteristic
56#95×C271C2211yescharacteristic
57C23×D271C2311yescharacteristic
58C23×D271C2311yescharacteristic
59C23×D271C2311yescharacteristic
60#16864654—232no—
61C2×Dic542—22no—
62C2×Dic542—22no—
63C2×Dic542D414yescomplemented
64C2×D1084—22no—
65C2×D1084—22no—
66C2×D1084—22no—
67C2×D1084—22no—
68C2×D1084—22no—
69C2×D1084—22no—
70C2×D1084—41no—
71C2×D1084—41no—
72C2×D1084—41no—
73C2×C22⋊D271—22no—
74C2×C22⋊D271—22no—
75C2×C22⋊D271—22no—
76C2×C22⋊D271—22no—
77C2×C22⋊D271—22no—
78C2×C22⋊D271—22no—
79C2×C22⋊D271—22no—
80C2×C22⋊D271—22no—
81C2×C22⋊D271—22no—
82C2×C22⋊D271—22no—
83C2×C22⋊D271—22no—
84C2×C22⋊D271—22no—
85C2×C22⋊D271—22no—
86C2×C22⋊D271—41no—
87C2×C22⋊D271—41no—
88C2×C22⋊D271—41no—
89C2×C22⋊D271—41no—
90C2×C22⋊D271D414yescomplemented
91C22×Dic271C2311yescharacteristic
92C22×Dic271C2311yescharacteristic
93C22×Dic271C2311yescharacteristic
94C22×Dic271C2311yescharacteristic
95C22×Dic271C2311yescharacteristic
96C2×C4×D271—22no—
97C2×C4×D271—22no—
98C2×C4×D271—22no—
99C2×C4×D271—22no—
100C2×C4×D271—22no—
101C2×C4×D271D414yescomplemented
102Dic271⋊C4—22no—
103C22⋊Dic271—22no—
104C22⋊Dic271—41no—
105(C2×D271)⋊C4—22no—
106(C2×D271)⋊C4—22no—
107(C2×D271)⋊C4—22no—
108(C2×D271)⋊C4—22no—
109(C2×D271)⋊C4—22no—
110(C2×D271)⋊C4—41no—
111(C2×D271)⋊C4—41no—
112C4×Dic271—22no—
113C4×Dic271—22no—
114C24×C271C2311yescharacteristic
115C24×C271C2311yescharacteristic
116C2×D4×C271—22no—
117C2×D4×C271—22no—
118C2×D4×C271—41no—
119C22×C4×C271—22no—
120C22×C4×C271—22no—
121C22×C4×C271—22no—
122C22×C4×C271C2311yescharacteristic
123C22×C4×C271C2311yescharacteristic
124C22×C4×C271C2311yescharacteristic
125C22×C4×C271C2311yescharacteristic
126C22×C4×C271C2311yescharacteristic
127C22⋊C4×C271—22no—
128C22⋊C4×C271—22no—
129C22⋊C4×C271—41no—
130C22×D271—21no—
131C22×D271—21no—
132C22×D271—21no—
133C22×D271—21no—
134C22×D271—21no—
135C22×D271—21no—
136C22×D271—21no—
137C22×D271—21no—
138C22×D271—21no—
139C22×D271—21no—
140C22×D271—21no—
141C22×D271—21no—
142C22×D271—21no—
143C22×D271—21no—
144C22×D271—21no—
145C22×D271—21no—
146C22×D271—21no—
147C22×D271—21no—
148C22×D271—21no—
149C22×D271C2×D412yes—
150C22×D271C2×D412yes—
151Dic542—28no—
152D1084—28no—
153D1084—44no—
154D1084—44no—
155C22⋊D271—28no—
156C22⋊D271—28no—
157C22⋊D271—44no—
158C22⋊D271—44no—
159C22⋊D271—44no—
160C22⋊D271—44no—
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×Dic271—21no—
171C2×Dic271C2×D412yes—
172C2×Dic271C2×D412yes—
173C2×Dic271C2×D412yes—
174C2×Dic271C2×D412yes—
175C2×Dic271#4112yescomplemented
176C4×D271—28no—
177C4×D271—28no—
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×C271C2×D412yes—
192C23×C271C2411yescharacteristic
193D4×C271—44no—
194C2×C4×C271—28no—
195C2×C4×C271—21no—
196C2×C4×C271—21no—
197C2×C4×C271—21no—
198C2×C4×C271—21no—
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×C271C2×D412yes—
208C2×C4×C271C2×D412yes—
209C2×C4×C271#4112yescomplemented
210C22×C271—22no—
211C22×C271—22no—
212C22×C271—22no—
213C22×C271—41no—
214C22×C271—41no—
215C22×C271—41no—
216C22×C271—41no—
217C22×C271—41no—
218C22×C271—41no—
219C22×C271—41no—
220C22×C271—41no—
221C22×C271—41no—
222C22×C271—41no—
223C22×C271—41no—
224C22×C271C22×D411yescharacteristic
225C22×C271C22×D411yescharacteristic
226C22×C271C22×D411yescharacteristic
227C22×C271C2×#4111yescharacteristic
228C22×C271C2×#4111yescharacteristic
229C22×C271H(5,𝔽2)11yescharacteristic
230C22×C271H(5,𝔽2)11yescharacteristic
231C2×D271—22no—
232C2×D271—22no—
233C2×D271—22no—
234C2×D271—22no—
235C2×D271—22no—
236C2×D271—22no—
237C2×D271—41no—
238C2×D271—41no—
239C2×D271—41no—
240C2×D271—41no—
241C2×D271—41no—
242C2×D271—41no—
243C2×D271—41no—
244C2×D271—41no—
245C2×D271—41no—
246C2×D271—41no—
247C2×D271—41no—
248C2×D271—41no—
249C2×D271—41no—
250C2×D271—41no—
251C2×D271—41no—
252C4×C271—22no—
253C4×C271—22no—
254C4×C271—22no—
255C4×C271—41no—
256C4×C271—41no—
257Dic271—22no—
258Dic271—22no—
259Dic271—41no—
260Dic271—41no—
261Dic271#12014yes—
262C2×C271—42no—
263C2×C271—81no—
264C2×C271C2×#12011yescharacteristic
265C2×C271#53211yescharacteristic
266C2×C271#53511yescharacteristic
267C2×C271#54611yescharacteristic
268C2×C271#54611yescharacteristic
269C2×C271#55011yescharacteristic
270C2×C271#55011yescharacteristic
271D271—42no—
272D271—42no—
273D271—81no—
274C271#243111yescharacteristic, complemented, Sylow
275#2431—2711nomaximal, Sylow
276C22×#41—2711no—
277C2×#127—2711no—
278C2×#124—2711no—
279C2×#121—2711no—
280#392—2711no—
281#392—2711no—
282#392—2711no—
283#392—2711no—
284#391—2711no—
285#390—2711no—
286#386—2711no—
287#386—2711no—
288#386—2711no—
289#386D27111yescharacteristic, complemented
290C4×C22⋊C4—2711no—
291C2×#41—5424no—
292C2×#41—5424no—
293C2×#41—5424no—
294C22×Q8—2711no—
295C22×D4—2711no—
296C22×D4—2711no—
297C22×D4—2711no—
298C22×D4—2711no—
299C22×D4—2711no—
300C22×D4—2711no—
301C22×D4—2711no—
302C22×D4—2711no—
303C22×D4—2711no—
304C22×D4C2×D27111yescharacteristic
305C23×C4—2711no—
306C23×C4—2711no—
307C23×C4C2×D27111yescharacteristic
308#127—5424no—
309#124—5424no—
310#121—5424no—
311C2×C4⋊C4—2711no—
312C2×C22⋊C4—2711no—
313C2×C22⋊C4—2711no—
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⋊C4C2×D27111yescharacteristic
322C2×C22⋊C4C2×D27111yescharacteristic
323C2×C22⋊C4C2×D27111yescharacteristic
324C2×C42—2711no—
325C2×C42—2711no—
326#95—2711no—
327#95—2711no—
328#95—2711no—
329#95—2711no—
330#95C2×D27111yescharacteristic
331#95C2×D27111yescharacteristic
332C24—2711no—
333C24—2711no—
334C24—2711no—
335C24C22×D27111yescharacteristic
336C24C22×D27111yescharacteristic
337#41—54232no—
338C2×Q8—2714no—
339C2×Q8—5422no—
340C2×Q8—5422no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—41no—
344C2×D4—2714no—
345C2×D4—5422no—
346C2×D4—5422no—
347C2×D4—5422no—
348C2×D4—5422no—
349C2×D4—5422no—
350C2×D4—5422no—
351C2×D4—5422no—
352C2×D4—5422no—
353C2×D4—5422no—
354C2×D4—5422no—
355C2×D4—5422no—
356C2×D4—5422no—
357C2×D4—5422no—
358C2×D4—5422no—
359C2×D4—5422no—
360C2×D4—5422no—
361C2×D4—5422no—
362C2×D4—5422no—
363C2×D4—5422no—
364C2×D4—10841no—
365C2×D4—10841no—
366C2×D4—10841no—
367C2×D4—10841no—
368C2×D4—10841no—
369C2×D4—10841no—
370C2×D4—10841no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—2711no—
375C22×C4—2711no—
376C22×C4—2711no—
377C22×C4—2714no—
378C22×C4—2711no—
379C22×C4—2711no—
380C22×C4—5422no—
381C22×C4—5422no—
382C22×C4—5422no—
383C22×C4—5422no—
384C22×C4—5422no—
385C22×C4C22×D27111yescharacteristic
386C22×C4C22×D27111yescharacteristic
387C22×C4C22×D27111yescharacteristic
388C22×C4C22×D27111yescharacteristic
389C22×C4C22×D27111yescharacteristic
390C4⋊C4—5422no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—41no—
394C22⋊C4—5422no—
395C22⋊C4—5422no—
396C22⋊C4—5422no—
397C22⋊C4—5422no—
398C22⋊C4—5422no—
399C22⋊C4—5422no—
400C22⋊C4—10841no—
401C22⋊C4—10841no—
402C22⋊C4—10841no—
403C42—5422no—
404C42—5422no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—2712no—
419C23—2712no—
420C23—5421no—
421C23—5421no—
422C23—5421no—
423C23—5421no—
424C23—5421no—
425C23—5421no—
426C23—5421no—
427C23—5421no—
428C23—5421no—
429C23—5421no—
430C23—5421no—
431C23—5421no—
432C23—5421no—
433C23—5421no—
434C23—5421no—
435C23—5421no—
436C23—5421no—
437C23—5421no—
438C23—5421no—
439C23D4×D27112yes—
440C23C23×D27111yescharacteristic
441Q8—5428no—
442D4—44no—
443D4—5428no—
444D4—5428no—
445D4—5428no—
446D4—10844no—
447D4—10844no—
448D4—10844no—
449D4—10844no—
450D4—10844no—
451D4—10844no—
452C2×C4—28no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
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—2712no—
466C2×C4—2712no—
467C2×C4—2712no—
468C2×C4—2712no—
469C2×C4—2712no—
470C2×C4—5428no—
471C2×C4—5428no—
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×C4D4×D27112yes—
483C2×C4D4×D27112yes—
484C2×C4#1686465412yescomplemented
485C22—22no—
486C22—22no—
487C22—22no—
488C22—41no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—5422no—
500C22—5422no—
501C22—5422no—
502C22—5422no—
503C22—5422no—
504C22—5422no—
505C22—10841no—
506C22—10841no—
507C22—10841no—
508C22—10841no—
509C22—10841no—
510C22—10841no—
511C22—10841no—
512C22—10841no—
513C22—10841no—
514C22—10841no—
515C22—10841no—
516C22—10841no—
517C22—10841no—
518C22—10841no—
519C22—10841no—
520C22C2×D4×D27111yescharacteristic
521C22C2×D4×D27111yescharacteristic
522C22C2×D4×D27111yescharacteristic
523C22#41×D27111yescharacteristic
524C22C2×#1686465411yescharacteristic
525C22#2257620511yescharacteristic
526C22#2257620611yescharacteristic
527C4—22no—
528C4—22no—
529C4—22no—
530C4—41no—
531C4—41no—
532C4—2714no—
533C4—5422no—
534C4—5422no—
535C4—10841no—
536C4—10841no—
537C2—42no—
538C2—81no—
539C2—10842no—
540C2—10842no—
541C2—21681no—
542C2#120×D27111yescharacteristic
543C2#3177444411yescharacteristic
544C2#3177446311yescharacteristic
545C2#3177453711yescharacteristic
546C2#3177453911yescharacteristic
547C2#3177456511yescharacteristic
548C2#3177456811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#386×C271C2
hypercenter#386D271
hypocenterC271#2431
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C22×#16864654C2semidirect
2. C2×#22576156C2semidirect
3. C2×#22576146C2semidirect
4. C2×#22576133C2semidirect
5. #31773807C2semidirect
6. #31773806C2semidirect
7. #31773806C2semidirect
8. #31773805C2semidirect
9. #31773801C2semidirect
10. #31773797C2semidirect
11. #31773776C2semidirect
12. #31773775C2semidirect
13. #31773773C2semidirect
14. #31773744C2semidirect
15. #386×C271C2semidirect
63. C2×Dic542D4semidirect
90. C2×C22⋊D271D4semidirect
101. C2×C4×D271D4semidirect
175. C2×Dic271#41semidirect
209. C2×C4×C271#41semidirect
274. C271#2431semidirect
289. #386D271semidirect
484. C2×C4#16864654semidirect

Maximal subgroups

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

subgroupindex
C22×#168646542
C2×#225761562
C2×#225761462
C2×#225761332
#317738072
#317738062
#317738052
#317738012
#317737972
#317737762
#317737752
#317737732
#317737442
#386×C2712
#2431271