Finite Group Databasebeta

← #46839744

Subgroups — #46839744

Order 34688.

786 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576145C211yesmaximal, characteristic, complemented
2C2×#22576130C211yesmaximal, characteristic
3C2×#22576118C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D271C211yesmaximal, characteristic, complemented
5#31773819C211yesmaximal, characteristic, complemented
6#31773814C211yesmaximal, characteristic, complemented
7#31773813C211yesmaximal, characteristic, complemented
8#31773798C211yesmaximal, characteristic, complemented
9#31773797C211yesmaximal, characteristic, complemented
10#31773786C211yesmaximal, characteristic, complemented
11#31773785C211yesmaximal, characteristic, complemented
12#31773775C211yesmaximal, characteristic, complemented
13#31773768C211yesmaximal, characteristic, complemented
14#31773752C211yesmaximal, characteristic, complemented
15#396×C271C211yesmaximal, characteristic, complemented
16C24×D271C2211yescharacteristic
17C22×D1084C2211yescharacteristic
18C22×D1084C2211yescharacteristic
19C22×C22⋊D271C2211yescharacteristic
20C22×C4×D271C2211yescharacteristic
21C22×C4×D271C2211yescharacteristic
22#22576145—24no—
23#22576130—24no—
24#22576118—24no—
25C2×C4⋊Dic271C2211yescharacteristic, complemented
26C2×Dic271⋊C4C2211yescharacteristic, complemented
27C2×Dic271⋊C4C2211yescharacteristic, complemented
28C2×C22⋊Dic271C2211yescharacteristic
29C2×C22⋊Dic271C2211yescharacteristic
30C2×(C2×D271)⋊C4C2211yescharacteristic
31C2×(C2×D271)⋊C4C2211yescharacteristic
32C2×(C2×D271)⋊C4C2211yescharacteristic
33C2×(C2×D271)⋊C4C2211yescharacteristic
34C2×(C2×D271)⋊C4C2211yescharacteristic
35C2×(C2×D271)⋊C4C2211yescharacteristic
36C2×(C2×D271)⋊C4C2211yescharacteristic
37C2×(C2×D271)⋊C4C2211yescharacteristic
38C2×(C2×D271)⋊C4C2211yescharacteristic, complemented
39C2×(C2×D271)⋊C4C2211yescharacteristic, complemented
40C2×(C2×D271)⋊C4C2211yescharacteristic, complemented
41C2×(C2×D271)⋊C4C2211yescharacteristic, complemented
42C22⋊C4×D271—24no—
43C2×C4×Dic271C2211yescharacteristic, complemented
44C2×C4×Dic271C2211yescharacteristic, complemented
45C271⋊#95C2211yescharacteristic, complemented
46C271⋊#95C2211yescharacteristic, complemented
47C271⋊#95C2211yescharacteristic, complemented
48C2×C4⋊C4×C271C2211yescharacteristic, complemented
49C2×C4⋊C4×C271C2211yescharacteristic, complemented
50C2×C4⋊C4×C271C2211yescharacteristic, complemented
51C2×C22⋊C4×C271C2211yescharacteristic
52C2×C22⋊C4×C271C2211yescharacteristic
53C2×C22⋊C4×C271C2211yescharacteristic
54#95×C271C2211yescharacteristic, complemented
55C23×D271—22no—
56C23×D271—22no—
57C23×D271—22no—
58C23×D271—22no—
59C23×D271—22no—
60C23×D271—41no—
61C23×D271—41no—
62C23×D271C2311yescharacteristic
63C23×D271C2311yescharacteristic
64C23×D271C2311yescharacteristic
65C2×D1084—22no—
66C2×D1084—22no—
67C2×D1084—22no—
68C2×D1084—22no—
69C2×D1084—22no—
70C2×D1084—41no—
71C2×C22⋊D271—22no—
72C2×C22⋊D271—22no—
73C2×C22⋊D271—41no—
74C22×Dic271C2311yescharacteristic
75C22×Dic271C2311yescharacteristic
76C22×Dic271C2311yescharacteristic
77C22×Dic271C2311yescharacteristic
78C22×Dic271C2311yescharacteristic
79C2×C4×D271—22no—
80C2×C4×D271—22no—
81C2×C4×D271—22no—
82C2×C4×D271—41no—
83C2×C4×D271—41no—
84C2×C4×D271D414yescomplemented
85C4⋊Dic271—22no—
86Dic271⋊C4—22no—
87Dic271⋊C4—41no—
88C22⋊Dic271—22no—
89C22⋊Dic271—22no—
90(C2×D271)⋊C4—22no—
91(C2×D271)⋊C4—22no—
92(C2×D271)⋊C4—22no—
93(C2×D271)⋊C4—22no—
94(C2×D271)⋊C4—22no—
95(C2×D271)⋊C4—22no—
96(C2×D271)⋊C4—22no—
97(C2×D271)⋊C4—22no—
98(C2×D271)⋊C4—41no—
99(C2×D271)⋊C4—41no—
100(C2×D271)⋊C4—41no—
101(C2×D271)⋊C4D414yes—
102C4×Dic271—22no—
103C4×Dic271—22no—
104C24×C271C2311yescharacteristic
105C22×C4×C271C2311yescharacteristic
106C22×C4×C271C2311yescharacteristic
107C22×C4×C271C2311yescharacteristic
108C22×C4×C271C2311yescharacteristic
109C22×C4×C271C2311yescharacteristic
110C22×C4×C271C2311yescharacteristic
111C4⋊C4×C271—22no—
112C4⋊C4×C271—22no—
113C4⋊C4×C271—41no—
114C22⋊C4×C271—22no—
115C22⋊C4×C271—22no—
116C22⋊C4×C271—41no—
117C22×D271—21no—
118C22×D271—21no—
119C22×D271—21no—
120C22×D271—21no—
121C22×D271—28no—
122C22×D271—21no—
123C22×D271—21no—
124C22×D271—21no—
125C22×D271—21no—
126C22×D271—21no—
127C22×D271—21no—
128C22×D271—21no—
129C22×D271—21no—
130C22×D271—21no—
131C22×D271—21no—
132C22×D271—21no—
133C22×D271—21no—
134C22×D271—21no—
135C22×D271—44no—
136C22×D271—44no—
137C22×D271—44no—
138C22×D271—44no—
139C22×D271—44no—
140C22×D271—44no—
141C22×D271C2×D412yes—
142C22×D271C2×D412yes—
143C22×D271#4112yes—
144C22×D271#4112yes—
145D1084—44no—
146D1084—44no—
147C22⋊D271—44no—
148C2×Dic271—21no—
149C2×Dic271—21no—
150C2×Dic271—21no—
151C2×Dic271—21no—
152C2×Dic271—21no—
153C2×Dic271—21no—
154C2×Dic271—21no—
155C2×Dic271—21no—
156C2×Dic271—21no—
157C2×Dic271—21no—
158C2×Dic271—21no—
159C2×Dic271—21no—
160C2×Dic271—21no—
161C2×Dic271C2×D412yes—
162C2×Dic271C2×D412yes—
163C4×D271—28no—
164C4×D271—44no—
165C23×C271—21no—
166C23×C271—21no—
167C23×C271—21no—
168C23×C271—21no—
169C23×C271—21no—
170C23×C271—21no—
171C23×C271#4112yes—
172C23×C271C2411yescharacteristic
173C2×C4×C271—21no—
174C2×C4×C271—21no—
175C2×C4×C271—21no—
176C2×C4×C271—21no—
177C2×C4×C271—21no—
178C2×C4×C271—21no—
179C2×C4×C271—21no—
180C2×C4×C271—21no—
181C2×C4×C271—21no—
182C2×C4×C271—21no—
183C2×C4×C271—21no—
184C2×C4×C271—21no—
185C2×C4×C271—21no—
186C2×C4×C271—21no—
187C2×C4×C271—21no—
188C2×C4×C271—21no—
189C2×C4×C271C2×D412yes—
190C2×C4×C271C2×D412yes—
191C22×C271—22no—
192C22×C271—22no—
193C22×C271—22no—
194C22×C271—41no—
195C22×C271—41no—
196C22×C271—41no—
197C22×C271—41no—
198C22×C271C22×D411yescharacteristic
199C22×C271C22×D411yescharacteristic
200C22×C271C2×#4111yescharacteristic
201C22×C271C2×#4111yescharacteristic
202C22×C271C2×#4111yescharacteristic
203C22×C271H(5,𝔽2)11yescharacteristic
204C22×C271H(5,𝔽2)11yescharacteristic
205C2×D271—22no—
206C2×D271—22no—
207C2×D271—22no—
208C2×D271—22no—
209C2×D271—22no—
210C2×D271—22no—
211C2×D271—22no—
212C2×D271—22no—
213C2×D271—22no—
214C2×D271—416no—
215C2×D271—41no—
216C2×D271—41no—
217C2×D271—41no—
218C2×D271—41no—
219C2×D271—41no—
220C2×D271—41no—
221C2×D271—41no—
222C2×D271—41no—
223C2×D271—41no—
224C2×D271—41no—
225C2×D271—41no—
226C2×D271#12114yes—
227C4×C271—22no—
228C4×C271—22no—
229C4×C271—41no—
230C4×C271—41no—
231C4×C271—41no—
232C4×C271—41no—
233Dic271—22no—
234Dic271—22no—
235Dic271—41no—
236Dic271—41no—
237Dic271—41no—
238C2×C271—42no—
239C2×C271C2×#12111yescharacteristic
240C2×C271#52511yescharacteristic
241C2×C271#53511yescharacteristic
242C2×C271#54611yescharacteristic
243C2×C271#54611yescharacteristic
244C2×C271#55111yescharacteristic
245C2×C271#55111yescharacteristic
246D271—24no—
247D271—42no—
248D271—81no—
249C271#241111yescharacteristic, complemented, Sylow
250#2411—2711nomaximal, Sylow
251C2×#123—2711no—
252C2×#120—2711no—
253C2×#117—2711no—
254C22×C22⋊C4—2711no—
255#396—2711no—
256#396D27111yescharacteristic, complemented
257#394—2711no—
258#394—2711no—
259#390—2711no—
260#390—2711no—
261#388—2711no—
262#388—2711no—
263#386—2711no—
264#385—2711no—
265#379—2711no—
266C25—2711no—
267C22×D4—2711no—
268C22×D4—2711no—
269C22×D4—2711no—
270C23×C4—2711no—
271C23×C4—2711no—
272#123—5424no—
273#120—5424no—
274#117—5424no—
275C2×C4⋊C4—2711no—
276C2×C4⋊C4—2711no—
277C2×C4⋊C4—2711no—
278C2×C4⋊C4C2×D27111yescharacteristic, complemented
279C2×C4⋊C4C2×D27111yescharacteristic, complemented
280C2×C4⋊C4C2×D27111yescharacteristic, complemented
281C2×C22⋊C4—2711no—
282C2×C22⋊C4—2711no—
283C2×C22⋊C4—2711no—
284C2×C22⋊C4—2711no—
285C2×C22⋊C4—2711no—
286C2×C22⋊C4—2711no—
287C2×C22⋊C4—2711no—
288C2×C22⋊C4—2711no—
289C2×C22⋊C4—2711no—
290C2×C22⋊C4—2711no—
291C2×C22⋊C4—2711no—
292C2×C22⋊C4—2711no—
293C2×C22⋊C4—2711no—
294C2×C22⋊C4—2711no—
295C2×C22⋊C4—5424no—
296C2×C22⋊C4C2×D27111yescharacteristic
297C2×C22⋊C4C2×D27111yescharacteristic
298C2×C22⋊C4C2×D27111yescharacteristic
299C2×C42—2711no—
300C2×C42—2711no—
301#95—2711no—
302#95—2711no—
303#95—2711no—
304#95C2×D27111yescharacteristic, complemented
305C24—2711no—
306C24—2711no—
307C24—2711no—
308C24—5422no—
309C24—5422no—
310C24—5422no—
311C24—5422no—
312C24—5422no—
313C24—10841no—
314C24—10841no—
315C24C22×D27111yescharacteristic
316C2×D4—5422no—
317C2×D4—5422no—
318C2×D4—5422no—
319C2×D4—5422no—
320C2×D4—5422no—
321C2×D4—5422no—
322C2×D4—5422no—
323C2×D4—10841no—
324C2×D4—10841no—
325C22×C4—2711no—
326C22×C4—2711no—
327C22×C4—2714no—
328C22×C4—2711no—
329C22×C4—2711no—
330C22×C4—2711no—
331C22×C4—5422no—
332C22×C4—5422no—
333C22×C4—5422no—
334C22×C4—10841no—
335C22×C4—10841no—
336C22×C4C22×D27111yescharacteristic
337C22×C4C22×D27111yescharacteristic
338C22×C4C22×D27111yescharacteristic
339C22×C4C22×D27111yescharacteristic
340C22×C4C22×D27111yescharacteristic
341C22×C4C22×D27111yescharacteristic
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—41no—
345C4⋊C4—5422no—
346C4⋊C4—5422no—
347C4⋊C4—10841no—
348C22⋊C4—22no—
349C22⋊C4—22no—
350C22⋊C4—41no—
351C22⋊C4—2714no—
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—5422no—
361C22⋊C4—5422no—
362C22⋊C4—10841no—
363C22⋊C4—10841no—
364C22⋊C4—10841no—
365C42—5422no—
366C42—5422no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—2712no—
374C23—2712no—
375C23—2712no—
376C23—2712no—
377C23—5428no—
378C23—5421no—
379C23—5421no—
380C23—5421no—
381C23—5421no—
382C23—5421no—
383C23—5421no—
384C23—5421no—
385C23—5421no—
386C23—5421no—
387C23—5421no—
388C23—5421no—
389C23—5421no—
390C23—5421no—
391C23—5421no—
392C23—5421no—
393C23—5421no—
394C23—5421no—
395C23—10844no—
396C23—10844no—
397C23—10844no—
398C23—10844no—
399C23—10844no—
400C23—10844no—
401C23#1686465512yes—
402C23C23×D27111yescharacteristic
403D4—10844no—
404D4—10844no—
405D4—10844no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—2712no—
423C2×C4—2712no—
424C2×C4—5428no—
425C2×C4—5421no—
426C2×C4—5421no—
427C2×C4—5421no—
428C2×C4—5421no—
429C2×C4—5421no—
430C2×C4—5421no—
431C2×C4—5421no—
432C2×C4—5421no—
433C2×C4—5421no—
434C2×C4—5421no—
435C2×C4—5421no—
436C2×C4—5421no—
437C2×C4—5421no—
438C2×C4—10844no—
439C2×C4D4×D27112yes—
440C2×C4D4×D27112yes—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—2714no—
449C22—5422no—
450C22—5422no—
451C22—5422no—
452C22—5422no—
453C22—5422no—
454C22—5422no—
455C22—5422no—
456C22—5422no—
457C22—5422no—
458C22—108416no—
459C22—10841no—
460C22—10841no—
461C22—10841no—
462C22—10841no—
463C22—10841no—
464C22—10841no—
465C22—10841no—
466C22—10841no—
467C22—10841no—
468C22—10841no—
469C22—10841no—
470C22C2×D4×D27111yescharacteristic
471C22C2×D4×D27111yescharacteristic
472C22#41×D27111yescharacteristic
473C22#41×D27111yescharacteristic
474C22C2×#1686465511yescharacteristic
475C22#2257620511yescharacteristic
476C22#2257620611yescharacteristic
477C4—22no—
478C4—22no—
479C4—41no—
480C4—41no—
481C4—41no—
482C4—41no—
483C4—5422no—
484C4—5422no—
485C4—10841no—
486C4—10841no—
487C4—10841no—
488C2—42no—
489C2—5424no—
490C2—10842no—
491C2—21681no—
492C2#121×D27111yescharacteristic
493C2#3177441011yescharacteristic
494C2#3177446311yescharacteristic
495C2#3177453611yescharacteristic
496C2#3177453811yescharacteristic
497C2#3177457211yescharacteristic
498C2#3177457411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#396×C271C2
prime radicalC2×#22576130C2
hypercenter#396D271
hypocenterC271#2411
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#22576145C2semidirect
3. C2×#22576118C2semidirect
4. C2×C22⋊C4×D271C2semidirect
5. #31773819C2semidirect
6. #31773814C2semidirect
7. #31773813C2semidirect
8. #31773798C2semidirect
9. #31773797C2semidirect
10. #31773786C2semidirect
11. #31773785C2semidirect
12. #31773775C2semidirect
13. #31773768C2semidirect
14. #31773752C2semidirect
15. #396×C271C2semidirect
25. C2×C4⋊Dic271C22semidirect
26. C2×Dic271⋊C4C22semidirect
27. C2×Dic271⋊C4C22semidirect
38. C2×(C2×D271)⋊C4C22semidirect
39. C2×(C2×D271)⋊C4C22semidirect
40. C2×(C2×D271)⋊C4C22semidirect
41. C2×(C2×D271)⋊C4C22semidirect
43. C2×C4×Dic271C22semidirect
44. C2×C4×Dic271C22semidirect
45. C271⋊#95C22semidirect
46. C271⋊#95C22semidirect
47. C271⋊#95C22semidirect
48. C2×C4⋊C4×C271C22semidirect
49. C2×C4⋊C4×C271C22semidirect
50. C2×C4⋊C4×C271C22semidirect
54. #95×C271C22semidirect
84. C2×C4×D271D4semidirect
249. C271#2411semidirect
256. #396D271semidirect
278. C2×C4⋊C4C2×D271semidirect
279. C2×C4⋊C4C2×D271semidirect
280. C2×C4⋊C4C2×D271semidirect
304. #95C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761452
C2×#225761302
C2×#225761182
C2×C22⋊C4×D2712
#317738192
#317738142
#317738132
#317737982
#317737972
#317737862
#317737852
#317737752
#317737682
#317737522
#396×C2712
#2411271