Finite Group Databasebeta

← #46839826

Subgroups — #46839826

Order 34688.

800 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576142C211yesmaximal, characteristic, complemented
2C2×#22576129C211yesmaximal, characteristic
3C2×C4×C22⋊D271C211yesmaximal, characteristic, complemented
4#31773833C211yesmaximal, characteristic, complemented
5#31773825C211yesmaximal, characteristic, complemented
6#31773813C211yesmaximal, characteristic, complemented
7#31773812C211yesmaximal, characteristic, complemented
8#31773791C211yesmaximal, characteristic, complemented
9#31773790C211yesmaximal, characteristic, complemented
10#31773776C211yesmaximal, characteristic, complemented
11#31773773C211yesmaximal, characteristic, complemented
12#31773770C211yesmaximal, characteristic, complemented
13#31773757C211yesmaximal, characteristic, complemented
14#31773753C211yesmaximal, characteristic, complemented
15C22×C22⋊C4×C271C211yesmaximal, characteristic, complemented
16C22×C22⋊D271C2211yescharacteristic
17C22×C22⋊D271C2211yescharacteristic
18C22×C22⋊D271C2211yescharacteristic
19C22×C4×D271C2211yescharacteristic, complemented
20#22576142—24no—
21#22576129—24no—
22C4×C22⋊D271—24no—
23C2×C4⋊Dic271C2211yescharacteristic, complemented
24C2×C4⋊Dic271C2211yescharacteristic, complemented
25C2×Dic271⋊C4C2211yescharacteristic, complemented
26C2×Dic271⋊C4C2211yescharacteristic, complemented
27C2×Dic271⋊C4C2211yescharacteristic, complemented
28C2×C22⋊Dic271C2211yescharacteristic
29C2×C22⋊Dic271C2211yescharacteristic
30C2×C22⋊Dic271C2211yescharacteristic
31C2×C22⋊Dic271C2211yescharacteristic
32C2×C22⋊Dic271C2211yescharacteristic
33C2×C22⋊Dic271C2211yescharacteristic
34C2×C22⋊Dic271C2211yescharacteristic
35C2×C22⋊Dic271C2211yescharacteristic
36C2×C22⋊Dic271C2211yescharacteristic
37C2×(C2×D271)⋊C4C2211yescharacteristic, complemented
38C2×(C2×D271)⋊C4C2211yescharacteristic, complemented
39C2×(C2×D271)⋊C4C2211yescharacteristic, complemented
40C2×C4×Dic271C2211yescharacteristic, complemented
41C271⋊#95C2211yescharacteristic, complemented
42C271⋊#95C2211yescharacteristic, complemented
43C271⋊#95C2211yescharacteristic, complemented
44C271⋊#95C2211yescharacteristic, complemented
45C271⋊#95C2211yescharacteristic, complemented
46C271⋊#95C2211yescharacteristic, complemented
47C25×C271C2211yescharacteristic
48C23×C4×C271C2211yescharacteristic
49C23×C4×C271C2211yescharacteristic
50C2×C22⋊C4×C271—24no—
51C2×C22⋊C4×C271C2211yescharacteristic
52C2×C22⋊C4×C271C2211yescharacteristic
53C2×C22⋊C4×C271C2211yescharacteristic
54C2×C22⋊C4×C271C2211yescharacteristic
55C23×D271C2311yescharacteristic
56C2×C22⋊D271—22no—
57C2×C22⋊D271—22no—
58C2×C22⋊D271—22no—
59C2×C22⋊D271—22no—
60C2×C22⋊D271—22no—
61C2×C22⋊D271—22no—
62C2×C22⋊D271—41no—
63C2×C22⋊D271—41no—
64C2×C22⋊D271D414yes—
65C22×Dic271C2311yescharacteristic
66C22×Dic271C2311yescharacteristic
67C22×Dic271C2311yescharacteristic
68C22×Dic271C2311yescharacteristic
69C22×Dic271C2311yescharacteristic
70C22×Dic271C2311yescharacteristic
71C22×Dic271C2311yescharacteristic
72C2×C4×D271—22no—
73C2×C4×D271—22no—
74C2×C4×D271—41no—
75C4⋊Dic271—22no—
76C4⋊Dic271—41no—
77Dic271⋊C4—22no—
78Dic271⋊C4—22no—
79Dic271⋊C4—41no—
80C22⋊Dic271—22no—
81C22⋊Dic271—22no—
82C22⋊Dic271—22no—
83C22⋊Dic271—22no—
84C22⋊Dic271—22no—
85C22⋊Dic271—22no—
86C22⋊Dic271—22no—
87C22⋊Dic271—41no—
88C22⋊Dic271—41no—
89(C2×D271)⋊C4—22no—
90(C2×D271)⋊C4—22no—
91(C2×D271)⋊C4—41no—
92C4×Dic271—22no—
93C24×C271—22no—
94C24×C271—22no—
95C24×C271—22no—
96C24×C271—22no—
97C24×C271—22no—
98C24×C271—41no—
99C24×C271—41no—
100C24×C271C2311yescharacteristic
101C24×C271C2311yescharacteristic
102C24×C271C2311yescharacteristic
103C22×C4×C271—22no—
104C22×C4×C271—22no—
105C22×C4×C271—22no—
106C22×C4×C271—22no—
107C22×C4×C271—41no—
108C22×C4×C271—41no—
109C22×C4×C271C2311yescharacteristic
110C22×C4×C271C2311yescharacteristic
111C22×C4×C271C2311yescharacteristic
112C22×C4×C271C2311yescharacteristic
113C22⋊C4×C271—22no—
114C22⋊C4×C271—22no—
115C22⋊C4×C271—22no—
116C22⋊C4×C271D414yescomplemented
117C22×D271—21no—
118C22×D271—21no—
119C22×D271—21no—
120C22×D271—21no—
121C22×D271—21no—
122C22×D271—21no—
123C22×D271C2×D412yes—
124C22⋊D271—28no—
125C22⋊D271—44no—
126C22⋊D271—44no—
127C2×Dic271—21no—
128C2×Dic271—21no—
129C2×Dic271—21no—
130C2×Dic271—21no—
131C2×Dic271—21no—
132C2×Dic271—21no—
133C2×Dic271—21no—
134C2×Dic271—21no—
135C2×Dic271—21no—
136C2×Dic271—21no—
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×Dic271C2×D412yes—
148C4×D271—44no—
149C23×C271—21no—
150C23×C271—21no—
151C23×C271—21no—
152C23×C271—21no—
153C23×C271—28no—
154C23×C271—21no—
155C23×C271—21no—
156C23×C271—21no—
157C23×C271—21no—
158C23×C271—21no—
159C23×C271—21no—
160C23×C271—21no—
161C23×C271—21no—
162C23×C271—21no—
163C23×C271—21no—
164C23×C271—21no—
165C23×C271—21no—
166C23×C271—44no—
167C23×C271—44no—
168C23×C271—44no—
169C23×C271—44no—
170C23×C271—44no—
171C23×C271—44no—
172C23×C271C2×D412yes—
173C23×C271C2×D412yes—
174C23×C271#4112yes—
175C23×C271#4112yes—
176C23×C271#4112yes—
177C23×C271C2411yescharacteristic
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—44no—
189C2×C4×C271—44no—
190C2×C4×C271C2×D412yes—
191C2×C4×C271C2×D412yes—
192C22×C271—22no—
193C22×C271—22no—
194C22×C271—22no—
195C22×C271—22no—
196C22×C271—22no—
197C22×C271—22no—
198C22×C271—22no—
199C22×C271—22no—
200C22×C271—22no—
201C22×C271—22no—
202C22×C271—22no—
203C22×C271—22no—
204C22×C271—416no—
205C22×C271—41no—
206C22×C271—41no—
207C22×C271—41no—
208C22×C271—41no—
209C22×C271—41no—
210C22×C271—41no—
211C22×C271—41no—
212C22×C271—41no—
213C22×C271#12314yescomplemented
214C22×C271C22×D411yescharacteristic
215C22×C271C22×D411yescharacteristic
216C22×C271C2×#4111yescharacteristic
217C22×C271C2×#4111yescharacteristic
218C22×C271C2×#4111yescharacteristic
219C22×C271H(5,𝔽2)11yescharacteristic
220C22×C271H(5,𝔽2)11yescharacteristic
221C2×D271—22no—
222C2×D271—22no—
223C2×D271—22no—
224C2×D271—41no—
225C2×D271—41no—
226C2×D271—41no—
227C2×D271—41no—
228C4×C271—22no—
229C4×C271—22no—
230C4×C271—41no—
231C4×C271—41no—
232Dic271—22no—
233Dic271—41no—
234Dic271—41no—
235Dic271—41no—
236Dic271—41no—
237Dic271—41no—
238Dic271—41no—
239C2×C271—24no—
240C2×C271—42no—
241C2×C271—42no—
242C2×C271C2×#12311yescharacteristic
243C2×C271#52511yescharacteristic
244C2×C271#53811yescharacteristic
245C2×C271D4211yescharacteristic
246C2×C271#54611yescharacteristic
247C2×C271#54611yescharacteristic
248C2×C271#55111yescharacteristic
249D271—42no—
250C271#241811yescharacteristic, complemented, Sylow
251#2418—2711nomaximal, Sylow
252C2×#123—2711no—
253C2×#120—2711no—
254C2×C4×D4—2711no—
255C22×C22⋊C4D27111yescharacteristic, complemented
256#399—2711no—
257#397—2711no—
258#394—2711no—
259#394—2711no—
260#388—2711no—
261#388—2711no—
262#386—2711no—
263#386—2711no—
264#385—2711no—
265#380—2711no—
266#379—2711no—
267C25C2×D27111yescharacteristic
268C22×D4—2711no—
269C22×D4—2711no—
270C22×D4—2711no—
271C23×C4—2711no—
272C23×C4C2×D27111yescharacteristic
273C23×C4C2×D27111yescharacteristic
274#123—5424no—
275#120—5424no—
276C4×D4—5424no—
277C2×C4⋊C4—2711no—
278C2×C4⋊C4—2711no—
279C2×C4⋊C4—2711no—
280C2×C4⋊C4—2711no—
281C2×C4⋊C4—2711no—
282C2×C22⋊C4—24no—
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⋊C4C2×D27111yescharacteristic
296C2×C22⋊C4C2×D27111yescharacteristic
297C2×C22⋊C4C2×D27111yescharacteristic
298C2×C22⋊C4C2×D27111yescharacteristic
299C2×C42—2711no—
300#95—2711no—
301#95—2711no—
302#95—2711no—
303#95—2711no—
304#95—2711no—
305#95—2711no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—41no—
312C24—41no—
313C24—2711no—
314C24C22×D27111yescharacteristic
315C24C22×D27111yescharacteristic
316C24C22×D27111yescharacteristic
317C2×D4—2714no—
318C2×D4—5422no—
319C2×D4—5422no—
320C2×D4—5422no—
321C2×D4—5422no—
322C2×D4—5422no—
323C2×D4—5422no—
324C2×D4—10841no—
325C2×D4—10841no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—2711no—
333C22×C4—2711no—
334C22×C4—2711no—
335C22×C4—2711no—
336C22×C4—2711no—
337C22×C4—2711no—
338C22×C4—2711no—
339C22×C4—5422no—
340C22×C4—5422no—
341C22×C4—10841no—
342C22×C4C22×D27111yescharacteristic
343C22×C4C22×D27111yescharacteristic
344C22×C4C22×D27111yescharacteristic
345C22×C4C22×D27111yescharacteristic
346C4⋊C4—5422no—
347C4⋊C4—5422no—
348C4⋊C4—5422no—
349C4⋊C4—10841no—
350C4⋊C4—10841no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—22no—
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—5422no—
363C22⋊C4—10841no—
364C22⋊C4—10841no—
365C22⋊C4—10841no—
366C22⋊C4C22⋊D27114yescomplemented
367C42—5422no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—28no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—44no—
386C23—44no—
387C23—44no—
388C23—44no—
389C23—44no—
390C23—44no—
391C23—2712no—
392C23—5421no—
393C23—5421no—
394C23—5421no—
395C23—5421no—
396C23—5421no—
397C23—5421no—
398C23D4×D27112yes—
399C23C2×C22⋊D27112yes—
400C23#1686465312yes—
401C23#1686465412yes—
402C23#1686465412yes—
403C23C23×D27111yescharacteristic
404D4—5428no—
405D4—10844no—
406D4—10844no—
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—44no—
418C2×C4—44no—
419C2×C4—2712no—
420C2×C4—5421no—
421C2×C4—5421no—
422C2×C4—5421no—
423C2×C4—5421no—
424C2×C4—5421no—
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—5421no—
439C2×C4—5421no—
440C2×C4—10844no—
441C2×C4C2×C22⋊D27112yes—
442C2×C4C2×C22⋊D27112yes—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—416no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—5422no—
465C22—5422no—
466C22—5422no—
467C22—10841no—
468C22—10841no—
469C22—10841no—
470C22—10841no—
471C22#2257614214yescomplemented
472C22C2×D4×D27111yescharacteristic
473C22C22×C22⋊D27111yescharacteristic
474C22C2×#1686465311yescharacteristic
475C22C2×#1686465411yescharacteristic
476C22C2×#1686465411yescharacteristic
477C22#2257620511yescharacteristic
478C22#2257620511yescharacteristic
479C4—22no—
480C4—22no—
481C4—41no—
482C4—41no—
483C4—5422no—
484C4—10841no—
485C4—10841no—
486C4—10841no—
487C4—10841no—
488C4—10841no—
489C4—10841no—
490C2—24no—
491C2—42no—
492C2—42no—
493C2—10842no—
494C2C2×#2257614211yescharacteristic
495C2#3177440811yescharacteristic
496C2#3177448511yescharacteristic
497C2D4×C22⋊D27111yescharacteristic
498C2#3177453711yescharacteristic
499C2#3177454111yescharacteristic
500C2#3177457311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
FittingC22×C22⋊C4×C271C2
prime radicalC2×#22576129C2
hypercenterC22×C22⋊C4D271
hypocenterC271#2418
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#22576142C2semidirect
3. C2×C4×C22⋊D271C2semidirect
4. #31773833C2semidirect
5. #31773825C2semidirect
6. #31773813C2semidirect
7. #31773812C2semidirect
8. #31773791C2semidirect
9. #31773790C2semidirect
10. #31773776C2semidirect
11. #31773773C2semidirect
12. #31773770C2semidirect
13. #31773757C2semidirect
14. #31773753C2semidirect
15. C22×C22⋊C4×C271C2semidirect
19. C22×C4×D271C22semidirect
23. C2×C4⋊Dic271C22semidirect
24. C2×C4⋊Dic271C22semidirect
25. C2×Dic271⋊C4C22semidirect
26. C2×Dic271⋊C4C22semidirect
27. C2×Dic271⋊C4C22semidirect
37. C2×(C2×D271)⋊C4C22semidirect
38. C2×(C2×D271)⋊C4C22semidirect
39. C2×(C2×D271)⋊C4C22semidirect
40. C2×C4×Dic271C22semidirect
41. C271⋊#95C22semidirect
42. C271⋊#95C22semidirect
43. C271⋊#95C22semidirect
44. C271⋊#95C22semidirect
45. C271⋊#95C22semidirect
46. C271⋊#95C22semidirect
116. C22⋊C4×C271D4semidirect
213. C22×C271#123semidirect
250. C271#2418semidirect
255. C22×C22⋊C4D271semidirect
366. C22⋊C4C22⋊D271semidirect
471. C22#22576142semidirect

Maximal subgroups

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

subgroupindex
C2×#225761422
C2×#225761292
C2×C4×C22⋊D2712
#317738332
#317738252
#317738132
#317738122
#317737912
#317737902
#317737762
#317737732
#317737702
#317737572
#317737532
C22×C22⋊C4×C2712
#2418271