Finite Group Databasebeta

← #46847900

Subgroups — #46847900

Order 34688.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576186C211yesmaximal, characteristic, complemented
2C2×#22576180C211yesmaximal, characteristic, complemented
3#31774568C211yesmaximal, characteristic, complemented
4#31774287C211yesmaximal, characteristic, complemented
5#31774145C211yesmaximal, characteristic, complemented
6#31774131C211yesmaximal, characteristic, complemented
7#31774116C211yesmaximal, characteristic, complemented
8#31774092C211yesmaximal, characteristic, complemented
9#31774057C211yesmaximal, characteristic, complemented
10#31774042C211yesmaximal, characteristic, complemented
11#31774040C211yesmaximal, characteristic, complemented
12#31774031C211yesmaximal, characteristic, complemented
13#31773997C211yesmaximal, characteristic, complemented
14#31773969C211yesmaximal, characteristic, complemented
15#546×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic
17C2×#16864654C2211yescharacteristic, complemented
18#22576186—24no—
19#22576180—24no—
20C2×#16864644C2211yescharacteristic, complemented
21C2×#16864642C2211yescharacteristic
22C2×#16864642C2211yescharacteristic
23C2×#16864641C2211yescharacteristic, complemented
24C2×#16864641C2211yescharacteristic, complemented
25C2×#16864639C2211yescharacteristic
26C2×#16864639C2211yescharacteristic
27C2×#16864639C2211yescharacteristic
28C2×#16864639C2211yescharacteristic
29C2×#16864638C2211yescharacteristic
30C22×C271⋊C8C2211yescharacteristic, complemented
31#22576157C2211yescharacteristic, complemented
32#22576156—21no—
33#22576134C2211yescharacteristic
34#22576134C2211yescharacteristic, complemented
35#22576133—21no—
36#22576133—21no—
37C4×Dic542C2211yescharacteristic, complemented
38#22576122—21no—
39#22576093C2211yescharacteristic, complemented
40C4⋊(C271⋊C8)C2211yescharacteristic, complemented
41#22576084C2211yescharacteristic, complemented
42#22576083C2211yescharacteristic, complemented
43#22576081C2211yescharacteristic
44#22576081C2211yescharacteristic
45#22576080C2211yescharacteristic, complemented
46#22576080C2211yescharacteristic, complemented
47C22⋊(C271⋊C8)C2211yescharacteristic
48C22⋊(C271⋊C8)C2211yescharacteristic, complemented
49#22576071C2211yescharacteristic, complemented
50C2×#41×C271C2211yescharacteristic
51C22×D4×C271C2211yescharacteristic
52#124×C271C2211yescharacteristic
53#123×C271—21no—
54C22⋊Q8×C271C2211yescharacteristic
55#121×C271—21no—
56#121×C271C2211yescharacteristic
57#120×C271—21no—
58C4×D4×C271C2211yescharacteristic
59C4×D4×C271C2211yescharacteristic
60C2×C22⋊C4×C271—21no—
61#16864654—22no—
62#16864654—22no—
63#16864654—41no—
64#16864654—41no—
65C2×Dic542C2311yescharacteristic
66C2×D1084—21no—
67C2×D1084—21no—
68C2×D1084C2311yescharacteristic
69C2×D1084C2311yescharacteristic
70C2×C22⋊D271—21no—
71C2×C22⋊D271—21no—
72C2×C22⋊D271—21no—
73C2×C22⋊D271—21no—
74C2×C4×D271—21no—
75C2×C4×D271—21no—
76#16864644—22no—
77#16864642—22no—
78#16864642D414yescomplemented
79#16864641—22no—
80#16864641—41no—
81#16864639—22no—
82#16864639—22no—
83#16864639—22no—
84#16864639—41no—
85#16864638—22no—
86C2×C271⋊C8—22no—
87C2×C271⋊C8C2311yescharacteristic
88C2×C271⋊C8C2311yescharacteristic
89C2×C271⋊C8C2311yescharacteristic
90C2×C271⋊C8C2311yescharacteristic
91C4⋊Dic271C2311yescharacteristic
92Dic271⋊C4—21no—
93(C2×D271)⋊C4—21no—
94(C2×D271)⋊C4—21no—
95C4×Dic271—21no—
96C24×C271—21no—
97#41×C271—22no—
98#41×C271—41no—
99C2×Q8×C271C2311yescharacteristic
100C2×D4×C271—22no—
101C2×D4×C271—21no—
102C2×D4×C271—21no—
103C2×D4×C271—21no—
104C2×D4×C271—41no—
105C2×D4×C271D411yescharacteristic
106C2×D4×C271C2311yescharacteristic
107C2×D4×C271C2311yescharacteristic
108C22×C4×C271—21no—
109C22×C4×C271—21no—
110C22×C4×C271C2311yescharacteristic
111C22×C4×C271C2311yescharacteristic
112C4⋊C4×C271—21no—
113C4⋊C4×C271D411yescharacteristic
114C4⋊C4×C271C2311yescharacteristic
115C22⋊C4×C271—21no—
116C22⋊C4×C271—21no—
117C22⋊C4×C271—21no—
118C22⋊C4×C271—21no—
119C22⋊C4×C271—21no—
120C22⋊C4×C271D411yescharacteristic
121C22⋊C4×C271D411yescharacteristic
122C42×C271C2311yescharacteristic
123C22×D271—21no—
124C22×D271—21no—
125Dic542—21no—
126Dic542C2×D412yes—
127D1084—21no—
128D1084—21no—
129D1084—21no—
130D1084—21no—
131D1084—42no—
132D1084—81no—
133C22⋊D271—42no—
134C22⋊D271—42no—
135C22⋊D271—81no—
136C22⋊D271—81no—
137C2×Dic271—21no—
138C2×Dic271—21no—
139C4×D271—42no—
140C4×D271—42no—
141C271⋊C8—21no—
142C271⋊C8—21no—
143C271⋊C8—21no—
144C271⋊C8C2×D412yes—
145C23×C271—21no—
146C23×C271—21no—
147C23×C271—42no—
148C23×C271—42no—
149C23×C271—81no—
150C23×C271C2×D411yescharacteristic
151C23×C271C2×D411yescharacteristic
152Q8×C271—21no—
153Q8×C271—21no—
154D4×C271—21no—
155D4×C271—21no—
156D4×C271—21no—
157D4×C271—41no—
158D4×C271—42no—
159D4×C271—42no—
160D4×C271—81no—
161D4×C271C2×D412yes—
162C2×C4×C271—21no—
163C2×C4×C271—21no—
164C2×C4×C271—21no—
165C2×C4×C271—21no—
166C2×C4×C271—21no—
167C2×C4×C271—21no—
168C2×C4×C271—42no—
169C2×C4×C271—81no—
170C2×C4×C271C2×D411yescharacteristic
171C2×C4×C271C2×D411yescharacteristic
172C2×C4×C271C2×D411yescharacteristic
173C2×C4×C271C2×D411yescharacteristic
174C2×C4×C271C2411yescharacteristic
175C22×C271—21no—
176C22×C271—21no—
177C22×C271—22no—
178C22×C271—21no—
179C22×C271—21no—
180C22×C271—21no—
181C22×C271—21no—
182C22×C271—41no—
183C22×C271—41no—
184C22×C271—41no—
185C22×C271—41no—
186C22×C271—41no—
187C22×C271—84no—
188C22×C271C22×D411yescharacteristic
189C2×D271—41no—
190C2×D271—41no—
191C2×D271—41no—
192C2×D271—41no—
193C2×D271—41no—
194C2×D271—41no—
195C4×C271—21no—
196C4×C271—21no—
197C4×C271—21no—
198C4×C271—21no—
199C4×C271—41no—
200C4×C271—41no—
201C4×C271C22×D411yescharacteristic
202C4×C271H(5,𝔽2)11yescharacteristic
203Dic271—22no—
204Dic271—41no—
205C2×C271—41no—
206C2×C271—41no—
207C2×C271—42no—
208C2×C271—81no—
209C2×C271D4211yescharacteristic
210C2×C271#57511yescharacteristic
211C2×C271#57611yescharacteristic
212D271—81no—
213D271—81no—
214C271#326811yescharacteristic, complemented, Sylow
215#3268—2711nomaximal, Sylow
216C2×AGL(1,ℤ/8)—2711no—
217C2×#135—2711no—
218#550—2711no—
219#546D27111yescharacteristic, complemented
220#496—2711no—
221#468—2711no—
222#466—2711no—
223#464—2711no—
224#459—2711no—
225#452—2711no—
226#449—2711no—
227#449—2711no—
228#447—2711no—
229#440—2711no—
230#435—2711no—
231C2×#41—2711no—
232C2×#41—2711no—
233C2×#41C2×D27111yescharacteristic
234C22×D4C2×D27111yescharacteristic
235AGL(1,ℤ/8)—5424no—
236#135—5424no—
237C2×Q16—2711no—
238C2×SD16—2711no—
239C2×SD16—2711no—
240C2×SD16—2711no—
241C2×SD16—2711no—
242C2×D8—2711no—
243C2×D8—2711no—
244C2×D8—2711no—
245C2×D8—2711no—
246C2×M16—2711no—
247C22×C8—2711no—
248#127—2711no—
249#127—5421no—
250#124C2×D27111yescharacteristic
251#123—21no—
252C22⋊Q8C2×D27111yescharacteristic
253#121—21no—
254#121—2711no—
255#121—2711no—
256#121—5421no—
257#121—5421no—
258#121C2×D27111yescharacteristic
259#120—21no—
260C4×Q8—2711no—
261C4×D4—5421no—
262C4×D4C2×D27111yescharacteristic
263C4×D4C2×D27111yescharacteristic
264C2×C22⋊C4—21no—
265#107—2711no—
266C4⋊C8—2711no—
267#103—2711no—
268#103—2711no—
269#102—2711no—
270#102—2711no—
271#102—2711no—
272#102—2711no—
273C22⋊C8—2711no—
274C22⋊C8—2711no—
275#97—2711no—
276C24—21no—
277#41—22no—
278#41—41no—
279#41—5422no—
280#41—5422no—
281#41—10841no—
282#41—10841no—
283C2×Q8—2711no—
284C2×Q8C22×D27111yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—2711no—
291C2×D4—2711no—
292C2×D4—5421no—
293C2×D4—5421no—
294C2×D4—5421no—
295C2×D4—5421no—
296C2×D4—5421no—
297C2×D4—5421no—
298C2×D4C22⋊D27111yescharacteristic
299C2×D4C22×D27111yescharacteristic
300C2×D4C22×D27111yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—5421no—
304C22×C4—5421no—
305C22×C4C22×D27111yescharacteristic
306C22×C4C22×D27111yescharacteristic
307Q16—5422no—
308SD16—2714no—
309SD16—5422no—
310SD16—5422no—
311SD16—10841no—
312D8—5422no—
313D8—5422no—
314D8—5422no—
315D8—10841no—
316M16—5422no—
317C2×C8—2711no—
318C2×C8—2711no—
319C2×C8—2711no—
320C2×C8—2711no—
321C2×C8—5422no—
322C4⋊C4—21no—
323C4⋊C4—2711no—
324C4⋊C4—5421no—
325C4⋊C4C22⋊D27111yescharacteristic
326C4⋊C4C22×D27111yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—21no—
330C22⋊C4—21no—
331C22⋊C4—21no—
332C22⋊C4—5421no—
333C22⋊C4—5421no—
334C22⋊C4C22⋊D27111yescharacteristic
335C22⋊C4C22⋊D27111yescharacteristic
336C42—5421no—
337C42C22×D27111yescharacteristic
338C23—21no—
339C23—21no—
340C23—42no—
341C23—42no—
342C23—81no—
343C23—5421no—
344C23—5421no—
345C23C2×C22⋊D27111yescharacteristic
346C23C2×C22⋊D27111yescharacteristic
347Q8—21no—
348Q8—21no—
349Q8—2712no—
350Q8—5421no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—41no—
355D4—42no—
356D4—42no—
357D4—81no—
358D4—5421no—
359D4—5421no—
360D4—5421no—
361D4—5421no—
362D4—10842no—
363D4—10842no—
364D4—10842no—
365D4—21681no—
366D4—21681no—
367D4—21681no—
368D4D4×D27112yes—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—81no—
377C2×C4—5421no—
378C2×C4—5421no—
379C2×C4—10842no—
380C2×C4—10842no—
381C2×C4C2×C22⋊D27111yescharacteristic
382C2×C4C2×C22⋊D27111yescharacteristic
383C2×C4C2×C22⋊D27111yescharacteristic
384C2×C4C2×C22⋊D27111yescharacteristic
385C2×C4C23×D27111yescharacteristic
386C8—2712no—
387C8—5421no—
388C8—5421no—
389C8—5421no—
390C22—21no—
391C22—21no—
392C22—22no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—84no—
403C22—10841no—
404C22—10841no—
405C22—10841no—
406C22—10841no—
407C22—10841no—
408C22—10841no—
409C22C22×C22⋊D27111yescharacteristic
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—41no—
415C4—41no—
416C4—5422no—
417C4—10841no—
418C4C2×D4×D27111yescharacteristic
419C4#2257620511yescharacteristic
420C2—41no—
421C2—41no—
422C2—42no—
423C2—81no—
424C2—21681no—
425C2—21681no—
426C2D4×C22⋊D27111yescharacteristic
427C2#3177469511yescharacteristic
428C2#3177470011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#546×C271C2
socleC22×C271C22×D4
hypercenter#546D271
hypocenterC271#3268
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#22576186C2semidirect
2. C2×#22576180C2semidirect
3. #31774568C2semidirect
4. #31774287C2semidirect
5. #31774145C2semidirect
6. #31774131C2semidirect
7. #31774116C2semidirect
8. #31774092C2semidirect
9. #31774057C2semidirect
10. #31774042C2semidirect
11. #31774040C2semidirect
12. #31774031C2semidirect
13. #31773997C2semidirect
14. #31773969C2semidirect
15. #546×C271C2semidirect
17. C2×#16864654C22semidirect
20. C2×#16864644C22semidirect
23. C2×#16864641C22semidirect
24. C2×#16864641C22semidirect
30. C22×C271⋊C8C22semidirect
31. #22576157C22semidirect
34. #22576134C22semidirect
37. C4×Dic542C22semidirect
39. #22576093C22semidirect
40. C4⋊(C271⋊C8)C22semidirect
41. #22576084C22semidirect
42. #22576083C22semidirect
45. #22576080C22semidirect
46. #22576080C22semidirect
48. C22⋊(C271⋊C8)C22semidirect
49. #22576071C22semidirect
78. #16864642D4semidirect
214. C271#3268semidirect
219. #546D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761862
C2×#225761802
#317745682
#317742872
#317741452
#317741312
#317741162
#317740922
#317740572
#317740422
#317740402
#317740312
#317739972
#317739692
#546×C2712
#3268271