Finite Group Databasebeta

← #46847885

Subgroups — #46847885

Order 34688.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 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#31774552C211yesmaximal, characteristic, complemented
4#31774292C211yesmaximal, characteristic, complemented
5#31774153C211yesmaximal, characteristic, complemented
6#31774124C211yesmaximal, characteristic, complemented
7#31774109C211yesmaximal, characteristic, complemented
8#31774102C211yesmaximal, characteristic, complemented
9#31774059C211yesmaximal, characteristic, complemented
10#31774057C211yesmaximal, characteristic, complemented
11#31774048C211yesmaximal, characteristic, complemented
12#31774040C211yesmaximal, characteristic, complemented
13#31774007C211yesmaximal, 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×#16864644C2211yescharacteristic, complemented
22C2×#16864642C2211yescharacteristic
23C2×#16864642C2211yescharacteristic
24C2×#16864642C2211yescharacteristic
25C2×#16864642C2211yescharacteristic
26C2×#16864641C2211yescharacteristic, complemented
27C2×#16864639C2211yescharacteristic
28C2×#16864639C2211yescharacteristic
29C2×#16864638C2211yescharacteristic
30C22×C271⋊C8C2211yescharacteristic, complemented
31#22576160C2211yescharacteristic, complemented
32#22576142—21no—
33#22576142—21no—
34#22576139C2211yescharacteristic
35#22576139C2211yescharacteristic, complemented
36#22576131—21no—
37C4×D1084C2211yescharacteristic, complemented
38C4⋊C4×D271—21no—
39#22576091C2211yescharacteristic, complemented
40C4⋊(C271⋊C8)C2211yescharacteristic, complemented
41#22576084C2211yescharacteristic, complemented
42#22576083C2211yescharacteristic, complemented
43#22576083C2211yescharacteristic, complemented
44#22576081C2211yescharacteristic
45#22576081C2211yescharacteristic
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×Dic542C2311yescharacteristic
67C2×D1084C2311yescharacteristic
68C2×C22⋊D271—21no—
69C2×C22⋊D271—21no—
70C2×C4×D271—21no—
71C2×C4×D271—21no—
72C2×C4×D271—21no—
73#16864644—22no—
74#16864644—41no—
75#16864642—22no—
76#16864642—22no—
77#16864642—22no—
78#16864642—41no—
79#16864641—22no—
80#16864639—22no—
81#16864639D414yescomplemented
82#16864638—22no—
83C2×C271⋊C8—22no—
84C2×C271⋊C8C2311yescharacteristic
85C2×C271⋊C8C2311yescharacteristic
86C2×C271⋊C8C2311yescharacteristic
87C2×C271⋊C8C2311yescharacteristic
88C4⋊Dic271—21no—
89C4⋊Dic271—21no—
90C4⋊Dic271C2311yescharacteristic
91Dic271⋊C4—21no—
92Dic271⋊C4—21no—
93C22⋊Dic271—21no—
94C22⋊Dic271—21no—
95(C2×D271)⋊C4—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—
124Dic542—21no—
125Dic542—21no—
126Dic542—21no—
127Dic542—21no—
128D1084—21no—
129D1084C2×D412yes—
130C22⋊D271—42no—
131C22⋊D271—42no—
132C2×Dic271—21no—
133C2×Dic271—21no—
134C2×Dic271—21no—
135C4×D271—42no—
136C4×D271—42no—
137C4×D271—42no—
138C271⋊C8—21no—
139C271⋊C8—21no—
140C271⋊C8—21no—
141C271⋊C8C2×D412yes—
142C23×C271—21no—
143C23×C271—21no—
144C23×C271—42no—
145C23×C271—42no—
146C23×C271—81no—
147C23×C271C2×D411yescharacteristic
148C23×C271C2×D411yescharacteristic
149Q8×C271—21no—
150Q8×C271—21no—
151D4×C271—21no—
152D4×C271—21no—
153D4×C271—21no—
154D4×C271—41no—
155D4×C271—42no—
156D4×C271—42no—
157D4×C271—81no—
158D4×C271C2×D412yes—
159C2×C4×C271—21no—
160C2×C4×C271—21no—
161C2×C4×C271—21no—
162C2×C4×C271—21no—
163C2×C4×C271—21no—
164C2×C4×C271—21no—
165C2×C4×C271—42no—
166C2×C4×C271—81no—
167C2×C4×C271C2×D411yescharacteristic
168C2×C4×C271C2×D411yescharacteristic
169C2×C4×C271C2×D411yescharacteristic
170C2×C4×C271C2×D411yescharacteristic
171C2×C4×C271C2411yescharacteristic
172C22×C271—21no—
173C22×C271—21no—
174C22×C271—22no—
175C22×C271—21no—
176C22×C271—21no—
177C22×C271—21no—
178C22×C271—21no—
179C22×C271—41no—
180C22×C271—41no—
181C22×C271—41no—
182C22×C271—41no—
183C22×C271—41no—
184C22×C271—84no—
185C22×C271C22×D411yescharacteristic
186C2×D271—22no—
187C2×D271—41no—
188C2×D271—41no—
189C4×C271—21no—
190C4×C271—21no—
191C4×C271—21no—
192C4×C271—21no—
193C4×C271—41no—
194C4×C271—41no—
195C4×C271C22×D411yescharacteristic
196C4×C271H(5,𝔽2)11yescharacteristic
197Dic271—41no—
198Dic271—41no—
199Dic271—41no—
200C2×C271—41no—
201C2×C271—41no—
202C2×C271—42no—
203C2×C271—81no—
204C2×C271D4211yescharacteristic
205C2×C271#57511yescharacteristic
206C2×C271#57711yescharacteristic
207D271—42no—
208C271#326711yescharacteristic, complemented, Sylow
209#3267—2711nomaximal, Sylow
210C2×AGL(1,ℤ/8)—2711no—
211C2×#135—2711no—
212#547—2711no—
213#546D27111yescharacteristic, complemented
214#497—2711no—
215#470—2711no—
216#465—2711no—
217#463—2711no—
218#461—2711no—
219#452—2711no—
220#452—2711no—
221#450—2711no—
222#449—2711no—
223#442—2711no—
224#435—2711no—
225C2×#41—2711no—
226C2×#41—2711no—
227C2×#41C2×D27111yescharacteristic
228C22×D4C2×D27111yescharacteristic
229AGL(1,ℤ/8)—5424no—
230#135—5424no—
231C2×Q16—2711no—
232C2×Q16—2711no—
233C2×SD16—2711no—
234C2×SD16—2711no—
235C2×SD16—2711no—
236C2×SD16—2711no—
237C2×SD16—2711no—
238C2×D8—2711no—
239C2×D8—2711no—
240C2×M16—2711no—
241C22×C8—2711no—
242C4⋊Q8—2711no—
243#124C2×D27111yescharacteristic
244#123—21no—
245#123—5421no—
246#123—5421no—
247C22⋊Q8—2711no—
248C22⋊Q8—2711no—
249C22⋊Q8C2×D27111yescharacteristic
250#121—21no—
251#121—5421no—
252#121C2×D27111yescharacteristic
253#120—21no—
254C4×D4—2711no—
255C4×D4C2×D27111yescharacteristic
256C4×D4C2×D27111yescharacteristic
257C2×C4⋊C4—5421no—
258C2×C22⋊C4—21no—
259#106—2711no—
260C4⋊C8—2711no—
261#103—2711no—
262#103—2711no—
263#103—2711no—
264#102—2711no—
265#102—2711no—
266#102—2711no—
267C22⋊C8—2711no—
268C22⋊C8—2711no—
269#97—2711no—
270C24—21no—
271#41—22no—
272#41—41no—
273#41—5422no—
274#41—5422no—
275#41—10841no—
276#41—10841no—
277C2×Q8—2711no—
278C2×Q8—2711no—
279C2×Q8C22×D27111yescharacteristic
280C2×D4—22no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—41no—
285C2×D4—2711no—
286C2×D4—5421no—
287C2×D4—5421no—
288C2×D4C22⋊D27111yescharacteristic
289C2×D4C22×D27111yescharacteristic
290C2×D4C22×D27111yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—5421no—
294C22×C4—5421no—
295C22×C4—5421no—
296C22×C4C22×D27111yescharacteristic
297C22×C4C22×D27111yescharacteristic
298Q16—5422no—
299Q16—10841no—
300SD16—5422no—
301SD16—5422no—
302SD16—5422no—
303SD16—5422no—
304SD16—10841no—
305D8—2714no—
306D8—5422no—
307M16—5422no—
308C2×C8—2711no—
309C2×C8—2711no—
310C2×C8—2711no—
311C2×C8—2711no—
312C2×C8—5422no—
313C4⋊C4—21no—
314C4⋊C4—2711no—
315C4⋊C4—5421no—
316C4⋊C4—5421no—
317C4⋊C4—5421no—
318C4⋊C4—5421no—
319C4⋊C4C22⋊D27111yescharacteristic
320C4⋊C4C22×D27111yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—21no—
326C22⋊C4—5421no—
327C22⋊C4—5421no—
328C22⋊C4—5421no—
329C22⋊C4C22⋊D27111yescharacteristic
330C22⋊C4C22⋊D27111yescharacteristic
331C42C22×D27111yescharacteristic
332C23—21no—
333C23—21no—
334C23—42no—
335C23—42no—
336C23—81no—
337C23—5421no—
338C23C2×C22⋊D27111yescharacteristic
339C23C2×C22⋊D27111yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—5421no—
343Q8—5421no—
344Q8—5421no—
345Q8—5421no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—41no—
350D4—42no—
351D4—42no—
352D4—81no—
353D4—2712no—
354D4—5421no—
355D4—10842no—
356D4—10842no—
357D4D4×D27112yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—42no—
365C2×C4—81no—
366C2×C4—5421no—
367C2×C4—5421no—
368C2×C4—5421no—
369C2×C4—10842no—
370C2×C4—10842no—
371C2×C4—10842no—
372C2×C4C2×C22⋊D27111yescharacteristic
373C2×C4C2×C22⋊D27111yescharacteristic
374C2×C4C2×C22⋊D27111yescharacteristic
375C2×C4C2×C22⋊D27111yescharacteristic
376C2×C4C23×D27111yescharacteristic
377C8—2712no—
378C8—5421no—
379C8—5421no—
380C8—5421no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—84no—
394C22—5422no—
395C22—10841no—
396C22—10841no—
397C22C22×C22⋊D27111yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—41no—
403C4—41no—
404C4—10841no—
405C4—10841no—
406C4—10841no—
407C4C2×D4×D27111yescharacteristic
408C4#2257620511yescharacteristic
409C2—41no—
410C2—41no—
411C2—42no—
412C2—81no—
413C2—10842no—
414C2D4×C22⋊D27111yescharacteristic
415C2#3177469511yescharacteristic
416C2#3177470811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#546×C271C2
socleC22×C271C22×D4
hypercenter#546D271
hypocenterC271#3267
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. #31774552C2semidirect
4. #31774292C2semidirect
5. #31774153C2semidirect
6. #31774124C2semidirect
7. #31774109C2semidirect
8. #31774102C2semidirect
9. #31774059C2semidirect
10. #31774057C2semidirect
11. #31774048C2semidirect
12. #31774040C2semidirect
13. #31774007C2semidirect
14. #31773969C2semidirect
15. #546×C271C2semidirect
17. C2×#16864654C22semidirect
20. C2×#16864644C22semidirect
21. C2×#16864644C22semidirect
26. C2×#16864641C22semidirect
30. C22×C271⋊C8C22semidirect
31. #22576160C22semidirect
35. #22576139C22semidirect
37. C4×D1084C22semidirect
39. #22576091C22semidirect
40. C4⋊(C271⋊C8)C22semidirect
41. #22576084C22semidirect
42. #22576083C22semidirect
43. #22576083C22semidirect
46. #22576080C22semidirect
48. C22⋊(C271⋊C8)C22semidirect
49. #22576071C22semidirect
81. #16864639D4semidirect
208. C271#3267semidirect
213. #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
#317745522
#317742922
#317741532
#317741242
#317741092
#317741022
#317740592
#317740572
#317740482
#317740402
#317740072
#317739692
#546×C2712
#3267271