Finite Group Databasebeta

← #46847881

Subgroups — #46847881

Order 34688.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576193C211yesmaximal, characteristic, complemented
2C2×SD16×D271C211yesmaximal, characteristic, complemented
3#31774560C211yesmaximal, characteristic, complemented
4#31774544C211yesmaximal, characteristic
5#31774288C211yesmaximal, characteristic, complemented
6#31774139C211yesmaximal, characteristic, complemented
7#31774118C211yesmaximal, characteristic, complemented
8#31774101C211yesmaximal, characteristic, complemented
9#31774057C211yesmaximal, characteristic, complemented
10#31774054C211yesmaximal, characteristic, complemented
11#31774049C211yesmaximal, characteristic, complemented
12#31774035C211yesmaximal, characteristic, complemented
13#31773996C211yesmaximal, characteristic, complemented
14#31773966C211yesmaximal, characteristic, complemented
15#468×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic
17C22×Dic542C2211yescharacteristic, complemented
18C2×Q8×D271C2211yescharacteristic
19C2×D4×D271C2211yescharacteristic
20#22576193—24no—
21C2×Dic1084C2211yescharacteristic, complemented
22C2×Dic1084C2211yescharacteristic, complemented
23C2×#16864644C2211yescharacteristic, complemented
24C2×#16864643C2211yescharacteristic
25C2×#16864643C2211yescharacteristic
26C2×#16864642C2211yescharacteristic
27C2×#16864642C2211yescharacteristic
28C2×#16864641C2211yescharacteristic
29SD16×D271—24no—
30C2×C8×D271C2211yescharacteristic
31#22576159—21no—
32#22576158C2211yescharacteristic, complemented
33#22576147C2211yescharacteristic
34#22576142—21no—
35#22576140C2211yescharacteristic
36#22576140C2211yescharacteristic
37#22576138—21no—
38#22576137—21no—
39#22576131—21no—
40#22576128—21no—
41#22576125C2211yescharacteristic, complemented
42#22576122C2211yescharacteristic
43C4×C22⋊D271C2211yescharacteristic
44(C22⋊D271)⋊C4—21no—
45C22⋊C4×D271—21no—
46Dic271⋊C8C2211yescharacteristic, complemented
47#22576083C2211yescharacteristic, complemented
48#22576083C2211yescharacteristic, complemented
49#22576083C2211yescharacteristic, complemented
50#22576080C2211yescharacteristic
51C22⋊(C271⋊C8)C2211yescharacteristic, complemented
52(C2×D271)⋊C8C2211yescharacteristic
53#22576070C2211yescharacteristic, complemented
54C2×SD16×C271C2211yescharacteristic
55C2×M16×C271C2211yescharacteristic, complemented
56C22⋊Q8×C271C2211yescharacteristic, complemented
57#121×C271C2211yescharacteristic
58#107×C271C2211yescharacteristic, complemented
59#103×C271C2211yescharacteristic, complemented
60#102×C271C2211yescharacteristic
61C23×D271—21no—
62#16864654—22no—
63#16864654—41no—
64C2×Dic542—21no—
65C2×Dic542—22no—
66C2×Dic542—21no—
67C2×Dic542—41no—
68C2×Dic542C2311yescharacteristic
69C2×Dic542C2311yescharacteristic
70C2×Dic542C2311yescharacteristic
71Q8×D271—22no—
72Q8×D271—41no—
73C2×D1084C2311yescharacteristic
74C2×C22⋊D271—21no—
75C2×C22⋊D271—21no—
76C2×C22⋊D271D411yescharacteristic
77D4×D271—22no—
78D4×D271—41no—
79C22×Dic271—21no—
80C2×C4×D271—21no—
81C2×C4×D271—21no—
82C2×C4×D271—21no—
83C2×C4×D271C2311yescharacteristic
84Dic1084—22no—
85Dic1084—22no—
86#16864644—41no—
87#16864643—22no—
88#16864643D414yescomplemented
89#16864642—22no—
90#16864642—41no—
91#16864641—22no—
92C2×C271⋊C8C2311yescharacteristic
93C2×C271⋊C8C2311yescharacteristic
94C8×D271—22no—
95C4⋊Dic271—21no—
96C4⋊Dic271—21no—
97Dic271⋊C4—21no—
98Dic271⋊C4—21no—
99Dic271⋊C4—21no—
100Dic271⋊C4D411yescharacteristic
101C22⋊Dic271—21no—
102C22⋊Dic271—21no—
103C22⋊Dic271D411yescharacteristic
104(C2×D271)⋊C4—21no—
105(C2×D271)⋊C4—21no—
106(C2×D271)⋊C4—21no—
107(C2×D271)⋊C4D411yescharacteristic
108C4×Dic271—21no—
109C4×Dic271C2311yescharacteristic
110C2×Q8×C271C2311yescharacteristic
111C2×D4×C271—21no—
112C2×D4×C271C2311yescharacteristic
113C22×C4×C271C2311yescharacteristic
114SD16×C271—22no—
115M16×C271—22no—
116C2×C8×C271C2311yescharacteristic
117C2×C8×C271C2311yescharacteristic
118C4⋊C4×C271—21no—
119C4⋊C4×C271C2311yescharacteristic
120C4⋊C4×C271C2311yescharacteristic
121C22⋊C4×C271—21no—
122C22⋊C4×C271—21no—
123C22×D271—21no—
124C22×D271—42no—
125C22×D271—42no—
126C22×D271—81no—
127C22×D271C2×D411yescharacteristic
128Dic542—21no—
129Dic542—21no—
130Dic542—21no—
131Dic542—21no—
132Dic542—21no—
133Dic542—42no—
134Dic542—42no—
135Dic542C2×D412yes—
136D1084—21no—
137D1084C2×D412yes—
138C22⋊D271—22no—
139C22⋊D271—42no—
140C22⋊D271—42no—
141C2×Dic271—21no—
142C2×Dic271—21no—
143C2×Dic271—21no—
144C2×Dic271—42no—
145C2×Dic271C2×D411yescharacteristic
146C2×Dic271C2×D411yescharacteristic
147C2×Dic271C2×D411yescharacteristic
148C4×D271—21no—
149C4×D271—21no—
150C4×D271—42no—
151C4×D271—42no—
152C4×D271—81no—
153C271⋊C8—21no—
154C271⋊C8—21no—
155C23×C271—21no—
156C23×C271C2×D411yescharacteristic
157Q8×C271—21no—
158Q8×C271—21no—
159D4×C271—21no—
160D4×C271—21no—
161D4×C271—81no—
162C2×C4×C271—21no—
163C2×C4×C271—21no—
164C2×C4×C271—21no—
165C2×C4×C271—21no—
166C2×C4×C271—21no—
167C2×C4×C271C2×D411yescharacteristic
168C2×C4×C271C2411yescharacteristic
169C8×C271—21no—
170C8×C271C2×D412yes—
171C22×C271—21no—
172C22×C271—21no—
173C22×C271—21no—
174C22×C271—41no—
175C22×C271—41no—
176C22×C271—41no—
177C22×C271C22×D411yescharacteristic
178C2×D271—21no—
179C2×D271—21no—
180C2×D271—22no—
181C2×D271—41no—
182C2×D271—41no—
183C2×D271—84no—
184C2×D271#13712yescomplemented
185C4×C271—21no—
186C4×C271—41no—
187C4×C271—41no—
188C4×C271—41no—
189C4×C271C22×D411yescharacteristic
190C4×C271H(5,𝔽2)11yescharacteristic
191Dic271—21no—
192Dic271—22no—
193Dic271—21no—
194Dic271—21no—
195Dic271—41no—
196Dic271—41no—
197C2×C271—41no—
198C2×C271—81no—
199C2×C271D4211yescharacteristic
200C2×C271C2×#13711yescharacteristic
201C2×C271#57711yescharacteristic
202D271—22no—
203D271—42no—
204C271#326611yescharacteristic, complemented, Sylow
205#3266—2711nomaximal, Sylow
206C2×#137—2711no—
207C22×SD16—2711no—
208D4×Q8—2711no—
209#546—2711no—
210#497—2711no—
211#468D27111yescharacteristic, complemented
212#467—2711no—
213#464—2711no—
214#461—2711no—
215#452—2711no—
216#451—2711no—
217#450—2711no—
218#448—2711no—
219#440—2711no—
220#435—2711no—
221C2×#41—2711no—
222C22×Q8—2711no—
223C22×Q8—2711no—
224C22×D4—2711no—
225#137—5424no—
226C2×Q16—2711no—
227C2×Q16—2711no—
228C2×Q16—2711no—
229C2×SD16—2711no—
230C2×SD16—2711no—
231C2×SD16—2711no—
232C2×SD16—2711no—
233C2×SD16—2711no—
234C2×SD16—5424no—
235C2×SD16C2×D27111yescharacteristic
236C2×M16C2×D27111yescharacteristic, complemented
237C22×C8—2711no—
238C4⋊Q8—2711no—
239C4⋊Q8—5421no—
240#124—2711no—
241#123—5421no—
242C22⋊Q8—2711no—
243C22⋊Q8—2711no—
244C22⋊Q8—5421no—
245C22⋊Q8—5421no—
246C22⋊Q8C2×D27111yescharacteristic, complemented
247#121—5421no—
248#121C2×D27111yescharacteristic
249#120—5421no—
250C4×Q8—2711no—
251C4×D4—2711no—
252C4×D4—2711no—
253C4×D4—5421no—
254C2×C22⋊C4—5421no—
255#107C2×D27111yescharacteristic, complemented
256C4⋊C8—2711no—
257#103—2711no—
258#103—2711no—
259#103—2711no—
260#103C2×D27111yescharacteristic, complemented
261#102—2711no—
262#102C2×D27111yescharacteristic
263C22⋊C8—2711no—
264C22⋊C8—2711no—
265#97—2711no—
266C24—5421no—
267#41—5422no—
268#41—10841no—
269C2×Q8—2711no—
270C2×Q8—2711no—
271C2×Q8—2711no—
272C2×Q8—5421no—
273C2×Q8—5422no—
274C2×Q8—5421no—
275C2×Q8—5422no—
276C2×Q8—10841no—
277C2×Q8—10841no—
278C2×Q8C22×D27111yescharacteristic
279C2×D4—21no—
280C2×D4—2711no—
281C2×D4—2711no—
282C2×D4—5421no—
283C2×D4—5422no—
284C2×D4—5421no—
285C2×D4—10841no—
286C2×D4C22×D27111yescharacteristic
287C22×C4—2711no—
288C22×C4—5421no—
289C22×C4—5421no—
290C22×C4—5421no—
291C22×C4—5421no—
292C22×C4C22×D27111yescharacteristic
293Q16—5422no—
294Q16—5422no—
295Q16—10841no—
296SD16—22no—
297SD16—2714no—
298SD16—5422no—
299SD16—5422no—
300SD16—5422no—
301SD16—10841no—
302M16—22no—
303C2×C8—2711no—
304C2×C8—2711no—
305C2×C8—5422no—
306C2×C8C22×D27111yescharacteristic
307C2×C8C22×D27111yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—2711no—
310C4⋊C4—5421no—
311C4⋊C4—5421no—
312C4⋊C4—5421no—
313C4⋊C4—5421no—
314C4⋊C4—5421no—
315C4⋊C4C22×D27111yescharacteristic
316C4⋊C4C22×D27111yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—2711no—
320C22⋊C4—2711no—
321C22⋊C4—5421no—
322C22⋊C4—5421no—
323C22⋊C4—5421no—
324C22⋊C4—5421no—
325C22⋊C4—5421no—
326C42—2711no—
327C42—5421no—
328C23—21no—
329C23—2711no—
330C23—5421no—
331C23—10842no—
332C23—10842no—
333C23—21681no—
334C23D4×D27111yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—2712no—
338Q8—5421no—
339Q8—5421no—
340Q8—5421no—
341Q8—5421no—
342Q8—5421no—
343Q8—10842no—
344Q8—10842no—
345D4—21no—
346D4—21no—
347D4—81no—
348D4—2712no—
349D4—5422no—
350D4—5421no—
351D4—10842no—
352D4—10842no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—2711no—
359C2×C4—2711no—
360C2×C4—2711no—
361C2×C4—5421no—
362C2×C4—5421no—
363C2×C4—5421no—
364C2×C4—5421no—
365C2×C4—5421no—
366C2×C4—10842no—
367C2×C4—10842no—
368C2×C4—10842no—
369C2×C4—21681no—
370C2×C4D4×D27111yescharacteristic
371C2×C4C23×D27111yescharacteristic
372C8—21no—
373C8—5421no—
374C8—5421no—
375C8D4×D27112yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—2712no—
383C22—5421no—
384C22—5421no—
385C22—5422no—
386C22—10841no—
387C22—10841no—
388C22—21684no—
389C22C2×D4×D27111yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—5421no—
395C4—5422no—
396C4—5421no—
397C4—5421no—
398C4—10841no—
399C4—10841no—
400C4C2×D4×D27111yescharacteristic
401C4#2257620611yescharacteristic
402C2—41no—
403C2—81no—
404C2—5422no—
405C2—10842no—
406C2#3177453111yescharacteristic
407C2#137×D27111yescharacteristic
408C2#3177470711yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#468×C271C2
socleC22×C271C22×D4
prime radical#31774544C2
hypercenter#468D271
hypocenterC271#3266
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#22576193C2semidirect
2. C2×SD16×D271C2semidirect
3. #31774560C2semidirect
5. #31774288C2semidirect
6. #31774139C2semidirect
7. #31774118C2semidirect
8. #31774101C2semidirect
9. #31774057C2semidirect
10. #31774054C2semidirect
11. #31774049C2semidirect
12. #31774035C2semidirect
13. #31773996C2semidirect
14. #31773966C2semidirect
15. #468×C271C2semidirect
17. C22×Dic542C22semidirect
21. C2×Dic1084C22semidirect
22. C2×Dic1084C22semidirect
23. C2×#16864644C22semidirect
32. #22576158C22semidirect
41. #22576125C22semidirect
46. Dic271⋊C8C22semidirect
47. #22576083C22semidirect
48. #22576083C22semidirect
49. #22576083C22semidirect
51. C22⋊(C271⋊C8)C22semidirect
53. #22576070C22semidirect
55. C2×M16×C271C22semidirect
56. C22⋊Q8×C271C22semidirect
58. #107×C271C22semidirect
59. #103×C271C22semidirect
88. #16864643D4semidirect
184. C2×D271#137semidirect
204. C271#3266semidirect
211. #468D271semidirect
236. C2×M16C2×D271semidirect
246. C22⋊Q8C2×D271semidirect
255. #107C2×D271semidirect
260. #103C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761932
C2×SD16×D2712
#317745602
#317745442
#317742882
#317741392
#317741182
#317741012
#317740572
#317740542
#317740492
#317740352
#317739962
#317739662
#468×C2712
#3266271