Finite Group Databasebeta

← #46847878

Subgroups — #46847878

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×#22576189C211yesmaximal, characteristic, complemented
2C22×#16864642C211yesmaximal, characteristic, complemented
3#31774560C211yesmaximal, characteristic, complemented
4#31774541C211yesmaximal, characteristic
5#31774288C211yesmaximal, characteristic, complemented
6#31774142C211yesmaximal, characteristic, complemented
7#31774136C211yesmaximal, characteristic, complemented
8#31774118C211yesmaximal, characteristic, complemented
9#31774100C211yesmaximal, characteristic, complemented
10#31774060C211yesmaximal, characteristic, complemented
11#31774055C211yesmaximal, characteristic, complemented
12#31774034C211yesmaximal, characteristic, complemented
13#31773994C211yesmaximal, characteristic, complemented
14#31773965C211yesmaximal, characteristic, complemented
15#450×C271C211yesmaximal, characteristic, complemented
16C2×#16864653C2211yescharacteristic
17C22×Dic542C2211yescharacteristic
18C2×Q8×D271C2211yescharacteristic, complemented
19#22576189—24no—
20C2×Dic1084C2211yescharacteristic, complemented
21C2×Dic1084C2211yescharacteristic, complemented
22C2×#16864644C2211yescharacteristic, complemented
23C2×#16864642—24no—
24C2×#16864642C2211yescharacteristic
25C2×#16864642C2211yescharacteristic
26C2×#16864642C2211yescharacteristic
27C2×#16864642C2211yescharacteristic
28C2×#16864637C2211yescharacteristic, complemented
29C22×C271⋊C8C2211yescharacteristic
30#22576159—21no—
31#22576158C2211yescharacteristic, complemented
32#22576147C2211yescharacteristic
33#22576141—21no—
34#22576140C2211yescharacteristic, complemented
35#22576139C2211yescharacteristic
36#22576138—21no—
37#22576137—21no—
38#22576132—21no—
39#22576131C2211yescharacteristic
40#22576129—21no—
41#22576125C2211yescharacteristic, complemented
42D4×Dic271C2211yescharacteristic
43C4×C22⋊D271C2211yescharacteristic
44(C22⋊D271)⋊C4—21no—
45C2×C22⋊Dic271—21no—
46#22576093C2211yescharacteristic, complemented
47Dic271⋊C8C2211yescharacteristic, complemented
48#22576085C2211yescharacteristic, complemented
49#22576085C2211yescharacteristic, complemented
50#22576084C2211yescharacteristic, complemented
51#22576083C2211yescharacteristic, complemented
52(C2×D271)⋊C8C2211yescharacteristic, complemented
53#22576070C2211yescharacteristic, complemented
54C22×D4×C271C2211yescharacteristic
55C2×SD16×C271C2211yescharacteristic
56C2×SD16×C271C2211yescharacteristic
57C22⋊Q8×C271C2211yescharacteristic
58#102×C271C2211yescharacteristic
59#102×C271C2211yescharacteristic
60C22⋊C8×C271C2211yescharacteristic
61#16864653—22no—
62#16864653—41no—
63C2×Dic542—21no—
64C2×Dic542—22no—
65C2×Dic542—21no—
66C2×Dic542—41no—
67C2×Dic542C2311yescharacteristic
68C2×Dic542C2311yescharacteristic
69C2×Dic542C2311yescharacteristic
70Q8×D271—22no—
71Q8×D271—41no—
72C2×C22⋊D271—21no—
73C2×C22⋊D271—21no—
74C2×C22⋊D271D411yescharacteristic
75C22×Dic271—21no—
76C22×Dic271—21no—
77C22×Dic271—21no—
78C2×C4×D271—21no—
79C2×C4×D271C2311yescharacteristic
80Dic1084—22no—
81Dic1084—41no—
82#16864644—22no—
83#16864642—22no—
84#16864642—22no—
85#16864642—22no—
86#16864642D414yescomplemented
87#16864637—22no—
88C2×C271⋊C8—22no—
89C2×C271⋊C8C2311yescharacteristic
90C2×C271⋊C8C2311yescharacteristic
91C4⋊Dic271—21no—
92C4⋊Dic271C2311yescharacteristic
93Dic271⋊C4—21no—
94Dic271⋊C4—21no—
95Dic271⋊C4—21no—
96Dic271⋊C4—21no—
97Dic271⋊C4D411yescharacteristic
98C22⋊Dic271—21no—
99C22⋊Dic271—21no—
100C22⋊Dic271—21no—
101C22⋊Dic271—21no—
102C22⋊Dic271—21no—
103C22⋊Dic271D411yescharacteristic
104(C2×D271)⋊C4—21no—
105(C2×D271)⋊C4D411yescharacteristic
106C4×Dic271—21no—
107C4×Dic271C2311yescharacteristic
108C24×C271—21no—
109C2×Q8×C271C2311yescharacteristic
110C2×D4×C271—22no—
111C2×D4×C271—21no—
112C2×D4×C271—41no—
113C2×D4×C271C2311yescharacteristic
114C2×D4×C271C2311yescharacteristic
115C22×C4×C271C2311yescharacteristic
116SD16×C271—22no—
117SD16×C271—41no—
118C2×C8×C271C2311yescharacteristic
119C2×C8×C271C2311yescharacteristic
120C4⋊C4×C271—21no—
121C4⋊C4×C271C2311yescharacteristic
122C22⋊C4×C271—21no—
123C22×D271C2×D411yescharacteristic
124Dic542—21no—
125Dic542—21no—
126Dic542—21no—
127Dic542—21no—
128Dic542—21no—
129Dic542—42no—
130Dic542—42no—
131Dic542C2×D412yes—
132C22⋊D271—22no—
133C22⋊D271—42no—
134C22⋊D271—81no—
135C2×Dic271—21no—
136C2×Dic271—21no—
137C2×Dic271—21no—
138C2×Dic271—21no—
139C2×Dic271—42no—
140C2×Dic271—42no—
141C2×Dic271—81no—
142C2×Dic271C2×D411yescharacteristic
143C2×Dic271C2×D411yescharacteristic
144C2×Dic271C2×D411yescharacteristic
145C4×D271—21no—
146C4×D271—21no—
147C4×D271—42no—
148C271⋊C8—21no—
149C271⋊C8C2×D412yes—
150C23×C271—21no—
151C23×C271—21no—
152C23×C271—42no—
153C23×C271—42no—
154C23×C271—81no—
155C23×C271C2×D411yescharacteristic
156Q8×C271—21no—
157Q8×C271—21no—
158D4×C271—21no—
159D4×C271—21no—
160D4×C271—21no—
161D4×C271—42no—
162D4×C271C2×D412yes—
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×C271—21no—
171C22×C271—21no—
172C22×C271—21no—
173C22×C271—22no—
174C22×C271—41no—
175C22×C271—41no—
176C22×C271—41no—
177C22×C271—41no—
178C22×C271—41no—
179C22×C271—84no—
180C22×C271#13712yescomplemented
181C22×C271C22×D411yescharacteristic
182C2×D271—21no—
183C2×D271—21no—
184C2×D271—21no—
185C4×C271—21no—
186C4×C271—41no—
187C4×C271—41no—
188C4×C271C22×D411yescharacteristic
189C4×C271H(5,𝔽2)11yescharacteristic
190Dic271—21no—
191Dic271—22no—
192Dic271—21no—
193Dic271—21no—
194Dic271—41no—
195Dic271—41no—
196Dic271—41no—
197C2×C271—22no—
198C2×C271—42no—
199C2×C271—81no—
200C2×C271D4211yescharacteristic
201C2×C271C2×#13711yescharacteristic
202C2×C271#57711yescharacteristic
203D271—41no—
204C271#326611yescharacteristic, complemented, Sylow
205#3266—2711nomaximal, Sylow
206C2×#137—2711no—
207C22×SD16—2711no—
208D4×Q8—2711no—
209#546—2711no—
210#497—2711no—
211#468—2711no—
212#467—2711no—
213#464—2711no—
214#461—2711no—
215#452—2711no—
216#451—2711no—
217#450D27111yescharacteristic, complemented
218#448—2711no—
219#440—2711no—
220#435—2711no—
221C2×#41—2711no—
222C22×Q8—2711no—
223C22×Q8—2711no—
224C22×D4C2×D27111yescharacteristic
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—5424no—
234C2×SD16C2×D27111yescharacteristic
235C2×SD16C2×D27111yescharacteristic
236C2×M16—2711no—
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
247#121—2711no—
248#121—5421no—
249#120—5421no—
250C4×Q8—2711no—
251C4×D4—2711no—
252C4×D4—2711no—
253C4×D4—5421no—
254C2×C22⋊C4—5421no—
255#107—2711no—
256C4⋊C8—2711no—
257#103—2711no—
258#103—2711no—
259#103—2711no—
260#103—2711no—
261#102C2×D27111yescharacteristic
262#102C2×D27111yescharacteristic
263C22⋊C8—2711no—
264C22⋊C8C2×D27111yescharacteristic
265#97—2711no—
266C24—21no—
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—22no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—2711no—
283C2×D4—5421no—
284C2×D4—5421no—
285C2×D4C22×D27111yescharacteristic
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—41no—
298SD16—2714no—
299SD16—5422no—
300SD16—5422no—
301SD16—5422no—
302M16—5422no—
303C2×C8—2711no—
304C2×C8—2711no—
305C2×C8—5422no—
306C2×C8C22×D27111yescharacteristic
307C2×C8C22×D27111yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—2711no—
310C4⋊C4—2711no—
311C4⋊C4—5421no—
312C4⋊C4—5421no—
313C4⋊C4—5421no—
314C4⋊C4—5421no—
315C4⋊C4—5421no—
316C4⋊C4C22×D27111yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—2711no—
319C22⋊C4—2711no—
320C22⋊C4—5421no—
321C22⋊C4—5421no—
322C22⋊C4—5421no—
323C22⋊C4—5421no—
324C22⋊C4—5421no—
325C22⋊C4—5421no—
326C42—2711no—
327C42—5421no—
328C23—21no—
329C23—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23—2711no—
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—21no—
348D4—42no—
349D4—5422no—
350D4—10842no—
351D4—21681no—
352D4D4×D27112yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—2711no—
358C2×C4—2711no—
359C2×C4—2711no—
360C2×C4—5421no—
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—21no—
374C8—2712no—
375C8—5421no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—84no—
385C22—5421no—
386C22—5421no—
387C22—5421no—
388C22#2257618912yescomplemented
389C22C2×D4×D27111yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—5421no—
394C4—5422no—
395C4—5421no—
396C4—5421no—
397C4—10841no—
398C4—10841no—
399C4—10841no—
400C4C2×D4×D27111yescharacteristic
401C4#2257620511yescharacteristic
402C2—22no—
403C2—42no—
404C2—81no—
405C2—10841no—
406C2#3177452911yescharacteristic
407C2C2×#2257618911yescharacteristic
408C2#3177470711yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#450×C271C2
socleC22×C271C22×D4
prime radical#31774541C2
hypercenter#450D271
hypocenterC271#3266
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#22576189C2semidirect
2. C22×#16864642C2semidirect
3. #31774560C2semidirect
5. #31774288C2semidirect
6. #31774142C2semidirect
7. #31774136C2semidirect
8. #31774118C2semidirect
9. #31774100C2semidirect
10. #31774060C2semidirect
11. #31774055C2semidirect
12. #31774034C2semidirect
13. #31773994C2semidirect
14. #31773965C2semidirect
15. #450×C271C2semidirect
18. C2×Q8×D271C22semidirect
20. C2×Dic1084C22semidirect
21. C2×Dic1084C22semidirect
22. C2×#16864644C22semidirect
28. C2×#16864637C22semidirect
31. #22576158C22semidirect
34. #22576140C22semidirect
41. #22576125C22semidirect
46. #22576093C22semidirect
47. Dic271⋊C8C22semidirect
48. #22576085C22semidirect
49. #22576085C22semidirect
50. #22576084C22semidirect
51. #22576083C22semidirect
52. (C2×D271)⋊C8C22semidirect
53. #22576070C22semidirect
86. #16864642D4semidirect
180. C22×C271#137semidirect
204. C271#3266semidirect
217. #450D271semidirect
388. C22#22576189semidirect

Maximal subgroups

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

subgroupindex
C2×#225761892
C22×#168646422
#317745602
#317745412
#317742882
#317741422
#317741362
#317741182
#317741002
#317740602
#317740552
#317740342
#317739942
#317739652
#450×C2712
#3266271