Finite Group Databasebeta

← #46845744

Subgroups — #46845744

Order 34688.

704 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 145 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774733C211yesmaximal, characteristic
2C2×#22576193C211yesmaximal, characteristic, complemented
3C2×#22576190C211yesmaximal, characteristic, complemented
4#31774473C211yesmaximal, characteristic, complemented
5#31774081C214yesmaximal, complemented
6#31774076C214yesmaximal, complemented
7#31773915C211yesmaximal, characteristic, complemented
8#31773879C211yesmaximal, characteristic, complemented
9#421×C271C211yesmaximal, characteristic, complemented
10#22576208—21no—
11#22576207—21no—
12#22576207C2212yes—
13#22576206—21no—
14#22576206C2212yes—
15#22576205—22no—
16#22576205—21no—
17C2×#16864654—22no—
18C2×#16864654C2211yescharacteristic
19C2×#16864654C2211yescharacteristic
20#41×D271—22no—
21#41×D271—22no—
22C22×Dic542C2211yescharacteristic, complemented
23#22576193—22no—
24#22576193C2214yescomplemented
25#22576190—22no—
26#22576190C2214yescomplemented
27C2×Dic1084—21no—
28C2×#16864644—21no—
29C2×#16864643—21no—
30C2×#16864642—21no—
31C2×#16864638C2211yescharacteristic, complemented
32#22576160C2212yescomplemented
33#22576148C2212yescomplemented
34#22576144—22no—
35#22576140—22no—
36#22576088C2214yescomplemented
37#22576078C2212yescomplemented
38#22576076C2212yescomplemented
39C2×#41×C271C2211yescharacteristic
40C2×M16×C271C2211yescharacteristic, complemented
41#117×C271C2211yescharacteristic, complemented
42#104×C271C2214yescomplemented
43#16864655—22no—
44#16864655—22no—
45#16864654—24no—
46#16864654—24no—
47#16864654—22no—
48#16864654—22no—
49#16864654—21no—
50#16864654—21no—
51#16864654—21no—
52#16864654—21no—
53#16864654—21no—
54#16864654—21no—
55#16864654D412yes—
56#16864654C2312yes—
57#16864653—24no—
58#16864653—22no—
59#16864653—24no—
60#16864653—22no—
61C2×Dic542—22no—
62C2×Dic542—21no—
63C2×Dic542—21no—
64C2×Dic542—21no—
65C2×Dic542—21no—
66C2×Dic542—21no—
67C2×Dic542D411yescharacteristic
68C2×Dic542C2312yes—
69C2×Dic542C2311yescharacteristic
70Q8×D271—22no—
71Q8×D271—22no—
72C2×D1084—22no—
73C2×D1084D411yescharacteristic
74C2×D1084C2311yescharacteristic
75C2×C22⋊D271—24no—
76C2×C22⋊D271—21no—
77C2×C22⋊D271D412yes—
78D4×D271—24no—
79D4×D271—22no—
80D4×D271—24no—
81D4×D271—22no—
82C22×Dic271—21no—
83C2×C4×D271—24no—
84C2×C4×D271—21no—
85C2×C4×D271D412yes—
86Dic1084—24no—
87#16864644—24no—
88#16864643—24no—
89#16864642—24no—
90#16864638—21no—
91#16864638C2312yes—
92C2×C271⋊C8—21no—
93C4⋊Dic271—24no—
94(C2×D271)⋊C4—24no—
95#41×C271—21no—
96#41×C271—21no—
97#41×C271D412yes—
98#41×C271C2312yes—
99C2×Q8×C271D411yescharacteristic
100C2×D4×C271—21no—
101C2×D4×C271D411yescharacteristic
102C22×C4×C271—21no—
103C22×C4×C271C2311yescharacteristic
104M16×C271—21no—
105M16×C271C2312yes—
106C2×C8×C271—21no—
107C4⋊C4×C271—21no—
108C22⋊C4×C271—21no—
109C42×C271C2312yes—
110C22×D271—24no—
111C22×D271—21no—
112C22×D271C2×D412yes—
113Dic542—24no—
114Dic542—21no—
115Dic542—21no—
116Dic542—21no—
117Dic542—22no—
118Dic542—22no—
119Dic542—41no—
120Dic542—41no—
121Dic542C2×D412yes—
122D1084—24no—
123D1084—21no—
124D1084—21no—
125D1084—21no—
126D1084C2×D412yes—
127C22⋊D271—24no—
128C22⋊D271—24no—
129C22⋊D271—22no—
130C22⋊D271—22no—
131C22⋊D271—22no—
132C22⋊D271—22no—
133C2×Dic271—24no—
134C2×Dic271—21no—
135C2×Dic271—22no—
136C2×Dic271—41no—
137C2×Dic271—41no—
138C2×Dic271C2×D412yes—
139C4×D271—24no—
140C4×D271—24no—
141C4×D271—22no—
142C4×D271—22no—
143C4×D271—22no—
144C4×D271—22no—
145C271⋊C8—22no—
146C23×C271—21no—
147C23×C271C2×D411yescharacteristic
148Q8×C271—21no—
149Q8×C271C2×D412yes—
150D4×C271—22no—
151D4×C271—22no—
152D4×C271—21no—
153D4×C271C2×D412yes—
154C2×C4×C271—21no—
155C2×C4×C271—22no—
156C2×C4×C271—22no—
157C2×C4×C271—22no—
158C2×C4×C271C2×D411yescharacteristic
159C2×C4×C271C2×D411yescharacteristic
160C2×C4×C271C2×D411yescharacteristic
161C2×C4×C271C2×D411yescharacteristic
162C2×C4×C271C2×D411yescharacteristic
163C2×C4×C271C2411yescharacteristic
164C8×C271—22no—
165C22×C271—22no—
166C22×C271—41no—
167C22×C271—81no—
168C22×C271#12011yescharacteristic
169C22×C271#12011yescharacteristic
170C22×C271C22×D411yescharacteristic
171C2×D271—22no—
172C2×D271—22no—
173C2×D271—42no—
174C2×D271—81no—
175C2×D271—84no—
176C4×C271—22no—
177C4×C271—42no—
178C4×C271#12011yescharacteristic
179C4×C271#12011yescharacteristic
180C4×C271C22×D411yescharacteristic
181C4×C271C22×D411yescharacteristic
182Dic271—22no—
183Dic271—22no—
184Dic271—42no—
185C2×C271—21no—
186C2×C271—21no—
187C2×C271—21no—
188C2×C271—42no—
189C2×C271C2×#12011yescharacteristic
190D271—42no—
191D271—42no—
192C271#302211yescharacteristic, complemented, Sylow
193#3022—2711nomaximal, Sylow
194#585—2711no—
195C2×#137—2711no—
196C2×#137—2711no—
197#536—2711no—
198#456—2714no—
199#455—2714no—
200#421—2711no—
201#421D27111yescharacteristic, complemented
202#413—2711no—
2032−1+4—2712no—
2042−1+4—5421no—
2052−1+4—5421no—
206H(5,𝔽2)—2712no—
207H(5,𝔽2)—5422no—
208H(5,𝔽2)—5421no—
209H(5,𝔽2)—5421no—
210C2×#41—2711no—
211C2×#41—2711no—
212C2×#41—5422no—
213C2×#41—5422no—
214C2×#41—5422no—
215C2×#41C2×D27111yescharacteristic
216C22×Q8—2711no—
217#137—2714no—
218#137—2714no—
219#137—5422no—
220#137—5422no—
221C2×Q16—5421no—
222C2×Q16—5421no—
223C2×SD16—5421no—
224C2×SD16—5421no—
225C2×M16—2711no—
226C2×M16C2×D27111yescharacteristic, complemented
227C4⋊Q8—2712no—
228#124—2712no—
229#123—5422no—
230C22⋊Q8—5422no—
231#117C2×D27111yescharacteristic, complemented
232#104—2714no—
233#104C2×D27114yescomplemented
234#101—2712no—
235#100—2712no—
236#41—21no—
237#41—21no—
238#41—2712no—
239#41—2712no—
240#41—5422no—
241#41—5422no—
242#41—5424no—
243#41—5424no—
244#41—5424no—
245#41—5424no—
246#41—5422no—
247#41—5422no—
248#41—5422no—
249#41—5422no—
250#41—5421no—
251#41—5421no—
252#41—5421no—
253#41—5421no—
254#41—5421no—
255#41—5421no—
256#41D108412yes—
257#41C22×D27112yes—
258C2×Q8—2711no—
259C2×Q8—2711no—
260C2×Q8—2712no—
261C2×Q8—5422no—
262C2×Q8—5422no—
263C2×Q8—5422no—
264C2×Q8—5421no—
265C2×Q8—5421no—
266C2×Q8—5421no—
267C2×Q8—5421no—
268C2×Q8—5421no—
269C2×Q8D108411yescharacteristic
270C2×D4—21no—
271C2×D4—2712no—
272C2×D4—2711no—
273C2×D4—2711no—
274C2×D4—5422no—
275C2×D4—5424no—
276C2×D4—5424no—
277C2×D4—5422no—
278C2×D4—5424no—
279C2×D4—5421no—
280C2×D4—5422no—
281C2×D4D108411yescharacteristic
282C22×C4—21no—
283C22×C4—2712no—
284C22×C4—5424no—
285C22×C4—5421no—
286C22×C4—5421no—
287C22×C4C22×D27111yescharacteristic
288Q16—5424no—
289Q16—5424no—
290SD16—5424no—
291SD16—5424no—
292M16—21no—
293M16—2712no—
294M16—5421no—
295M16C22×D27112yes—
296C2×C8—21no—
297C2×C8—5421no—
298C4⋊C4—21no—
299C4⋊C4—5424no—
300C22⋊C4—21no—
301C22⋊C4—5424no—
302C42C22×D27112yes—
303C23—21no—
304C23—2712no—
305C23—5424no—
306C23—5421no—
307C23D4×D27111yescharacteristic
308Q8—21no—
309Q8—2712no—
310Q8—5424no—
311Q8—5421no—
312Q8—5421no—
313Q8—5421no—
314Q8—5422no—
315Q8—5422no—
316Q8—10841no—
317Q8—10841no—
318Q8C2×D108412yes—
319D4—22no—
320D4—22no—
321D4—21no—
322D4—2712no—
323D4—5424no—
324D4—5424no—
325D4—5424no—
326D4—5422no—
327D4—5422no—
328D4—5422no—
329D4—5422no—
330D4—5421no—
331D4—5421no—
332D4—5421no—
333D4C2×D108412yes—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—2712no—
339C2×C4—5424no—
340C2×C4—5424no—
341C2×C4—5424no—
342C2×C4—5421no—
343C2×C4—5422no—
344C2×C4—5422no—
345C2×C4—5422no—
346C2×C4—5422no—
347C2×C4—5422no—
348C2×C4—10841no—
349C2×C4—10841no—
350C2×C4D4×D27111yescharacteristic
351C2×C4D4×D27111yescharacteristic
352C2×C4D4×D27111yescharacteristic
353C2×C4C2×D108411yescharacteristic
354C2×C4C2×D108411yescharacteristic
355C2×C4C23×D27111yescharacteristic
356C8—22no—
357C8—5422no—
358C22—22no—
359C22—41no—
360C22—81no—
361C22—5422no—
362C22—5422no—
363C22—10842no—
364C22—21681no—
365C22—21684no—
366C22#2257613011yescharacteristic
367C22#2257613011yescharacteristic
368C22C2×D4×D27111yescharacteristic
369C4—22no—
370C4—42no—
371C4—5422no—
372C4—5422no—
373C4—10842no—
374C4#2257613011yescharacteristic
375C4#2257613011yescharacteristic
376C4C2×D4×D27111yescharacteristic
377C4C22×D108411yescharacteristic
378C2—21no—
379C2—21no—
380C2—21no—
381C2—42no—
382C2—10842no—
383C2—10842no—
384C2C2×#2257613011yescharacteristic
subgroupG/H
centerC2C2×#22576130
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#421×C271C2
socleC2×C271C2×#120
prime radical#31774733C2
hypercenter#421D271
hypocenterC271#3022
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. C2×#22576193C2semidirect
3. C2×#22576190C2semidirect
4. #31774473C2semidirect
5. #31774081C2semidirect
6. #31774076C2semidirect
7. #31773915C2semidirect
8. #31773879C2semidirect
9. #421×C271C2semidirect
22. C22×Dic542C22semidirect
24. #22576193C22semidirect
26. #22576190C22semidirect
31. C2×#16864638C22semidirect
32. #22576160C22semidirect
33. #22576148C22semidirect
36. #22576088C22semidirect
37. #22576078C22semidirect
38. #22576076C22semidirect
40. C2×M16×C271C22semidirect
41. #117×C271C22semidirect
42. #104×C271C22semidirect
192. C271#3022semidirect
201. #421D271semidirect
226. C2×M16C2×D271semidirect
231. #117C2×D271semidirect
233. #104C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317747332
C2×#225761932
C2×#225761902
#317744732
#317740812
#317740762
#317739152
#317738792
#421×C2712
#3022271