Finite Group Databasebeta

← #46845737

Subgroups — #46845737

Order 34688.

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

normal subgroups only

This group has 736 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, complemented
2C2×#22576188C211yesmaximal, characteristic, complemented
3C2×#22576185C211yesmaximal, characteristic, complemented
4#31774466C211yesmaximal, characteristic, complemented
5#31774071C214yesmaximal, complemented
6#31774066C214yesmaximal, complemented
7#31773915C211yesmaximal, characteristic, complemented
8#31773879C211yesmaximal, characteristic, complemented
9#421×C271C211yesmaximal, characteristic, complemented
10#22576208—21no—
11#22576207—21no—
12#22576207C2212yescomplemented
13#22576206—21no—
14#22576206C2212yescomplemented
15#22576205—22no—
16#22576205—21no—
17C2×#16864654—22no—
18C2×#16864654C2211yescharacteristic
19C2×#16864654C2211yescharacteristic
20#41×D271—22no—
21#41×D271—22no—
22C22×D1084C2211yescharacteristic, complemented
23#22576188—22no—
24#22576188C2214yescomplemented
25#22576185—22no—
26#22576185C2214yescomplemented
27C2×#16864643—21no—
28C2×#16864641—21no—
29C2×D2168—21no—
30C2×#16864639—21no—
31C2×#16864638C2211yescharacteristic, complemented
32#22576157C2212yescomplemented
33#22576148C2212yescomplemented
34#22576135—22no—
35#22576130—22no—
36#22576088C2214yescomplemented
37#22576078C2212yescomplemented
38#22576076C2212yescomplemented
39C2×#41×C271C2211yescharacteristic
40C2×M16×C271C2211yescharacteristic, complemented
41#117×C271C2211yescharacteristic, complemented
42#104×C271C2214yescomplemented
43C23×D271—21no—
44#16864655—22no—
45#16864655—22no—
46#16864654—24no—
47#16864654—24no—
48#16864654—22no—
49#16864654—22no—
50#16864654—21no—
51#16864654—21no—
52#16864654—21no—
53#16864654—21no—
54#16864654—21no—
55#16864654—21no—
56#16864654D412yescomplemented
57#16864654C2312yes—
58#16864653—24no—
59#16864653—22no—
60#16864653—24no—
61#16864653—22no—
62C2×Dic542—22no—
63C2×Dic542D411yescharacteristic, complemented
64C2×Dic542C2311yescharacteristic
65Q8×D271—22no—
66Q8×D271—22no—
67C2×D1084—22no—
68C2×D1084—24no—
69C2×D1084—21no—
70C2×D1084—21no—
71C2×D1084—21no—
72C2×D1084—21no—
73C2×D1084—21no—
74C2×D1084D411yescharacteristic, complemented
75C2×D1084C2312yes—
76C2×D1084C2311yescharacteristic
77C2×C22⋊D271—24no—
78C2×C22⋊D271—21no—
79C2×C22⋊D271D412yes—
80D4×D271—24no—
81D4×D271—22no—
82D4×D271—24no—
83D4×D271—22no—
84C2×C4×D271—24no—
85C2×C4×D271—21no—
86C2×C4×D271D412yes—
87#16864643—24no—
88#16864641—24no—
89D2168—24no—
90#16864639—24no—
91#16864638—21no—
92#16864638C2312yes—
93C2×C271⋊C8—21no—
94(C2×D271)⋊C4—24no—
95#41×C271—21no—
96#41×C271—21no—
97#41×C271D412yescomplemented
98#41×C271C2312yes—
99C2×Q8×C271D411yescharacteristic, complemented
100C2×D4×C271—21no—
101C2×D4×C271D411yescharacteristic, complemented
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×D271—22no—
113C22×D271—41no—
114C22×D271—41no—
115C22×D271—82no—
116C22×D271C2×D412yes—
117Dic542—24no—
118Dic542—21no—
119Dic542—21no—
120Dic542—21no—
121Dic542C2×D412yes—
122D1084—24no—
123D1084—21no—
124D1084—21no—
125D1084—21no—
126D1084—22no—
127D1084—22no—
128D1084—41no—
129D1084—41no—
130D1084—84no—
131D1084C2×D412yes—
132C22⋊D271—24no—
133C22⋊D271—24no—
134C22⋊D271—22no—
135C22⋊D271—22no—
136C22⋊D271—22no—
137C22⋊D271—22no—
138C2×Dic271—24no—
139C2×Dic271—21no—
140C2×Dic271C2×D412yes—
141C4×D271—24no—
142C4×D271—24no—
143C4×D271—22no—
144C4×D271—22no—
145C4×D271—22no—
146C4×D271—22no—
147C271⋊C8—22no—
148C23×C271—21no—
149C23×C271C2×D411yescharacteristic
150Q8×C271—21no—
151Q8×C271C2×D412yes—
152D4×C271—22no—
153D4×C271—22no—
154D4×C271—21no—
155D4×C271C2×D412yes—
156C2×C4×C271—21no—
157C2×C4×C271—22no—
158C2×C4×C271—22no—
159C2×C4×C271—22no—
160C2×C4×C271C2×D411yescharacteristic
161C2×C4×C271C2×D411yescharacteristic
162C2×C4×C271C2×D411yescharacteristic
163C2×C4×C271C2×D411yescharacteristic
164C2×C4×C271C2×D411yescharacteristic
165C2×C4×C271C2411yescharacteristic
166C8×C271—22no—
167C22×C271—22no—
168C22×C271—41no—
169C22×C271—81no—
170C22×C271#12011yescharacteristic
171C22×C271#12011yescharacteristic
172C22×C271C22×D411yescharacteristic
173C2×D271—22no—
174C2×D271—22no—
175C2×D271—42no—
176C2×D271—44no—
177C2×D271—42no—
178C2×D271—81no—
179C2×D271—84no—
180C2×D271—82no—
181C2×D271—82no—
182C4×C271—22no—
183C4×C271—42no—
184C4×C271#12011yescharacteristic
185C4×C271#12011yescharacteristic
186C4×C271C22×D411yescharacteristic
187C4×C271C22×D411yescharacteristic
188Dic271—22no—
189Dic271—22no—
190C2×C271—21no—
191C2×C271—21no—
192C2×C271—21no—
193C2×C271—42no—
194C2×C271C2×#12011yescharacteristic
195D271—42no—
196D271—42no—
197D271—82no—
198C271#302111yescharacteristic, complemented, Sylow
199#3021—2711nomaximal, Sylow
200#585—2711no—
201C2×AGL(1,ℤ/8)—2711no—
202C2×AGL(1,ℤ/8)—2711no—
203#535—2711no—
204#454—2714no—
205#453—2714no—
206#421—2711no—
207#421D27111yescharacteristic, complemented
208#413—2711no—
2092−1+4—2712no—
2102−1+4—5421no—
2112−1+4—5421no—
212H(5,𝔽2)—2712no—
213H(5,𝔽2)—5422no—
214H(5,𝔽2)—5421no—
215H(5,𝔽2)—5421no—
216C2×#41—2711no—
217C2×#41—2711no—
218C2×#41—5422no—
219C2×#41—5422no—
220C2×#41—5422no—
221C2×#41C2×D27111yescharacteristic
222C22×D4—2711no—
223AGL(1,ℤ/8)—2714no—
224AGL(1,ℤ/8)—2714no—
225AGL(1,ℤ/8)—5422no—
226AGL(1,ℤ/8)—5422no—
227C2×SD16—5421no—
228C2×SD16—5421no—
229C2×D8—5421no—
230C2×D8—5421no—
231C2×M16—2711no—
232C2×M16C2×D27111yescharacteristic, complemented
233#127—2712no—
234#124—2712no—
235#121—5422no—
236#120—5422no—
237#117C2×D27111yescharacteristic, complemented
238#104—2714no—
239#104C2×D27114yescomplemented
240#101—2712no—
241#100—2712no—
242C24—5421no—
243#41—21no—
244#41—21no—
245#41—2712no—
246#41—2712no—
247#41—5422no—
248#41—5422no—
249#41—5424no—
250#41—5424no—
251#41—5424no—
252#41—5424no—
253#41—5422no—
254#41—5422no—
255#41—5422no—
256#41—5422no—
257#41—5421no—
258#41—5421no—
259#41—5421no—
260#41—5421no—
261#41—5421no—
262#41—5421no—
263#41D108412yescomplemented
264#41C22×D27112yes—
265C2×Q8—2711no—
266C2×Q8—2711no—
267C2×Q8—5422no—
268C2×Q8—5422no—
269C2×Q8—5422no—
270C2×Q8D108411yescharacteristic, complemented
271C2×D4—21no—
272C2×D4—2712no—
273C2×D4—2711no—
274C2×D4—2711no—
275C2×D4—2712no—
276C2×D4—5422no—
277C2×D4—5424no—
278C2×D4—5424no—
279C2×D4—5424no—
280C2×D4—5422no—
281C2×D4—5424no—
282C2×D4—5421no—
283C2×D4—5422no—
284C2×D4—5421no—
285C2×D4—5421no—
286C2×D4—5421no—
287C2×D4—5421no—
288C2×D4—5421no—
289C2×D4D108411yescharacteristic, complemented
290C22×C4—21no—
291C22×C4—2712no—
292C22×C4—5424no—
293C22×C4—5421no—
294C22×C4C22×D27111yescharacteristic
295SD16—5424no—
296SD16—5424no—
297D8—5424no—
298D8—5424no—
299M16—21no—
300M16—2712no—
301M16—5421no—
302M16C22×D27112yes—
303C2×C8—21no—
304C2×C8—5421no—
305C4⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—5424no—
308C42C22×D27112yes—
309C23—21no—
310C23—2712no—
311C23—5424no—
312C23—5421no—
313C23—5422no—
314C23—10841no—
315C23—10841no—
316C23—21682no—
317C23D4×D27111yescharacteristic
318Q8—21no—
319Q8—2712no—
320Q8—5424no—
321Q8—5421no—
322Q8—5421no—
323Q8—5421no—
324Q8C2×D108412yes—
325D4—22no—
326D4—22no—
327D4—21no—
328D4—2712no—
329D4—5424no—
330D4—5424no—
331D4—5424no—
332D4—5422no—
333D4—5422no—
334D4—5422no—
335D4—5422no—
336D4—5421no—
337D4—5421no—
338D4—5422no—
339D4—5421no—
340D4—5422no—
341D4—10841no—
342D4—10841no—
343D4—21684no—
344D4C2×D108412yes—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—2712no—
350C2×C4—5424no—
351C2×C4—5424no—
352C2×C4—5424no—
353C2×C4—5421no—
354C2×C4—5422no—
355C2×C4—5422no—
356C2×C4—5422no—
357C2×C4—5422no—
358C2×C4D4×D27111yescharacteristic
359C2×C4D4×D27111yescharacteristic
360C2×C4D4×D27111yescharacteristic
361C2×C4C2×D108411yescharacteristic
362C2×C4C2×D108411yescharacteristic
363C2×C4C23×D27111yescharacteristic
364C8—22no—
365C8—5422no—
366C22—22no—
367C22—41no—
368C22—81no—
369C22—5422no—
370C22—5422no—
371C22—10842no—
372C22—10844no—
373C22—10842no—
374C22—21681no—
375C22—21682no—
376C22—21682no—
377C22—21684no—
378C22#2257613011yescharacteristic
379C22#2257613011yescharacteristic
380C22C2×D4×D27111yescharacteristic
381C4—22no—
382C4—42no—
383C4—5422no—
384C4—5422no—
385C4#2257613011yescharacteristic
386C4#2257613011yescharacteristic
387C4C2×D4×D27111yescharacteristic
388C4C22×D108411yescharacteristic
389C2—21no—
390C2—21no—
391C2—21no—
392C2—42no—
393C2—10842no—
394C2—10842no—
395C2—21682no—
396C2C2×#2257613011yescharacteristic
subgroupG/H
centerC2C2×#22576130
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#421×C271C2
socleC2×C271C2×#120
hypercenter#421D271
hypocenterC271#3021
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
1. #31774733C2semidirect
2. C2×#22576188C2semidirect
3. C2×#22576185C2semidirect
4. #31774466C2semidirect
5. #31774071C2semidirect
6. #31774066C2semidirect
7. #31773915C2semidirect
8. #31773879C2semidirect
9. #421×C271C2semidirect
12. #22576207C22semidirect
14. #22576206C22semidirect
22. C22×D1084C22semidirect
24. #22576188C22semidirect
26. #22576185C22semidirect
31. C2×#16864638C22semidirect
32. #22576157C22semidirect
33. #22576148C22semidirect
36. #22576088C22semidirect
37. #22576078C22semidirect
38. #22576076C22semidirect
40. C2×M16×C271C22semidirect
41. #117×C271C22semidirect
42. #104×C271C22semidirect
56. #16864654D4semidirect
63. C2×Dic542D4semidirect
74. C2×D1084D4semidirect
97. #41×C271D4semidirect
99. C2×Q8×C271D4semidirect
101. C2×D4×C271D4semidirect
198. C271#3021semidirect
207. #421D271semidirect
232. C2×M16C2×D271semidirect
237. #117C2×D271semidirect
239. #104C2×D271semidirect
263. #41D1084semidirect
270. C2×Q8D1084semidirect
289. C2×D4D1084semidirect

Maximal subgroups

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

subgroupindex
#317747332
C2×#225761882
C2×#225761852
#317744662
#317740712
#317740662
#317739152
#317738792
#421×C2712
#3021271