Finite Group Databasebeta

← #456×D271 #46845703

Subgroups — #456×D271 #46845703

Order 34688.

708 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 135 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
12−1+4×D271C211yesmaximal, characteristic
2#137×D271C212yesmaximal, complemented
3C4⋊Q8×D271C211yesmaximal, characteristic, complemented
4#31774082C211yesmaximal, characteristic, complemented
5#31774081C212yesmaximal, complemented
6#31774080C211yesmaximal, characteristic, complemented
7#31774079C211yesmaximal, characteristic, complemented
8#31774078C212yesmaximal, complemented
9#104×D271C212yesmaximal, complemented
10#101×D271C211yesmaximal, characteristic, complemented
11#456×C271C211yesmaximal, characteristic, complemented
12#22576208—22no—
13#22576208C2211yescharacteristic
14#22576207—22no—
15#22576207—22no—
16#22576207C2212yes—
17#41×D271—22no—
18#41×D271—22no—
19#41×D271C2212yes—
20C2×Q8×D271—22no—
21C2×Q8×D271C2211yescharacteristic
22C2×Q8×D271C2211yescharacteristic
23#22576193C2212yescomplemented
24#22576192—22no—
25#22576191C2212yescomplemented
26#22576190C2212yescomplemented
27#22576189—22no—
28Q16×D271—22no—
29SD16×D271—22no—
30M16×D271C2212yes—
31#22576160C2211yescharacteristic, complemented
32#22576159—22no—
33#22576158C2211yescharacteristic, complemented
34#22576158C2211yescharacteristic, complemented
35C4⋊C4×D271—22no—
36C42×D271C2211yescharacteristic
37#22576088C2212yescomplemented
38#22576087C2212yescomplemented
39#22576086C2212yescomplemented
40#22576079C2211yescharacteristic, complemented
41#22576078C2212yescomplemented
422−1+4×C271C2211yescharacteristic
43#137×C271C2212yescomplemented
44C4⋊Q8×C271C2211yescharacteristic, complemented
45#104×C271C2212yescomplemented
46#101×C271C2211yescharacteristic, complemented
47#16864655—22no—
48#16864655—22no—
49#16864655D412yes—
50#16864654—24no—
51#16864654—24no—
52#16864654—22no—
53#16864654—22no—
54#16864654—22no—
55#16864654C2312yes—
56#16864653—24no—
57#16864653—22no—
58#16864653—24no—
59#16864653—22no—
60#16864653—22no—
61#16864653D412yes—
62C2×Dic542—24no—
63C2×Dic542—22no—
64C2×Dic542—21no—
65C2×Dic542D412yes—
66C2×Dic542C2311yescharacteristic
67C2×Dic542C2311yescharacteristic
68Q8×D271—22no—
69Q8×D271—24no—
70Q8×D271—22no—
71Q8×D271—22no—
72Q8×D271—22no—
73Q8×D271—22no—
74Q8×D271—22no—
75Q8×D271—22no—
76Q8×D271D412yes—
77D4×D271—22no—
78D4×D271—22no—
79D4×D271D412yes—
80C2×C4×D271—24no—
81C2×C4×D271—21no—
82C2×C4×D271—22no—
83C2×C4×D271—21no—
84C2×C4×D271D412yes—
85C2×C4×D271C2311yescharacteristic
86Dic1084—22no—
87#16864644—22no—
88#16864644—22no—
89#16864643—22no—
90#16864642—22no—
91#16864641—22no—
92#16864638C2312yes—
93#16864637—22no—
94C8×D271—22no—
95C4⋊Dic271—22no—
96Dic271⋊C4—22no—
97Dic271⋊C4—22no—
98C4×Dic271—21no—
99C4×Dic271C2311yescharacteristic
100#41×C271—22no—
101#41×C271—22no—
102#41×C271C2312yes—
103C2×Q8×C271—22no—
104C2×Q8×C271C2311yescharacteristic
105C2×Q8×C271C2311yescharacteristic
106Q16×C271—22no—
107SD16×C271—22no—
108M16×C271C2312yes—
109C4⋊C4×C271—22no—
110C42×C271C2311yescharacteristic
111C22×D271—22no—
112C22×D271C2×D411yescharacteristic
113Dic542—22no—
114Dic542—24no—
115Dic542—22no—
116Dic542—24no—
117Dic542—22no—
118Dic542—22no—
119Dic542—42no—
120Dic542C2×D412yes—
121D1084—22no—
122D1084—22no—
123D1084C2×D412yes—
124C22⋊D271—24no—
125C22⋊D271—22no—
126C22⋊D271—22no—
127C22⋊D271—22no—
128C2×Dic271—24no—
129C2×Dic271—21no—
130C2×Dic271—22no—
131C2×Dic271—21no—
132C2×Dic271C2×D412yes—
133C2×Dic271C2×D411yescharacteristic
134C4×D271—24no—
135C4×D271—24no—
136C4×D271—24no—
137C4×D271—22no—
138C4×D271—22no—
139C4×D271—22no—
140C4×D271—22no—
141C4×D271—22no—
142C4×D271—41no—
143C4×D271—41no—
144C4×D271C2×D412yes—
145C4×D271C2×D412yes—
146C271⋊C8—22no—
147Q8×C271—22no—
148Q8×C271—22no—
149Q8×C271—22no—
150Q8×C271C2×D412yes—
151D4×C271—22no—
152D4×C271—22no—
153D4×C271C2×D412yes—
154C2×C4×C271—24no—
155C2×C4×C271—21no—
156C2×C4×C271—22no—
157C2×C4×C271—21no—
158C2×C4×C271C2×D412yes—
159C2×C4×C271C2411yescharacteristic
160C8×C271—22no—
161C22×C271—22no—
162C22×C271C22×D411yescharacteristic
163C2×D271—22no—
164C2×D271—22no—
165C2×D271—44no—
166C2×D271#12011yescharacteristic
167C2×D271#12011yescharacteristic
168C4×C271—22no—
169C4×C271—22no—
170C4×C271—22no—
171C4×C271—41no—
172C4×C271C22×D412yes—
173Dic271—22no—
174Dic271—22no—
175Dic271—22no—
176Dic271—41no—
177Dic271#12012yes—
178C2×C271—21no—
179C2×C271—42no—
180C2×C271C2×#12011yescharacteristic
181D271—21no—
182D271—42no—
183D271#45612yesdirect product factor
184C271C2×#45611yescharacteristic, complemented, Sylow
185C2×#456—2711nomaximal, Sylow
186C2×2−1+4—2711no—
187C2×#137—2712no—
188C2×C4⋊Q8—2711no—
189#456—2711no—
190#456—2712no—
191#456—2711no—
192#456—2712no—
193#456—2711no—
194#456D27111yescharacteristic, direct product factor
195C2×#104—2712no—
196C2×#101—2711no—
1972−1+4—2712no—
1982−1+4—2711no—
1992−1+4—5422no—
2002−1+4—5422no—
2012−1+4—5422no—
2022−1+4C2×D27111yescharacteristic
203C2×#41—2712no—
204C2×#41—5422no—
205C2×#41—5422no—
206C22×Q8—2711no—
207C22×Q8—2711no—
208C22×Q8—5422no—
209#137—2712no—
210#137—2712no—
211#137—2712no—
212#137—5422no—
213#137—5422no—
214#137C2×D27112yescomplemented
215C2×Q16—5422no—
216C2×SD16—5422no—
217C2×M16—2712no—
218C4⋊Q8—2711no—
219C4⋊Q8—2711no—
220C4⋊Q8—2711no—
221C4⋊Q8—5422no—
222C4⋊Q8C2×D27111yescharacteristic, complemented
223C2×C4⋊C4—5422no—
224C2×C42—2711no—
225#104—2712no—
226#104—2712no—
227#104—2712no—
228#104C2×D27112yescomplemented
229#101—2711no—
230#101—2712no—
231#101C2×D27111yescharacteristic, complemented
232#41—22no—
233#41—22no—
234#41—2712no—
235#41—2712no—
236#41—2712no—
237#41—5422no—
238#41—5422no—
239#41—5424no—
240#41—5424no—
241#41—5422no—
242#41—5424no—
243#41—5422no—
244#41—5424no—
245#41—5422no—
246#41—5422no—
247#41—5422no—
248#41—5422no—
249#41C22×D27112yes—
250C2×Q8—22no—
251C2×Q8—2712no—
252C2×Q8—2711no—
253C2×Q8—2712no—
254C2×Q8—2711no—
255C2×Q8—5421no—
256C2×Q8—5422no—
257C2×Q8—5424no—
258C2×Q8—5422no—
259C2×Q8—5422no—
260C2×Q8—5422no—
261C2×Q8—5422no—
262C2×Q8—5422no—
263C2×Q8—5422no—
264C2×Q8—5424no—
265C2×Q8—5422no—
266C2×Q8C22×D27111yescharacteristic
267C2×Q8C22×D27111yescharacteristic
268C2×D4—2712no—
269C2×D4—5422no—
270C2×D4—5422no—
271C22×C4—2711no—
272C22×C4—2712no—
273C22×C4—5422no—
274C22×C4—5421no—
275C22×C4—5424no—
276C22×C4—5421no—
277Q16—22no—
278Q16—5422no—
279Q16—5422no—
280Q16—5422no—
281SD16—22no—
282SD16—5422no—
283SD16—5422no—
284SD16—5422no—
285M16—2712no—
286M16—5422no—
287M16C22×D27112yes—
288C2×C8—5422no—
289C4⋊C4—22no—
290C4⋊C4—5422no—
291C4⋊C4—5422no—
292C4⋊C4—5422no—
293C42—2711no—
294C42—5421no—
295C42C22×D27111yescharacteristic
296C23—2711no—
297C23—5422no—
298Q8—22no—
299Q8—22no—
300Q8—22no—
301Q8—2712no—
302Q8—5422no—
303Q8—5424no—
304Q8—5424no—
305Q8—5422no—
306Q8—5422no—
307Q8—5422no—
308Q8—10842no—
309Q8D4×D27112yes—
310D4—22no—
311D4—22no—
312D4—2712no—
313D4—5422no—
314D4—5422no—
315D4—5422no—
316D4—5422no—
317D4—5424no—
318D4—5422no—
319D4D4×D27112yes—
320C2×C4—24no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—2711no—
325C2×C4—2712no—
326C2×C4—2712no—
327C2×C4—2712no—
328C2×C4—5422no—
329C2×C4—5422no—
330C2×C4—5422no—
331C2×C4—5422no—
332C2×C4—5424no—
333C2×C4—5424no—
334C2×C4—5424no—
335C2×C4—5422no—
336C2×C4—5424no—
337C2×C4—5421no—
338C2×C4—5422no—
339C2×C4—5421no—
340C2×C4—10841no—
341C2×C4—10841no—
342C2×C4D4×D27112yes—
343C2×C4C23×D27111yescharacteristic
344C8—22no—
345C8—5422no—
346C22—22no—
347C22—2711no—
348C22—2711no—
349C22—5422no—
350C22—5422no—
351C22—10844no—
352C22C2×D4×D27111yescharacteristic
353C4—22no—
354C4—22no—
355C4—22no—
356C4—41no—
357C4—2712no—
358C4—5422no—
359C4—5422no—
360C4—5422no—
361C4—10841no—
362C4C2×D4×D27112yes—
363C2—21no—
364C2—42no—
365C2—2712no—
366C2—5421no—
367C2—10842no—
368C2#120×D27111yescharacteristic
subgroupG/H
centerC2#120×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#456×C271C2
socleC2×C271C2×#120
prime radical2−1+4×D271C2
hypercenter#456D271
hypocenterC271C2×#456
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
2. #137×D271C2semidirect
3. C4⋊Q8×D271C2semidirect
4. #31774082C2semidirect
5. #31774081C2semidirect
6. #31774080C2semidirect
7. #31774079C2semidirect
8. #31774078C2semidirect
9. #104×D271C2semidirect
10. #101×D271C2semidirect
11. #456×C271C2semidirect
23. #22576193C22semidirect
25. #22576191C22semidirect
26. #22576190C22semidirect
31. #22576160C22semidirect
33. #22576158C22semidirect
34. #22576158C22semidirect
37. #22576088C22semidirect
38. #22576087C22semidirect
39. #22576086C22semidirect
40. #22576079C22semidirect
41. #22576078C22semidirect
43. #137×C271C22semidirect
44. C4⋊Q8×C271C22semidirect
45. #104×C271C22semidirect
46. #101×C271C22semidirect
183. D271#456direct (N × H)
184. C271C2×#456semidirect
194. #456D271direct (N × H)
214. #137C2×D271semidirect
222. C4⋊Q8C2×D271semidirect
228. #104C2×D271semidirect
231. #101C2×D271semidirect

Maximal subgroups

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

subgroupindex
2−1+4×D2712
#137×D2712
C4⋊Q8×D2712
#317740822
#317740812
#317740802
#317740792
#317740782
#104×D2712
#101×D2712
#456×C2712
C2×#456271