Finite Group Databasebeta

← #32157655

Subgroups — #32157655

Order 18176.

2414 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22591289C216yesmaximal, complemented
2Q8×C4⋊Dic71C212yesmaximal, complemented
3#22591233C216yesmaximal, complemented
4#22591181C212yesmaximal
5#22590983C216yesmaximal, complemented
6#22590747C213yesmaximal, complemented
7#22590738C213yesmaximal, complemented
8#22590705C212yesmaximal, complemented
9C2×C4×#41×C71C211yesmaximal, characteristic
10C2×Q8×Dic71C2214yes—
11C2×#14556087C2216yes—
12#17985902C22112yes—
13#17985898C22112yes—
14#17985892C22124yes—
15#17985869C2216yes—
16#17985867C2216yes—
17#17985865C2211yescharacteristic
18#17985865C2213yes—
19#17985862C2216yes—
20#17985859C2216yes—
21#17985859C2216yes—
22#17985859C2216yes—
23C4⋊C4×Dic71C22112yes—
24C4×C4⋊Dic71C2216yes—
25C4×C4⋊Dic71C2216yes—
26C4×C4⋊Dic71C2216yes—
27C4×C22⋊Dic71C2216yes—
28#17985845C2216yes—
29C22×#41×C71C2211yescharacteristic
30C4×#41×C71C4116yescomplemented
31C2×C4×Q8×C71C2211yescharacteristic
32C2×C4×Q8×C71C2211yescharacteristic
33C2×C4×D4×C71C2213yes—
34C2×C4×D4×C71C2213yes—
35C2×#117×C71C2213yes—
36C22×C42×C71C2213yes—
37Q8×Dic71—216no—
38#14556087—412no—
39C2×C4⋊Dic71C2312yes—
40C2×C4⋊Dic71C23112yes—
41C2×C4⋊Dic71C2316yes—
42C2×C4⋊Dic71C23112yes—
43C2×C22⋊Dic71C23112yes—
44C2×C22⋊Dic71C23112yes—
45C2×C4×Dic71C23112yes—
46C2×C4×Dic71C23112yes—
47C71⋊#95C2312yes—
48C71⋊#95C2316yes—
49C71⋊#95C23112yes—
50C71⋊#95C2312yes—
51C71⋊#95C2316yes—
52C71⋊#95C23112yes—
53C2×#41×C71—28no—
54C2×#41×C71C2×C418yes—
55C22×Q8×C71C2311yescharacteristic
56C22×D4×C71C2313yes—
57C23×C4×C71C2313yes—
58C23×C4×C71C2313yes—
59C23×C4×C71C2313yes—
60C4×Q8×C71C2×C418yes—
61C4×Q8×C71C2×C418yes—
62C4×D4×C71C2×C4124yes—
63C4×D4×C71C2×C4124yes—
64#117×C71C2×C4124yes—
65C2×C4⋊C4×C71C2313yes—
66C2×C4⋊C4×C71C2313yes—
67C2×C22⋊C4×C71C2316yes—
68C2×C42×C71C2×C4124yes—
69C2×C42×C71C2313yes—
70C2×C42×C71C2313yes—
71C2×C42×C71C2313yes—
72C2×C42×C71C2311yescharacteristic
73C22×Dic71C2414yes—
74C22×Dic71C24112yes—
75C4⋊Dic71—224no—
76C4⋊Dic71—212no—
77C4⋊Dic71—24no—
78C4⋊Dic71—412no—
79C22⋊Dic71—412no—
80C22⋊Dic71—412no—
81C4×Dic71—224no—
82C4×Dic71—412no—
83C24×C71C2413yes—
84#41×C71—232no—
85C2×Q8×C71C22×C414yes—
86C2×Q8×C71#4118yes—
87C2×D4×C71—212no—
88C2×D4×C71C22×C4112yes—
89C22×C4×C71—212no—
90C22×C4×C71—26no—
91C22×C4×C71—26no—
92C22×C4×C71—46no—
93C22×C4×C71C22×C4112yes—
94C22×C4×C71C22×C4112yes—
95C22×C4×C71C22×C4112yes—
96C22×C4×C71C2411yescharacteristic
97C22×C4×C71C2411yescharacteristic
98C22×C4×C71C2413yes—
99C22×C4×C71C2413yes—
100C22×C4×C71C2413yes—
101C22×C4×C71C2411yescharacteristic
102C4⋊C4×C71C22×C4112yes—
103C4⋊C4×C71C22×C4112yes—
104C22⋊C4×C71C22×C4124yes—
105C42×C71C22×C414yes—
106C42×C71C22×C4112yes—
107C42×C71C22×C4112yes—
108C42×C71C22×C4112yes—
109C23×C71—26no—
110C23×C71—23no—
111C23×C71—23no—
112C23×C71—26no—
113C23×C71C23×C416yes—
114C23×C71C2511yescharacteristic
115Q8×C71#117116yescomplemented
116D4×C71—224no—
117C2×C4×C71—22no—
118C2×C4×C71—23no—
119C2×C4×C71—23no—
120C2×C4×C71—22no—
121C2×C4×C71—26no—
122C2×C4×C71—224no—
123C2×C4×C71—224no—
124C2×C4×C71—412no—
125C2×C4×C71C23×C412yes—
126C2×C4×C71C23×C412yes—
127C2×C4×C71C23×C416yes—
128C2×C4×C71C23×C416yes—
129C2×C4×C71C23×C416yes—
130C2×C4×C71C23×C412yes—
131C2×C4×C71C2×#41112yes—
132C2×C4×C71H(5,𝔽2)12yes—
133C2×C4×C712−1+412yes—
134C2×Dic71—24no—
135C2×Dic71—212no—
136C2×Dic71—212no—
137C2×Dic71—24no—
138C2×Dic71—24no—
139C2×Dic71—212no—
140C22×C71—26no—
141C22×C71—26no—
142C22×C71—26no—
143C22×C71—46no—
144C22×C71—46no—
145C22×C71C24×C411yescharacteristic
146C22×C71C22×#4112yes—
147C22×C71C2×H(5,𝔽2)11yescharacteristic
148C22×C71C2×2−1+411yescharacteristic
149C22×C71#58512yes—
150C4×C71—22no—
151C4×C71—26no—
152C4×C71—22no—
153C4×C71—26no—
154C4×C71C2×#117112yes—
155C4×C71#52014yes—
156Dic71—412no—
157Dic71—44no—
158#34193—711nomaximal, Sylow
159C2×C71—46no—
160C2×C71C22×#11711yescharacteristic
161C2×C71C2×#52011yescharacteristic
162C2×C71#342811yescharacteristic
163C2×C71#351712yes—
164C2×C71#351812yes—
165C2×C4×#41D7111yescharacteristic
166#2354—716no—
167Q8×C4⋊C4—712no—
168#2347—716no—
169#2341—712no—
170#2319—716no—
171#2285—713no—
172#2283—713no—
173#2276—712no—
174C71#3419311yescharacteristic, complemented, Sylow
175C22×#41C2×D7111yescharacteristic
176C4×#41Dic71116yescomplemented
177C2×C4×Q8—714no—
178C2×C4×Q8C2×D7111yescharacteristic
179C2×C4×Q8C2×D7111yescharacteristic
180C2×C4×D4C2×D7113yes—
181C2×C4×D4C2×D7113yes—
182C2×#117—716no—
183C2×#117C2×D7113yes—
184C22×C42C2×D7113yes—
185#390—7112no—
186#389—7112no—
187#388—7124no—
188#384—716no—
189#383—716no—
190#382—711no—
191#382—713no—
192#381—716no—
193#380—716no—
194#380—716no—
195#380—716no—
196C4×C4⋊C4—7112no—
197C4×C4⋊C4—716no—
198C4×C4⋊C4—716no—
199C4×C4⋊C4—716no—
200C4×C22⋊C4—716no—
201#376—716no—
202C2×#41—28no—
203C2×#41C2×Dic7118yes—
204C22×Q8C22×D7111yescharacteristic
205C22×D4C22×D7113yes—
206C23×C4C22×D7113yes—
207C23×C4C22×D7113yes—
208C23×C4C22×D7113yes—
209C4×Q8—14216no—
210C4×Q8C2×Dic7118yes—
211C4×Q8C2×Dic7118yes—
212C4×D4C2×Dic71124yes—
213C4×D4C2×Dic71124yes—
214#117—28412no—
215#117C2×Dic71124yes—
216C2×C4⋊C4—7112no—
217C2×C4⋊C4—716no—
218C2×C4⋊C4—7112no—
219C2×C4⋊C4—712no—
220C2×C4⋊C4C22×D7113yes—
221C2×C4⋊C4C22×D7113yes—
222C2×C22⋊C4—7112no—
223C2×C22⋊C4—7112no—
224C2×C22⋊C4C22×D7116yes—
225C2×C42—7112no—
226C2×C42—7112no—
227C2×C42C2×Dic71124yes—
228C2×C42C22×D7113yes—
229C2×C42C22×D7113yes—
230C2×C42C22×D7113yes—
231C2×C42C22×D7111yescharacteristic
232#95—712no—
233#95—716no—
234#95—7112no—
235#95—712no—
236#95—716no—
237#95—7112no—
238C24C23×D7113yes—
239#41—232no—
240C2×Q8C22×Dic7114yes—
241C2×Q8#1176586518yes—
242C2×D4—212no—
243C2×D4C22×Dic71112yes—
244C22×C4—212no—
245C22×C4—26no—
246C22×C4—26no—
247C22×C4—46no—
248C22×C4—714no—
249C22×C4—7112no—
250C22×C4C22×Dic71112yes—
251C22×C4C22×Dic71112yes—
252C22×C4C22×Dic71112yes—
253C22×C4C23×D7111yescharacteristic
254C22×C4C23×D7111yescharacteristic
255C22×C4C23×D7113yes—
256C22×C4C23×D7113yes—
257C22×C4C23×D7113yes—
258C22×C4C23×D7111yescharacteristic
259C4⋊C4—28412no—
260C4⋊C4—14224no—
261C4⋊C4—14212no—
262C4⋊C4—1424no—
263C4⋊C4C22×Dic71112yes—
264C4⋊C4C22×Dic71112yes—
265C22⋊C4—28412no—
266C22⋊C4—28412no—
267C22⋊C4C22×Dic71124yes—
268C42—28412no—
269C42—14224no—
270C42C22×Dic7114yes—
271C42C22×Dic71112yes—
272C42C22×Dic71112yes—
273C42C22×Dic71112yes—
274C23—26no—
275C23—23no—
276C23—23no—
277C23—26no—
278C23C23×Dic7116yes—
279C23C24×D7111yescharacteristic
280Q8#14556087116yescomplemented
281D4—224no—
282C2×C4—22no—
283C2×C4—23no—
284C2×C4—23no—
285C2×C4—26no—
286C2×C4—22no—
287C2×C4—224no—
288C2×C4—224no—
289C2×C4—412no—
290C2×C4—1424no—
291C2×C4—14212no—
292C2×C4—14212no—
293C2×C4—1424no—
294C2×C4—1424no—
295C2×C4—14212no—
296C2×C4C23×Dic7112yes—
297C2×C4C23×Dic7112yes—
298C2×C4C23×Dic7116yes—
299C2×C4C23×Dic7116yes—
300C2×C4C23×Dic7116yes—
301C2×C4C23×Dic7112yes—
302C2×C4C2×#11765865112yes—
303C2×C4#1455617412yes—
304C2×C4#1455617512yes—
305C22—26no—
306C22—26no—
307C22—26no—
308C22—46no—
309C22—46no—
310C22C24×Dic7111yescharacteristic
311C22C22×#1176586512yes—
312C22C2×#1455617411yescharacteristic
313C22C2×#1455617511yescharacteristic
314C22#1798683512yes—
315C4—22no—
316C4—26no—
317C4—22no—
318C4—26no—
319C4—28412no—
320C4—2844no—
321C4C2×#14556087112yes—
322C4#1798648214yes—
323C2—46no—
324C2C22×#1455608711yescharacteristic
325C2C2×#1798648211yescharacteristic
326C2#2260154311yescharacteristic
327C2#2260234612yes—
328C2#2260235712yes—
subgroupG/H
center, FrattiniC23C24×D71
commutatorC22×C71C24×C4
FittingC2×C4×#41×C71C2
socle, cosocleC23×C71C25
prime radicalC22×#41×C71C22
hypercenterC2×C4×#41D71
hypocenterC71#34193
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C4×#41

Complements

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

NH ≅ G/Ntype
1. #22591289C2semidirect
2. Q8×C4⋊Dic71C2semidirect
3. #22591233C2semidirect
5. #22590983C2semidirect
6. #22590747C2semidirect
7. #22590738C2semidirect
8. #22590705C2semidirect
30. C4×#41×C71C4semidirect
115. Q8×C71#117semidirect
174. C71#34193semidirect
176. C4×#41Dic71semidirect
280. Q8#14556087semidirect

Maximal subgroups

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

subgroupindex
#225912892
Q8×C4⋊Dic712
#225912332
#225911812
#225909832
#225907472
#225907382
#225907052
C2×C4×#41×C712
#3419371