Finite Group Databasebeta

← #31885489

Subgroups — #31885489

Order 18176.

522 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22588408C211yesmaximal, characteristic
2C4×C22⋊(C71⋊C8)C211yesmaximal, characteristic
3#22586166C212yesmaximal, complemented
4#22586163C212yesmaximal, complemented
5C4×C22⋊Q8×C71C211yesmaximal, characteristic
6C2×C22⋊(C71⋊C8)—21no—
7C2×C22⋊(C71⋊C8)—21no—
8C2×C4×C71⋊C8—21no—
9#17985754—21no—
10#17985754C2212yes—
11#17985754C2212yes—
12C2×C22⋊Q8×C71C411yescharacteristic
13C2×C4×Q8×C71C411yescharacteristic
14C22×C42×C71C2211yescharacteristic
15#391×C71C2211yescharacteristic
16#389×C71C2211yescharacteristic
17#387×C71—21no—
18#385×C71—21no—
19#380×C71C411yescharacteristic
20C4×C4⋊C4×C71—21no—
21C4×C4⋊C4×C71C411yescharacteristic
22C4×C22⋊C4×C71—21no—
23C22×C71⋊C8—22no—
24C22×C71⋊C8—22no—
25C22⋊(C71⋊C8)—24no—
26C22⋊(C71⋊C8)—42no—
27C4×C71⋊C8—42no—
28C22×Q8×C71C2×C411yescharacteristic
29C23×C4×C71C2×C411yescharacteristic
30C23×C4×C71D411yescharacteristic
31C23×C4×C71D411yescharacteristic
32C22⋊Q8×C71—22no—
33C22⋊Q8×C71C814yescomplemented
34C4×Q8×C71—42no—
35C2×C4⋊C4×C71—21no—
36C2×C4⋊C4×C71—21no—
37C2×C4⋊C4×C71—21no—
38C2×C4⋊C4×C71C2×C411yescharacteristic
39C2×C4⋊C4×C71C2×C411yescharacteristic
40C2×C4⋊C4×C71C2×C411yescharacteristic
41C2×C22⋊C4×C71—21no—
42C2×C22⋊C4×C71—21no—
43C2×C42×C71—21no—
44C2×C42×C71—21no—
45C2×C42×C71—42no—
46C2×C42×C71C2×C411yescharacteristic
47C2×C42×C71D411yescharacteristic
48C2×C42×C71D411yescharacteristic
49C2×C42×C71C2311yescharacteristic
50#95×C71—21no—
51#95×C71—21no—
52#95×C71—21no—
53#95×C71—21no—
54#95×C71—21no—
55C2×C71⋊C8—24no—
56C2×C71⋊C8—42no—
57C2×C71⋊C8—42no—
58C2×C71⋊C8—42no—
59C2×C71⋊C8—42no—
60C2×C71⋊C8—42no—
61C24×C71C22⋊C411yescharacteristic
62C2×Q8×C71—21no—
63C2×Q8×C71—41no—
64C2×Q8×C71—41no—
65C2×Q8×C71C2×C812yes—
66C22×C4×C71—21no—
67C22×C4×C71—21no—
68C22×C4×C71—22no—
69C22×C4×C71—22no—
70C22×C4×C71—22no—
71C22×C4×C71—22no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—41no—
75C22×C4×C71—41no—
76C22×C4×C71—41no—
77C22×C4×C71—41no—
78C22×C4×C71C22⋊C411yescharacteristic
79C22×C4×C71C22⋊C411yescharacteristic
80C22×C4×C71C22⋊C411yescharacteristic
81C22×C4×C71C2×C812yes—
82C22×C4×C71M1612yes—
83C22×C4×C71C22×C411yescharacteristic
84C22×C4×C71C2×D411yescharacteristic
85C22×C4×C71C2×D411yescharacteristic
86C4⋊C4×C71—24no—
87C4⋊C4×C71—21no—
88C4⋊C4×C71—41no—
89C4⋊C4×C71—41no—
90C4⋊C4×C71—42no—
91C4⋊C4×C71—42no—
92C4⋊C4×C71C2×C812yes—
93C22⋊C4×C71—24no—
94C22⋊C4×C71—42no—
95C42×C71—22no—
96C42×C71—22no—
97C42×C71—41no—
98C42×C71—41no—
99C42×C71—42no—
100C42×C71—42no—
101C23×C71—21no—
102C23×C71—21no—
103C23×C71—21no—
104C23×C71—21no—
105C23×C71—21no—
106C23×C71—21no—
107C23×C71C22⋊C812yes—
108C23×C71C2×C22⋊C411yescharacteristic
109Q8×C71—42no—
110Q8×C71—42no—
111C2×C4×C71—21no—
112C2×C4×C71—21no—
113C2×C4×C71—21no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—21no—
117C2×C4×C71—21no—
118C2×C4×C71—21no—
119C2×C4×C71—21no—
120C2×C4×C71—21no—
121C2×C4×C71—21no—
122C2×C4×C71—21no—
123C2×C4×C71—22no—
124C2×C4×C71—22no—
125C2×C4×C71—42no—
126C2×C4×C71—42no—
127C2×C4×C71—44no—
128C2×C4×C71—44no—
129C2×C4×C71—41no—
130C2×C4×C71—41no—
131C2×C4×C71—41no—
132C2×C4×C71—41no—
133C2×C4×C71—41no—
134C2×C4×C71—41no—
135C2×C4×C71—41no—
136C2×C4×C71—41no—
137C2×C4×C71—41no—
138C2×C4×C71—41no—
139C2×C4×C71C22⋊C812yes—
140C2×C4×C71#9912yes—
141C2×C4×C71#10412yes—
142C2×C4×C71C22×C811yescharacteristic
143C2×C4×C71C2×M1611yescharacteristic
144C2×C4×C71#13512yes—
145C2×C4×C71#13712yes—
146C71⋊C8—44no—
147C71⋊C8—82no—
148C22×C71—22no—
149C22×C71—22no—
150C22×C71—22no—
151C22×C71—41no—
152C22×C71—41no—
153C22×C71—41no—
154C22×C71—41no—
155C22×C71C2×C22⋊C811yescharacteristic
156C22×C71C2×#9911yescharacteristic
157C22×C71#41311yescharacteristic
158C22×C71#41611yescharacteristic
159C22×C71#41911yescharacteristic
160C22×C71C2×#10411yescharacteristic
161C22×C71#42111yescharacteristic
162C4×C71—22no—
163C4×C71—21no—
164C4×C71—21no—
165C4×C71—22no—
166C4×C71—22no—
167C4×C71—22no—
168C4×C71—22no—
169C4×C71—42no—
170C4×C71—42no—
171C4×C71—81no—
172C4×C71—81no—
173#7583—711nomaximal, Sylow
174C2×C71—42no—
175C2×C71#145911yescharacteristic
176C2×C71#147611yescharacteristic
177C2×C71#147911yescharacteristic
178C2×C71#150811yescharacteristic
179C2×C71#151611yescharacteristic
180C2×C71#152911yescharacteristic
181C2×C71#155411yescharacteristic
182C4×C22⋊Q8D7111yescharacteristic
183#1817—711no—
184C4×C22⋊C8—711no—
185#1272—712no—
186#1271—712no—
187C71#758311yescharacteristic, complemented, Sylow
188C2×C22⋊Q8Dic7111yescharacteristic
189C2×C4×Q8Dic7111yescharacteristic
190C22×C42C2×D7111yescharacteristic
191C2×C22⋊C8—1421no—
192C2×C22⋊C8—1421no—
193C2×C4×C8—1421no—
194#391C2×D7111yescharacteristic
195#389C2×D7111yescharacteristic
196#387—21no—
197#385—21no—
198#380Dic7111yescharacteristic
199C4×C4⋊C4—21no—
200C4×C4⋊C4Dic7111yescharacteristic
201C4×C22⋊C4—21no—
202#336—712no—
203#336—712no—
204#336—1421no—
205C22×Q8C2×Dic7111yescharacteristic
206C23×C4C2×Dic7111yescharacteristic
207C23×C4C22⋊D7111yescharacteristic
208C23×C4C22⋊D7111yescharacteristic
209C22×C8—1422no—
210C22×C8—1422no—
211C22⋊Q8—22no—
212C22⋊Q8C71⋊C814yescomplemented
213C4×Q8—42no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4C2×Dic7111yescharacteristic
218C2×C4⋊C4C2×Dic7111yescharacteristic
219C2×C4⋊C4C2×Dic7111yescharacteristic
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—42no—
225C2×C42C2×Dic7111yescharacteristic
226C2×C42C22⋊D7111yescharacteristic
227C2×C42C22⋊D7111yescharacteristic
228C2×C42C22×D7111yescharacteristic
229C22⋊C8—2842no—
230C22⋊C8—1424no—
231C4×C8—2842no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237C24C22⋊Dic7111yescharacteristic
238C2×Q8—21no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8C2×C71⋊C812yes—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4C22⋊Dic7111yescharacteristic
255C22×C4C22⋊Dic7111yescharacteristic
256C22×C4C22⋊Dic7111yescharacteristic
257C22×C4C2×C71⋊C812yes—
258C22×C4#1176584912yes—
259C22×C4C22×Dic7111yescharacteristic
260C22×C4C2×C22⋊D7111yescharacteristic
261C22×C4C2×C22⋊D7111yescharacteristic
262C2×C8—2842no—
263C2×C8—2842no—
264C2×C8—2842no—
265C2×C8—2842no—
266C2×C8—2842no—
267C2×C8—1424no—
268C4⋊C4—24no—
269C4⋊C4—21no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4C2×C71⋊C812yes—
275C22⋊C4—24no—
276C22⋊C4—42no—
277C42—22no—
278C42—22no—
279C42—41no—
280C42—41no—
281C42—42no—
282C42—42no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23C22⋊(C71⋊C8)12yes—
290C23C2×C22⋊Dic7111yescharacteristic
291Q8—42no—
292Q8—42no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4C22⋊(C71⋊C8)12yes—
322C2×C4#1455604312yes—
323C2×C4#1455605512yes—
324C2×C4C22×C71⋊C811yescharacteristic
325C2×C4C2×#1176584911yescharacteristic
326C2×C4#1455614812yes—
327C2×C4#1455615912yes—
328C8—2844no—
329C8—5682no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22C2×C22⋊(C71⋊C8)11yescharacteristic
338C22C2×#1455604311yescharacteristic
339C22#1798598311yescharacteristic
340C22#1798599511yescharacteristic
341C22#1798600811yescharacteristic
342C22C2×#1455605511yescharacteristic
343C22#1798601611yescharacteristic
344C4—22no—
345C4—22no—
346C4—22no—
347C4—21no—
348C4—21no—
349C4—22no—
350C4—22no—
351C4—42no—
352C4—42no—
353C4—81no—
354C4—81no—
355C2—42no—
356C2#2258665811yescharacteristic
357C2#2258672211yescharacteristic
358C2#2258673511yescharacteristic
359C2#2258686011yescharacteristic
360C2#2258689611yescharacteristic
361C2#2258694711yescharacteristic
362C2#2258705011yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC2×C4×C71C22×C8
FrattiniC2×C42C22×D71
FittingC4×C22⋊Q8×C71C2
socleC23×C71C2×C22⋊C4
prime radicalC24×C71C22⋊C4
cosocleC2×C42×C71C23
hypercenterC4×C22⋊Q8D71
hypocenterC71#7583
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #22586166C2semidirect
4. #22586163C2semidirect
33. C22⋊Q8×C71C8semidirect
187. C71#7583semidirect
212. C22⋊Q8C71⋊C8semidirect

Maximal subgroups

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

subgroupindex
#225884082
C4×C22⋊(C71⋊C8)2
#225861662
#225861632
C4×C22⋊Q8×C712
#758371