Finite Group Databasebeta

← #31885817

Subgroups — #31885817

Order 18176.

482 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22588320C211yesmaximal, characteristic
2C4×C22⋊(C71⋊C8)C211yesmaximal, characteristic
3#22586166C212yesmaximal, complemented
4#22586163C212yesmaximal, complemented
5C71×#2855C211yesmaximal, characteristic
6C2×C4⋊(C71⋊C8)—21no—
7C2×C22⋊(C71⋊C8)—21no—
8C2×C22⋊(C71⋊C8)—21no—
9C2×C4×C71⋊C8—21no—
10#17985754C2212yes—
11#17985754C2212yes—
12C2×C22⋊Q8×C71C411yescharacteristic
13C22×C42×C71C2211yescharacteristic
14#391×C71C411yescharacteristic
15#391×C71C2211yescharacteristic
16#389×C71C411yescharacteristic
17#389×C71C2211yescharacteristic
18#387×C71—21no—
19#387×C71—21no—
20#385×C71—21no—
21#381×C71—21no—
22#380×C71C411yescharacteristic
23C22×C71⋊C8—22no—
24C22×C71⋊C8—22no—
25C4⋊(C71⋊C8)—42no—
26C22⋊(C71⋊C8)—24no—
27C22⋊(C71⋊C8)—42no—
28C4×C71⋊C8—42no—
29C22×Q8×C71C2×C411yescharacteristic
30C23×C4×C71C2×C411yescharacteristic
31C23×C4×C71D411yescharacteristic
32C23×C4×C71D411yescharacteristic
33C22⋊Q8×C71—42no—
34C2×C4⋊C4×C71—21no—
35C2×C4⋊C4×C71—21no—
36C2×C4⋊C4×C71—21no—
37C2×C4⋊C4×C71C2×C411yescharacteristic
38C2×C4⋊C4×C71C2×C411yescharacteristic
39C2×C4⋊C4×C71C2×C411yescharacteristic
40C2×C22⋊C4×C71—21no—
41C2×C22⋊C4×C71—21no—
42C2×C42×C71—42no—
43C2×C42×C71C2×C411yescharacteristic
44C2×C42×C71D411yescharacteristic
45C2×C42×C71D411yescharacteristic
46C2×C42×C71C2311yescharacteristic
47#95×C71—21no—
48#95×C71—21no—
49#95×C71—21no—
50#95×C71—21no—
51#95×C71—21no—
52#95×C71—21no—
53#95×C71—21no—
54C2×C71⋊C8—24no—
55C2×C71⋊C8—42no—
56C2×C71⋊C8—42no—
57C2×C71⋊C8—42no—
58C2×C71⋊C8—42no—
59C2×C71⋊C8—42no—
60C24×C71C22⋊C411yescharacteristic
61C2×Q8×C71—22no—
62C2×Q8×C71—41no—
63C2×Q8×C71—41no—
64C22×C4×C71—22no—
65C22×C4×C71—22no—
66C22×C4×C71—22no—
67C22×C4×C71—22no—
68C22×C4×C71—21no—
69C22×C4×C71—21no—
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71—41no—
73C22×C4×C71—41no—
74C22×C4×C71—41no—
75C22×C4×C71—41no—
76C22×C4×C71C22⋊C411yescharacteristic
77C22×C4×C71C22⋊C411yescharacteristic
78C22×C4×C71C22⋊C411yescharacteristic
79C22×C4×C71SD1612yes—
80C22×C4×C71Q1612yes—
81C22×C4×C71C22×C411yescharacteristic
82C22×C4×C71C2×D411yescharacteristic
83C22×C4×C71C2×D411yescharacteristic
84C4⋊C4×C71—22no—
85C4⋊C4×C71—41no—
86C4⋊C4×C71—41no—
87C4⋊C4×C71—42no—
88C4⋊C4×C71—42no—
89C4⋊C4×C71—42no—
90C22⋊C4×C71—42no—
91C22⋊C4×C71—42no—
92C42×C71—22no—
93C42×C71—22no—
94C42×C71—41no—
95C42×C71—41no—
96C23×C71—21no—
97C23×C71—21no—
98C23×C71—21no—
99C23×C71—21no—
100C23×C71—21no—
101C23×C71—21no—
102C23×C71#10312yes—
103C23×C71C2×C22⋊C411yescharacteristic
104Q8×C71—82no—
105C2×C4×C71—21no—
106C2×C4×C71—21no—
107C2×C4×C71—21no—
108C2×C4×C71—21no—
109C2×C4×C71—21no—
110C2×C4×C71—21no—
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—42no—
118C2×C4×C71—42no—
119C2×C4×C71—44no—
120C2×C4×C71—44no—
121C2×C4×C71—41no—
122C2×C4×C71—41no—
123C2×C4×C71—41no—
124C2×C4×C71—41no—
125C2×C4×C71—41no—
126C2×C4×C71—41no—
127C2×C4×C71—41no—
128C2×C4×C71—41no—
129C2×C4×C71—41no—
130C2×C4×C71—41no—
131C2×C4×C71—41no—
132C2×C4×C71—41no—
133C2×C4×C71#9912yes—
134C2×C4×C71#10312yes—
135C2×C4×C71#10412yes—
136C2×C4×C71#13112yes—
137C2×C4×C71C2×SD1611yescharacteristic
138C2×C4×C71C2×Q1611yescharacteristic
139C2×C4×C71#13512yes—
140C71⋊C8—44no—
141C71⋊C8—82no—
142C22×C71—22no—
143C22×C71—22no—
144C22×C71—22no—
145C22×C71—41no—
146C22×C71—41no—
147C22×C71—41no—
148C22×C71—41no—
149C22×C71#40811yescharacteristic
150C22×C71C2×#9911yescharacteristic
151C22×C71#41311yescharacteristic
152C22×C71C2×#10311yescharacteristic
153C22×C71#41611yescharacteristic
154C22×C71C2×#10411yescharacteristic
155C22×C71#42111yescharacteristic
156C4×C71—22no—
157C4×C71—22no—
158C4×C71—22no—
159C4×C71—22no—
160C4×C71—21no—
161C4×C71—22no—
162C4×C71—21no—
163C4×C71—81no—
164C4×C71—81no—
165C4×C71—81no—
166C4×C71—81no—
167#7661—711nomaximal, Sylow
168C2×C71—42no—
169C2×C71#146811yescharacteristic
170C2×C71#148211yescharacteristic
171C2×C71#149311yescharacteristic
172C2×C71#150111yescharacteristic
173C2×C71#152011yescharacteristic
174C2×C71#152911yescharacteristic
175C2×C71#154211yescharacteristic
176#2855D7111yescharacteristic
177#1797—711no—
178C4×C22⋊C8—711no—
179#1272—712no—
180#1271—712no—
181C71#766111yescharacteristic, complemented, Sylow
182C2×C22⋊Q8Dic7111yescharacteristic
183C22×C42C2×D7111yescharacteristic
184C2×C4⋊C8—1421no—
185C2×C22⋊C8—1421no—
186C2×C22⋊C8—1421no—
187C2×C4×C8—1421no—
188#391Dic7111yescharacteristic
189#391C2×D7111yescharacteristic
190#389Dic7111yescharacteristic
191#389C2×D7111yescharacteristic
192#387—21no—
193#387—21no—
194#385—21no—
195#381—21no—
196#380Dic7111yescharacteristic
197#336—712no—
198#336—712no—
199C22×Q8C2×Dic7111yescharacteristic
200C23×C4C2×Dic7111yescharacteristic
201C23×C4C22⋊D7111yescharacteristic
202C23×C4C22⋊D7111yescharacteristic
203C22×C8—1422no—
204C22×C8—1422no—
205C22⋊Q8—42no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4C2×Dic7111yescharacteristic
210C2×C4⋊C4C2×Dic7111yescharacteristic
211C2×C4⋊C4C2×Dic7111yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C42—42no—
215C2×C42C2×Dic7111yescharacteristic
216C2×C42C22⋊D7111yescharacteristic
217C2×C42C22⋊D7111yescharacteristic
218C2×C42C22×D7111yescharacteristic
219C4⋊C8—2842no—
220C22⋊C8—2842no—
221C22⋊C8—1424no—
222C4×C8—2842no—
223#95—21no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230C24C22⋊Dic7111yescharacteristic
231C2×Q8—22no—
232C2×Q8—41no—
233C2×Q8—41no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4C22⋊Dic7111yescharacteristic
247C22×C4C22⋊Dic7111yescharacteristic
248C22×C4C22⋊Dic7111yescharacteristic
249C22×C4#1176585212yes—
250C22×C4#1176585512yes—
251C22×C4C22×Dic7111yescharacteristic
252C22×C4C2×C22⋊D7111yescharacteristic
253C22×C4C2×C22⋊D7111yescharacteristic
254C2×C8—2842no—
255C2×C8—2842no—
256C2×C8—2842no—
257C2×C8—2842no—
258C2×C8—2842no—
259C2×C8—1424no—
260C4⋊C4—22no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C42—22no—
269C42—22no—
270C42—41no—
271C42—41no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23#1455605212yes—
279C23C2×C22⋊Dic7111yescharacteristic
280Q8—82no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4#1455604312yes—
310C2×C4#1455605212yes—
311C2×C4#1455605512yes—
312C2×C4#1455613512yes—
313C2×C4C2×#1176585211yescharacteristic
314C2×C4C2×#1176585511yescharacteristic
315C2×C4#1455614812yes—
316C8—2844no—
317C8—5682no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22#1798596611yescharacteristic
326C22C2×#1455604311yescharacteristic
327C22#1798598311yescharacteristic
328C22C2×#1455605211yescharacteristic
329C22#1798599511yescharacteristic
330C22C2×#1455605511yescharacteristic
331C22#1798601611yescharacteristic
332C4—21no—
333C4—22no—
334C4—21no—
335C4—22no—
336C4—22no—
337C4—22no—
338C4—22no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4—81no—
343C2—42no—
344C2#2258668811yescharacteristic
345C2#2258674811yescharacteristic
346C2#2258679511yescharacteristic
347C2#2258683111yescharacteristic
348C2#2258691211yescharacteristic
349C2#2258694711yescharacteristic
350C2#2258699711yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2855C2
socleC23×C71C2×C22⋊C4
prime radicalC24×C71C22⋊C4
cosocleC2×C42×C71C23
hypercenter#2855D71
hypocenterC71#7661
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
3. #22586166C2semidirect
4. #22586163C2semidirect
181. C71#7661semidirect

Maximal subgroups

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

subgroupindex
#225883202
C4×C22⋊(C71⋊C8)2
#225861662
#225861632
C71×#28552
#766171