Finite Group Databasebeta

← #31891731

Subgroups — #31891731

Order 18176.

388 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22588511C211yesmaximal, characteristic, complemented
2#22587128C211yesmaximal, characteristic, complemented
3#22586779C211yesmaximal, characteristic
4#22586722C211yesmaximal, characteristic
5#22586229C211yesmaximal, characteristic, complemented
6#22586176C211yesmaximal, characteristic, complemented
7C71×#2991C211yesmaximal, characteristic
8#17986052C2211yescharacteristic
9C2×#14556063—22no—
10#17986023C2211yescharacteristic
11C4×#11765849C2211yescharacteristic
12C2×C4×C71⋊C8C2211yescharacteristic
13#17985751—21no—
14#17985749—21no—
15Q8⋊(C71⋊C8)—21no—
16Q8⋊(C71⋊C8)C411yescharacteristic
17Q8⋊(C71⋊C8)C411yescharacteristic
18D4⋊(C71⋊C8)—21no—
19D4⋊(C71⋊C8)C411yescharacteristic
20D4⋊(C71⋊C8)C411yescharacteristic
21C4×#41×C71C2211yescharacteristic
22C8×Q8×C71—21no—
23#436×C71—22no—
24#436×C71—21no—
25#431×C71—21no—
26#431×C71C2211yescharacteristic
27#423×C71—21no—
28#423×C71C2211yescharacteristic
29C2×#11765849—21no—
30C2×#11765849—21no—
31C22×C71⋊C8—21no—
32#14556063—44no—
33C4⋊(C71⋊C8)—21no—
34C4⋊(C71⋊C8)—21no—
35C4⋊(C71⋊C8)C2×C411yescharacteristic
36C4⋊(C71⋊C8)C2×C411yescharacteristic
37C22⋊(C71⋊C8)—21no—
38#14556039—21no—
39C4×C71⋊C8—21no—
40C4×C71⋊C8—21no—
41C4×C71⋊C8—21no—
42C4×C71⋊C8C2×C411yescharacteristic
43C4×C71⋊C8C2×C411yescharacteristic
44C2×#41×C71Q811yescharacteristic
45C2×M16×C71—22no—
46C2×M16×C71D412yes—
47C4×Q8×C71C2×C411yescharacteristic
48C4×Q8×C71D411yescharacteristic
49C4×D4×C71—21no—
50C4×D4×C71—21no—
51C4×D4×C71C2×C411yescharacteristic
52C4×D4×C71D411yescharacteristic
53#117×C71—21no—
54#117×C71Q811yescharacteristic
55C2×C42×C71—21no—
56C2×C42×C71C2311yescharacteristic
57C4⋊C8×C71—22no—
58C4⋊C8×C71—21no—
59C4⋊C8×C71—21no—
60C4⋊C8×C71D411yescharacteristic
61C4⋊C8×C71D411yescharacteristic
62C22⋊C8×C71—22no—
63C22⋊C8×C71—21no—
64C4×C8×C71—22no—
65C4×C8×C71—21no—
66#11765849—22no—
67#11765849—41no—
68#11765849—42no—
69C2×C71⋊C8—22no—
70C2×C71⋊C8—22no—
71C2×C71⋊C8—21no—
72C2×C71⋊C8—21no—
73C2×C71⋊C8—21no—
74C2×C71⋊C8—21no—
75C2×C71⋊C8—41no—
76C2×C71⋊C8—41no—
77#41×C71—22no—
78#41×C71—41no—
79C2×Q8×C71C4⋊C411yescharacteristic
80C2×D4×C71—21no—
81C2×D4×C71C4⋊C411yescharacteristic
82C22×C4×C71—21no—
83C22×C4×C71—21no—
84C22×C4×C71—21no—
85C22×C4×C71C2×D411yescharacteristic
86C22×C4×C71C2×Q811yescharacteristic
87C22×C4×C71#4111yescharacteristic
88M16×C71—42no—
89M16×C71—82no—
90C2×C8×C71—21no—
91C2×C8×C71—22no—
92C2×C8×C71—21no—
93C4⋊C4×C71—21no—
94C4⋊C4×C71—21no—
95C4⋊C4×C71C4⋊C411yescharacteristic
96C4⋊C4×C71C4⋊C411yescharacteristic
97C22⋊C4×C71—21no—
98C22⋊C4×C71—21no—
99C22⋊C4×C71—21no—
100C42×C71—21no—
101C42×C71—21no—
102C42×C71—21no—
103C42×C71C22×C411yescharacteristic
104C42×C71C2×D411yescharacteristic
105C42×C71C2×D411yescharacteristic
106C42×C71#4111yescharacteristic
107C23×C71—21no—
108C23×C71C22⋊Q811yescharacteristic
109Q8×C71—21no—
110Q8×C71#10812yescomplemented
111D4×C71—21no—
112D4×C71—21no—
113D4×C71—42no—
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—42no—
124C2×C4×C71—41no—
125C2×C4×C71—41no—
126C2×C4×C71—41no—
127C2×C4×C71—41no—
128C2×C4×C71C2×C4⋊C411yescharacteristic
129C2×C4×C71C4×D411yescharacteristic
130C2×C4×C71C4×D411yescharacteristic
131C2×C4×C71#12011yescharacteristic
132C2×C4×C71C22⋊Q811yescharacteristic
133C2×C4×C71#12311yescharacteristic
134C8×C71—42no—
135C8×C71—41no—
136C8×C71—41no—
137C71⋊C8—22no—
138C71⋊C8—22no—
139C71⋊C8—42no—
140C71⋊C8—41no—
141C71⋊C8—41no—
142C71⋊C8—81no—
143C22×C71—21no—
144C22×C71—21no—
145C22×C71—21no—
146C22×C71—41no—
147C22×C71—41no—
148C22×C71—41no—
149C22×C71#38511yescharacteristic
150C4×C71—21no—
151C4×C71—22no—
152C4×C71—21no—
153C4×C71—21no—
154C4×C71—41no—
155C4×C71—41no—
156C4×C71C2×#10811yescharacteristic
157C4×C71#43011yescharacteristic
158C4×C71#44311yescharacteristic
159C4×C71#44411yescharacteristic
160C4×C71#45311yescharacteristic
161C4×C71#45611yescharacteristic
162#8899—711nomaximal, Sylow
163C2×C71—41no—
164C2×C71—81no—
165C2×C71#185511yescharacteristic
166C2×C71#185911yescharacteristic
167C2×C71#186611yescharacteristic
168#2991D7111yescharacteristic
169#1848—711no—
170#1574—711no—
171#1489—711no—
172#1476—711no—
173#1299—711no—
174#1278—711no—
175C71#889911yescharacteristic, complemented, Sylow
176C4×#41C2×D7111yescharacteristic
177C8×Q8—21no—
178#436—22no—
179#436—21no—
180#431—21no—
181#431—711no—
182#431C2×D7111yescharacteristic
183C2×#108—1422no—
184#423—21no—
185#423—711no—
186#423C2×D7111yescharacteristic
187C4×M16—711no—
188C2×C4×C8—711no—
189#335—1421no—
190#334—1421no—
191Q8⋊C8—711no—
192Q8⋊C8—711no—
193Q8⋊C8—1421no—
194D4⋊C8—711no—
195D4⋊C8—711no—
196D4⋊C8—1421no—
197C2×#41Dic14211yescharacteristic
198C2×M16—22no—
199C2×M16—1421no—
200C2×M16—1421no—
201C2×M16C22⋊D7112yes—
202C22×C8—1421no—
203C4×Q8C4×D7111yescharacteristic
204C4×Q8C22⋊D7111yescharacteristic
205C4×D4—21no—
206C4×D4—21no—
207C4×D4C4×D7111yescharacteristic
208C4×D4C22⋊D7111yescharacteristic
209#117—21no—
210#117Dic14211yescharacteristic
211C2×C42—21no—
212C2×C42C22×D7111yescharacteristic
213#108—2844no—
214C4⋊C8—22no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—711no—
218C4⋊C8—711no—
219C4⋊C8—1421no—
220C4⋊C8—1421no—
221C4⋊C8C22⋊D7111yescharacteristic
222C4⋊C8C22⋊D7111yescharacteristic
223C22⋊C8—22no—
224C22⋊C8—21no—
225C22⋊C8—1421no—
226#97—1421no—
227C4×C8—22no—
228C4×C8—21no—
229C4×C8—711no—
230C4×C8—711no—
231C4×C8—1421no—
232C4×C8—1421no—
233C4×C8—1421no—
234#41—22no—
235#41—41no—
236C2×Q8Dic71⋊C411yescharacteristic
237C2×D4—21no—
238C2×D4Dic71⋊C411yescharacteristic
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4C2×C22⋊D7111yescharacteristic
243C22×C4C2×Dic14211yescharacteristic
244C22×C4#1176586511yescharacteristic
245M16—42no—
246M16—82no—
247M16—2841no—
248M16—2842no—
249M16—1422no—
250C2×C8—21no—
251C2×C8—22no—
252C2×C8—21no—
253C2×C8—2841no—
254C2×C8—2841no—
255C2×C8—1421no—
256C2×C8—1421no—
257C2×C8—1422no—
258C2×C8—1422no—
259C2×C8—1421no—
260C2×C8—1421no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4Dic71⋊C411yescharacteristic
264C4⋊C4Dic71⋊C411yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42C2×C4×D7111yescharacteristic
272C42C2×C22⋊D7111yescharacteristic
273C42C2×C22⋊D7111yescharacteristic
274C42#1176586511yescharacteristic
275C23—21no—
276C23#1455610711yescharacteristic
277Q8—21no—
278Q8#1455606312yescomplemented
279D4—21no—
280D4—21no—
281D4—42no—
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—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4C2×Dic71⋊C411yescharacteristic
297C2×C4C4×C22⋊D7111yescharacteristic
298C2×C4C4×C22⋊D7111yescharacteristic
299C2×C4#1455609711yescharacteristic
300C2×C4#1455610711yescharacteristic
301C2×C4#1455611111yescharacteristic
302C8—42no—
303C8—41no—
304C8—41no—
305C8—2842no—
306C8—2841no—
307C8—2841no—
308C8—5681no—
309C8—1422no—
310C8—1422no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22#1798587211yescharacteristic
318C4—21no—
319C4—22no—
320C4—21no—
321C4—21no—
322C4—41no—
323C4—41no—
324C4C2×#1455606311yescharacteristic
325C4#1798604711yescharacteristic
326C4#1798611411yescharacteristic
327C4#1798611911yescharacteristic
328C4#1798616711yescharacteristic
329C4#1798618211yescharacteristic
330C2—41no—
331C2—81no—
332C2#2258854511yescharacteristic
333C2#2258856611yescharacteristic
334C2#2258861511yescharacteristic
subgroupG/H
centerC2×C4#14556097
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2991C2
socleC22×C71#385
prime radicalC2×#41×C71Q8
cosocleC2×C42×C71C23
hypercenter#2991D71
hypocenterC71#8899
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2991

Complements

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

NH ≅ G/Ntype
1. #22588511C2semidirect
2. #22587128C2semidirect
5. #22586229C2semidirect
6. #22586176C2semidirect
110. Q8×C71#108semidirect
175. C71#8899semidirect
278. Q8#14556063semidirect

Maximal subgroups

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

subgroupindex
#225885112
#225871282
#225867792
#225867222
#225862292
#225861762
C71×#29912
#889971