Finite Group Databasebeta

← #31897501

Subgroups — #31897501

Order 18176.

572 conjugacy classes of subgroups.

The subgroup lattice has 396 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#22586183C212yesmaximal
4#22586183C212yesmaximal
5C2×#387×C71C211yesmaximal, characteristic
6C2×C4⋊(C71⋊C8)—21no—
7C2×C22⋊(C71⋊C8)C2212yes—
8C2×C22⋊(C71⋊C8)C2212yes—
9C2×C4×C71⋊C8—21no—
10#17985754—21no—
11#17985754—21no—
12C22×C4⋊C4×C71C411yescharacteristic
13C22×C4⋊C4×C71C411yescharacteristic
14C22×C42×C71C2211yescharacteristic
15#387×C71—21no—
16#387×C71—21no—
17#387×C71—21no—
18#387×C71—21no—
19C2×#95×C71C411yescharacteristic
20C2×#95×C71C411yescharacteristic
21C2×#95×C71C2211yescharacteristic
22C2×#95×C71C2211yescharacteristic
23C22×C71⋊C8—22no—
24C22×C71⋊C8—22no—
25C4⋊(C71⋊C8)—42no—
26C22⋊(C71⋊C8)—22no—
27C22⋊(C71⋊C8)—24no—
28C22⋊(C71⋊C8)Q814yes—
29C4×C71⋊C8—42no—
30C23×C4×C71C2×C411yescharacteristic
31C23×C4×C71C2×C411yescharacteristic
32C23×C4×C71C2×C411yescharacteristic
33C23×C4×C71C2×C411yescharacteristic
34C23×C4×C71C2×C411yescharacteristic
35C23×C4×C71C2×C411yescharacteristic
36C23×C4×C71C2311yescharacteristic
37C2×C4⋊C4×C71—21no—
38C2×C4⋊C4×C71—21no—
39C2×C4⋊C4×C71—22no—
40C2×C4⋊C4×C71—21no—
41C2×C4⋊C4×C71—21no—
42C2×C4⋊C4×C71—42no—
43C2×C4⋊C4×C71C814yescomplemented
44C2×C42×C71—42no—
45C2×C42×C71D411yescharacteristic
46C2×C42×C71D411yescharacteristic
47C2×C42×C71D411yescharacteristic
48C2×C42×C71D411yescharacteristic
49#95×C71—21no—
50#95×C71—21no—
51#95×C71—21no—
52#95×C71—21no—
53#95×C71—21no—
54#95×C71—21no—
55#95×C71—21no—
56#95×C71—21no—
57C2×C71⋊C8—24no—
58C2×C71⋊C8—42no—
59C2×C71⋊C8—42no—
60C2×C71⋊C8—42no—
61C2×C71⋊C8—42no—
62C2×C71⋊C8—42no—
63C24×C71C22×C411yescharacteristic
64C22×C4×C71—21no—
65C22×C4×C71—21no—
66C22×C4×C71—21no—
67C22×C4×C71—21no—
68C22×C4×C71—22no—
69C22×C4×C71—22no—
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71—22no—
73C22×C4×C71—22no—
74C22×C4×C71—41no—
75C22×C4×C71—41no—
76C22×C4×C71—41no—
77C22×C4×C71—41no—
78C22×C4×C71—41no—
79C22×C4×C71—41no—
80C22×C4×C71—41no—
81C22×C4×C71—41no—
82C22×C4×C71—41no—
83C22×C4×C71—41no—
84C22×C4×C71—41no—
85C22×C4×C71—41no—
86C22×C4×C71—41no—
87C22×C4×C71C22⋊C411yescharacteristic
88C22×C4×C71C22⋊C411yescharacteristic
89C22×C4×C71C22⋊C411yescharacteristic
90C22×C4×C71C22⋊C411yescharacteristic
91C22×C4×C71C2×C812yes—
92C22×C4×C71C2×C812yes—
93C22×C4×C71C2×C812yes—
94C22×C4×C71M1612yes—
95C22×C4×C71C2×D411yescharacteristic
96C22×C4×C71C2×D411yescharacteristic
97C22×C4×C71C2×Q812yes—
98C22×C4×C71#4112yes—
99C4⋊C4×C71—24no—
100C4⋊C4×C71—24no—
101C4⋊C4×C71—42no—
102C4⋊C4×C71—42no—
103C42×C71—22no—
104C42×C71—22no—
105C42×C71—41no—
106C42×C71—41no—
107C23×C71—21no—
108C23×C71—21no—
109C23×C71—21no—
110C23×C71—21no—
111C23×C71C2×C22⋊C411yescharacteristic
112C23×C71C4×Q812yes—
113C23×C71C22×C811yescharacteristic
114C23×C71C2×M1611yescharacteristic
115C23×C71#13112yes—
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—21no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—22no—
127C2×C4×C71—22no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—42no—
132C2×C4×C71—42no—
133C2×C4×C71—42no—
134C2×C4×C71—42no—
135C2×C4×C71—44no—
136C2×C4×C71—42no—
137C2×C4×C71—42no—
138C2×C4×C71—41no—
139C2×C4×C71—41no—
140C2×C4×C71—41no—
141C2×C4×C71—41no—
142C2×C4×C71—42no—
143C2×C4×C71—42no—
144C2×C4×C71—41no—
145C2×C4×C71—41no—
146C2×C4×C71—41no—
147C2×C4×C71—41no—
148C2×C4×C71—41no—
149C2×C4×C71—41no—
150C2×C4×C71—82no—
151C2×C4×C71—82no—
152C2×C4×C71C22⋊C812yes—
153C2×C4×C71C22⋊C812yes—
154C2×C4×C71#9912yes—
155C2×C4×C71C22⋊Q812yes—
156C2×C4×C71#12411yescharacteristic
157C2×C4×C71C4⋊Q811yescharacteristic
158C71⋊C8—44no—
159C71⋊C8—82no—
160C22×C71—21no—
161C22×C71—21no—
162C22×C71—21no—
163C22×C71—21no—
164C22×C71—22no—
165C22×C71—22no—
166C22×C71—22no—
167C22×C71—22no—
168C22×C71#39111yescharacteristic
169C22×C71C2×C22⋊C811yescharacteristic
170C22×C71#40811yescharacteristic
171C22×C71C2×#9911yescharacteristic
172C22×C71#41011yescharacteristic
173C22×C71#41311yescharacteristic
174C22×C71#41311yescharacteristic
175C22×C71C8×Q812yes—
176C22×C71#44612yes—
177C4×C71—22no—
178C4×C71—22no—
179C4×C71—22no—
180C4×C71—22no—
181C4×C71—21no—
182C4×C71—21no—
183C4×C71—42no—
184C4×C71—42no—
185C4×C71—41no—
186C4×C71—81no—
187C4×C71—81no—
188#9880—711nomaximal, Sylow
189C2×C71—22no—
190C2×C71—22no—
191C2×C71#145811yescharacteristic
192C2×C71#145911yescharacteristic
193C2×C71#146811yescharacteristic
194C2×C71#147311yescharacteristic
195C2×C71#198411yescharacteristic
196C2×C71#199611yescharacteristic
197C2×C71#199811yescharacteristic
198C2×#387D7111yescharacteristic
199#1797—711no—
200C4×C22⋊C8—711no—
201#1280—712no—
202#1280—712no—
203C71#988011yescharacteristic, complemented, Sylow
204C22×C4⋊C4Dic7111yescharacteristic
205C22×C4⋊C4Dic7111yescharacteristic
206C22×C42C2×D7111yescharacteristic
207C2×C4⋊C8—1421no—
208C2×C22⋊C8—712no—
209C2×C22⋊C8—712no—
210C2×C4×C8—1421no—
211#387—21no—
212#387—21no—
213#387—21no—
214#387—21no—
215C2×#95Dic7111yescharacteristic
216C2×#95Dic7111yescharacteristic
217C2×#95C2×D7111yescharacteristic
218C2×#95C2×D7111yescharacteristic
219#336—1421no—
220#336—1421no—
221C23×C4C2×Dic7111yescharacteristic
222C23×C4C2×Dic7111yescharacteristic
223C23×C4C2×Dic7111yescharacteristic
224C23×C4C2×Dic7111yescharacteristic
225C23×C4C2×Dic7111yescharacteristic
226C23×C4C2×Dic7111yescharacteristic
227C23×C4C22×D7111yescharacteristic
228C22×C8—1422no—
229C22×C8—1422no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4C71⋊C814yescomplemented
237C2×C42—42no—
238C2×C42C22⋊D7111yescharacteristic
239C2×C42C22⋊D7111yescharacteristic
240C2×C42C22⋊D7111yescharacteristic
241C2×C42C22⋊D7111yescharacteristic
242C4⋊C8—2842no—
243C22⋊C8—714no—
244C22⋊C8—1424no—
245C22⋊C8—1422no—
246C4×C8—2842no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255C24C22×Dic7111yescharacteristic
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4C22⋊Dic7111yescharacteristic
280C22×C4C22⋊Dic7111yescharacteristic
281C22×C4C22⋊Dic7111yescharacteristic
282C22×C4C22⋊Dic7111yescharacteristic
283C22×C4C2×C71⋊C812yes—
284C22×C4C2×C71⋊C812yes—
285C22×C4C2×C71⋊C812yes—
286C22×C4#1176584912yes—
287C22×C4C2×C22⋊D7111yescharacteristic
288C22×C4C2×C22⋊D7111yescharacteristic
289C22×C4Q8×D7112yes—
290C22×C4#1176586612yes—
291C2×C8—2842no—
292C2×C8—2842no—
293C2×C8—2842no—
294C2×C8—2842no—
295C2×C8—2842no—
296C2×C8—1424no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C42—22no—
302C42—22no—
303C42—41no—
304C42—41no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23C2×C22⋊Dic7111yescharacteristic
310C23Q8×Dic7112yes—
311C23C22×C71⋊C811yescharacteristic
312C23C2×#1176584911yescharacteristic
313C23#1455613512yes—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4C22⋊(C71⋊C8)12yes—
351C2×C4C22⋊(C71⋊C8)12yes—
352C2×C4#1455604312yes—
353C2×C4#1455610412yes—
354C2×C4#1455611711yescharacteristic
355C2×C4#1455612611yescharacteristic
356C8—2844no—
357C8—5682no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22#1798590611yescharacteristic
367C22C2×C22⋊(C71⋊C8)11yescharacteristic
368C22#1798596611yescharacteristic
369C22C2×#1455604311yescharacteristic
370C22#1798597311yescharacteristic
371C22#1798598311yescharacteristic
372C22#1798598311yescharacteristic
373C22Q8×C71⋊C812yes—
374C22#1798613012yes—
375C4—22no—
376C4—22no—
377C4—22no—
378C4—22no—
379C4—21no—
380C4—21no—
381C4—42no—
382C4—42no—
383C4—41no—
384C4—81no—
385C4—81no—
386C2—22no—
387C2—22no—
388C2#2258665411yescharacteristic
389C2#2258665811yescharacteristic
390C2#2258668811yescharacteristic
391C2#2258670911yescharacteristic
392C2#2258927711yescharacteristic
393C2#2258932611yescharacteristic
394C2#2258933411yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC23×C71C22×C8
FrattiniC23×C4C22×D71
FittingC2×#387×C71C2
socleC23×C71C2×C22⋊C4
prime radicalC24×C71C22×C4
cosocleC23×C4×C71C23
hypercenterC2×#387D71
hypocenterC71#9880
LCSG ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#387

Complements

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

NH ≅ G/Ntype
43. C2×C4⋊C4×C71C8semidirect
203. C71#9880semidirect
236. C2×C4⋊C4C71⋊C8semidirect

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
#225861832
C2×#387×C712
#988071