Finite Group Databasebeta

← #31906905

Subgroups — #31906905

Order 18176.

470 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22586681C211yesmaximal, characteristic
2#22586678C211yesmaximal, characteristic
3#22586288C212yesmaximal, complemented
4#22586270C212yesmaximal, complemented
5C4×#123×C71C211yesmaximal, characteristic
6#17986056—21no—
7#17985963—21no—
8#17985725—21no—
9#17985723—21no—
10#17985723C2212yes—
11#17985723C2212yes—
12C2×#123×C71C411yescharacteristic
13C2×C4×D4×C71C2211yescharacteristic
14C22×C42×C71C411yescharacteristic
15#388×C71—21no—
16#387×C71—21no—
17#386×C71C411yescharacteristic
18#385×C71C2211yescharacteristic
19#381×C71C2211yescharacteristic
20C4×C4⋊C4×C71—21no—
21C4×C22⋊C4×C71—21no—
22C4×C22⋊C4×C71C411yescharacteristic
23C2×#11765849—41no—
24C4⋊(C71⋊C8)—41no—
25C22⋊(C71⋊C8)—22no—
26C22⋊(C71⋊C8)—22no—
27C22⋊(C71⋊C8)—41no—
28#14556039—41no—
29C22×D4×C71C2×C411yescharacteristic
30C23×C4×C71C2×C411yescharacteristic
31C23×C4×C71C2×C411yescharacteristic
32C23×C4×C71C2×C411yescharacteristic
33C23×C4×C71D411yescharacteristic
34#123×C71—22no—
35#123×C71—22no—
36C4×D4×C71—41no—
37C4×D4×C71—41no—
38C2×C4⋊C4×C71—21no—
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71D411yescharacteristic
41C2×C22⋊C4×C71—21no—
42C2×C22⋊C4×C71—21no—
43C2×C22⋊C4×C71C2×C411yescharacteristic
44C2×C22⋊C4×C71C2×C411yescharacteristic
45C2×C22⋊C4×C71D411yescharacteristic
46C2×C22⋊C4×C71C2311yescharacteristic
47C2×C42×C71—21no—
48C2×C42×C71—21no—
49C2×C42×C71—21no—
50C2×C42×C71—41no—
51C2×C42×C71—41no—
52C2×C42×C71D411yescharacteristic
53#95×C71—21no—
54#95×C71—21no—
55#95×C71—21no—
56#95×C71—21no—
57#95×C71—21no—
58#11765849—82no—
59C2×C71⋊C8—42no—
60C2×C71⋊C8—42no—
61C24×C71C22⋊C411yescharacteristic
62C24×C71C22×C411yescharacteristic
63C2×D4×C71—21no—
64C2×D4×C71—21no—
65C2×D4×C71—41no—
66C2×D4×C71—41no—
67C22×C4×C71—21no—
68C22×C4×C71—21no—
69C22×C4×C71—21no—
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—21no—
75C22×C4×C71—21no—
76C22×C4×C71—42no—
77C22×C4×C71—42no—
78C22×C4×C71—42no—
79C22×C4×C71—41no—
80C22×C4×C71—41no—
81C22×C4×C71—41no—
82C22×C4×C71C22⋊C411yescharacteristic
83C22×C4×C71C22⋊C411yescharacteristic
84C22×C4×C71C22⋊C411yescharacteristic
85C22×C4×C71M1612yescomplemented
86C22×C4×C71C2×D411yescharacteristic
87C22×C4×C71C2×D411yescharacteristic
88C4⋊C4×C71—24no—
89C4⋊C4×C71—41no—
90C4⋊C4×C71—42no—
91C22⋊C4×C71—24no—
92C22⋊C4×C71—21no—
93C22⋊C4×C71—41no—
94C22⋊C4×C71—41no—
95C22⋊C4×C71—42no—
96C22⋊C4×C71—41no—
97C22⋊C4×C71M1612yescomplemented
98C42×C71—21no—
99C42×C71—21no—
100C42×C71—42no—
101C42×C71—42no—
102C42×C71—41no—
103C42×C71—41no—
104C23×C71—21no—
105C23×C71—21no—
106C23×C71—21no—
107C23×C71—21no—
108C23×C71—21no—
109C23×C71—41no—
110C23×C71—41no—
111C23×C71—41no—
112C23×C71—41no—
113C23×C71C2×C22⋊C411yescharacteristic
114C23×C71C2×M1611yescharacteristic
115C23×C71C2×M1611yescharacteristic
116D4×C71—42no—
117D4×C71—42no—
118C2×C4×C71—22no—
119C2×C4×C71—22no—
120C2×C4×C71—22no—
121C2×C4×C71—22no—
122C2×C4×C71—21no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—42no—
129C2×C4×C71—41no—
130C2×C4×C71—41no—
131C2×C4×C71—41no—
132C2×C4×C71—42no—
133C2×C4×C71—42no—
134C2×C4×C71—41no—
135C2×C4×C71—41no—
136C2×C4×C71—41no—
137C2×C4×C71—42no—
138C2×C4×C71—42no—
139C2×C4×C71—41no—
140C2×C4×C71—41no—
141C2×C4×C71—42no—
142C2×C4×C71—41no—
143C2×C4×C71—42no—
144C2×C4×C71—41no—
145C2×C4×C71—41no—
146C2×C4×C71—41no—
147C2×C4×C71—81no—
148C2×C4×C71#9911yescharacteristic
149C2×C4×C71#9911yescharacteristic
150C71⋊C8—82no—
151C71⋊C8—82no—
152C22×C71—21no—
153C22×C71—21no—
154C22×C71—21no—
155C22×C71—21no—
156C22×C71—41no—
157C22×C71—41no—
158C22×C71—42no—
159C22×C71—42no—
160C22×C71—41no—
161C22×C71—41no—
162C22×C71—41no—
163C22×C71—41no—
164C22×C71—41no—
165C22×C71—81no—
166C22×C71#40711yescharacteristic
167C22×C71C2×#9911yescharacteristic
168C22×C71#41311yescharacteristic
169C4×C71—21no—
170C4×C71—42no—
171C4×C71—41no—
172C4×C71—41no—
173C4×C71—41no—
174C4×C71—42no—
175C4×C71—41no—
176C4×C71—42no—
177C4×C71—81no—
178C4×C71#35212yes—
179#11526—711nomaximal, Sylow
180C2×C71—21no—
181C2×C71—21no—
182C2×C71—41no—
183C2×C71—41no—
184C2×C71—81no—
185C2×C71#146511yescharacteristic
186C2×C71C2×#35211yescharacteristic
187C2×C71#212011yescharacteristic
188C4×#123D7111yescharacteristic
189#1466—711no—
190#1465—711no—
191#1326—712no—
192#1317—712no—
193C71#1152611yescharacteristic, complemented, Sylow
194C2×#123Dic7111yescharacteristic
195C2×C4×D4C2×D7111yescharacteristic
196C22×C42Dic7111yescharacteristic
197#432—1421no—
198#407—1421no—
199#388—21no—
200#387—21no—
201#386Dic7111yescharacteristic
202#385C2×D7111yescharacteristic
203#381C2×D7111yescharacteristic
204C4×C4⋊C4—21no—
205C4×C22⋊C4—21no—
206C4×C22⋊C4Dic7111yescharacteristic
207#324—1421no—
208#323—712no—
209#323—712no—
210#323—1421no—
211C22×D4C2×Dic7111yescharacteristic
212C23×C4C2×Dic7111yescharacteristic
213C23×C4C2×Dic7111yescharacteristic
214C23×C4C2×Dic7111yescharacteristic
215C23×C4C22⋊D7111yescharacteristic
216C2×M16—2841no—
217#123—22no—
218#123—22no—
219C4×D4—41no—
220C4×D4—41no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4C22⋊D7111yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4C2×Dic7111yescharacteristic
227C2×C22⋊C4C2×Dic7111yescharacteristic
228C2×C22⋊C4C22⋊D7111yescharacteristic
229C2×C22⋊C4C22×D7111yescharacteristic
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—41no—
234C2×C42—41no—
235C2×C42C22⋊D7111yescharacteristic
236C4⋊C8—2841no—
237C22⋊C8—2841no—
238C22⋊C8—1422no—
239C22⋊C8—1422no—
240#97—2841no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95—21no—
246C24C22⋊Dic7111yescharacteristic
247C24C22×Dic7111yescharacteristic
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—41no—
251C2×D4—41no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4C22⋊Dic7111yescharacteristic
268C22×C4C22⋊Dic7111yescharacteristic
269C22×C4C22⋊Dic7111yescharacteristic
270C22×C4#1176584912yescomplemented
271C22×C4C2×C22⋊D7111yescharacteristic
272C22×C4C2×C22⋊D7111yescharacteristic
273M16—5682no—
274C2×C8—2842no—
275C2×C8—2842no—
276C4⋊C4—24no—
277C4⋊C4—41no—
278C4⋊C4—42no—
279C22⋊C4—24no—
280C22⋊C4—21no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—42no—
284C22⋊C4—41no—
285C22⋊C4#1176584912yescomplemented
286C42—21no—
287C42—21no—
288C42—42no—
289C42—42no—
290C42—41no—
291C42—41no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23C2×C22⋊Dic7111yescharacteristic
302C23C2×#1176584911yescharacteristic
303C23C2×#1176584911yescharacteristic
304D4—42no—
305D4—42no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—81no—
336C2×C4#1455604311yescharacteristic
337C2×C4#1455604311yescharacteristic
338C8—5682no—
339C8—5682no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—42no—
347C22—42no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—81no—
354C22#1798596311yescharacteristic
355C22C2×#1455604311yescharacteristic
356C22#1798598311yescharacteristic
357C4—21no—
358C4—42no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—42no—
363C4—41no—
364C4—42no—
365C4—81no—
366C4#1798578912yes—
367C2—21no—
368C2—21no—
369C2—41no—
370C2—41no—
371C2—81no—
372C2#2258667811yescharacteristic
373C2C2×#1798578911yescharacteristic
374C2#2258998111yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC24×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
FittingC4×#123×C71C2
socleC22×C71C2×#99
prime radicalC22×D4×C71C2×C4
cosocleC2×C22⋊C4×C71C23
hypercenterC4×#123D71
hypocenterC71#11526
LCSG ▷ C24×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#123

Complements

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

NH ≅ G/Ntype
3. #22586288C2semidirect
4. #22586270C2semidirect
85. C22×C4×C71M16semidirect
97. C22⋊C4×C71M16semidirect
193. C71#11526semidirect
270. C22×C4#11765849semidirect
285. C22⋊C4#11765849semidirect

Maximal subgroups

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

subgroupindex
#225866812
#225866782
#225862882
#225862702
C4×#123×C712
#1152671