Finite Group Databasebeta

← #32005351

Subgroups — #32005351

Order 18176.

898 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986130C211yesmaximal, characteristic, complemented
2C2×D4×C71⋊C8C211yesmaximal, characteristic
3#22589277C211yesmaximal, characteristic, complemented
4#22589135C211yesmaximal, characteristic
5#22588955C214yesmaximal, complemented
6#22588457C211yesmaximal, characteristic, complemented
7#22588118C214yesmaximal, complemented
8C4×#14556039C211yesmaximal, characteristic, complemented
9C4×#124×C71C211yesmaximal, characteristic
10C23×C71⋊C8C2212yes—
11#17986130—24no—
12D4×C71⋊C8—24no—
13C2×C4⋊(C71⋊C8)C2212yes—
14C2×C4⋊(C71⋊C8)C2211yescharacteristic
15C2×C4⋊(C71⋊C8)C2211yescharacteristic
16C2×C22⋊(C71⋊C8)C2212yes—
17C2×C22⋊(C71⋊C8)C2214yes—
18C2×#14556039C2212yes—
19C2×#14556039C2212yes—
20C2×#14556039C2212yes—
21C2×C4×C71⋊C8C2211yescharacteristic
22C2×C4×C71⋊C8C2211yescharacteristic
23#17985754C2214yes—
24#17985754C2214yes—
25C2×#124×C71C411yescharacteristic
26C2×C4×Q8×C71C2211yescharacteristic
27C2×C4×D4×C71C2211yescharacteristic
28#391×C71C411yescharacteristic
29#390×C71C411yescharacteristic
30#388×C71C414yes—
31#382×C71C411yescharacteristic
32C4×C22⋊C4×C71C2214yes—
33C43×C71C2211yescharacteristic
34C22×C71⋊C8—24no—
35C22×C71⋊C8—28no—
36C22×C71⋊C8C2312yes—
37C22×C71⋊C8C2312yes—
38C22×C71⋊C8C2311yescharacteristic
39C22×C71⋊C8C2312yes—
40C22×C71⋊C8C2311yescharacteristic
41C4⋊(C71⋊C8)—24no—
42C4⋊(C71⋊C8)—22no—
43C4⋊(C71⋊C8)—22no—
44C22⋊(C71⋊C8)—24no—
45C22⋊(C71⋊C8)—28no—
46#14556039—24no—
47#14556039—24no—
48#14556039—24no—
49C4×C71⋊C8—22no—
50C4×C71⋊C8D414yes—
51C22×Q8×C71C2×C411yescharacteristic
52C22×D4×C71C2×C411yescharacteristic
53C23×C4×C71C2312yes—
54#124×C71—24no—
55C4×Q8×C71—22no—
56C4×Q8×C71—22no—
57C4×D4×C71—22no—
58C4×D4×C71—22no—
59C2×C4⋊C4×C71C2×C411yescharacteristic
60C2×C4⋊C4×C71C2×C411yescharacteristic
61C2×C4⋊C4×C71C2×C412yes—
62C2×C22⋊C4×C71C2×C412yes—
63C2×C22⋊C4×C71C2×C414yes—
64C2×C22⋊C4×C71C2×C414yes—
65C2×C42×C71C2×C411yescharacteristic
66C2×C42×C71C2×C411yescharacteristic
67C2×C42×C71C2×C412yes—
68C2×C42×C71C2312yes—
69C2×C42×C71C2312yes—
70C2×C42×C71C2311yescharacteristic
71#95×C71C2×C414yes—
72#95×C71C2×C414yes—
73C2×C71⋊C8—22no—
74C2×C71⋊C8—22no—
75C2×C71⋊C8—22no—
76C2×C71⋊C8—22no—
77C2×C71⋊C8—22no—
78C2×C71⋊C8—24no—
79C2×C71⋊C8—24no—
80C2×C71⋊C8—48no—
81C2×C71⋊C8C2×D412yes—
82C2×C71⋊C8C2×D412yes—
83C2×C71⋊C8#4114yes—
84C2×C71⋊C8#4114yes—
85C24×C71C22×C412yes—
86C2×Q8×C71—22no—
87C2×Q8×C71—22no—
88C2×Q8×C71—22no—
89C2×D4×C71—22no—
90C2×D4×C71—22no—
91C2×D4×C71M1614yes—
92C22×C4×C71—22no—
93C22×C4×C71—22no—
94C22×C4×C71—24no—
95C22×C4×C71—24no—
96C22×C4×C71C22×C412yes—
97C22×C4×C71C22×C412yes—
98C22×C4×C71C22×C412yes—
99C22×C4×C71C22×C412yes—
100C22×C4×C71C22×C412yes—
101C22×C4×C71C22×C411yescharacteristic
102C22×C4×C71C22×C411yescharacteristic
103C22×C4×C71C2411yescharacteristic
104C4⋊C4×C71—22no—
105C4⋊C4×C71—22no—
106C4⋊C4×C71—24no—
107C22⋊C4×C71—28no—
108C22⋊C4×C71—24no—
109C22⋊C4×C71—44no—
110C42×C71—22no—
111C42×C71—22no—
112C42×C71—22no—
113C42×C71—22no—
114C42×C71—22no—
115C42×C71—22no—
116C42×C71—42no—
117C42×C71C2×D412yes—
118C42×C71#4112yes—
119C23×C71—22no—
120C23×C71—24no—
121C23×C71—22no—
122C23×C71—24no—
123C23×C71C2×M1614yes—
124C23×C71C23×C411yescharacteristic
125Q8×C71—44no—
126D4×C71—28no—
127C2×C4×C71—22no—
128C2×C4×C71—22no—
129C2×C4×C71—22no—
130C2×C4×C71—22no—
131C2×C4×C71—22no—
132C2×C4×C71—22no—
133C2×C4×C71—24no—
134C2×C4×C71—22no—
135C2×C4×C71—22no—
136C2×C4×C71—22no—
137C2×C4×C71—22no—
138C2×C4×C71—22no—
139C2×C4×C71—22no—
140C2×C4×C71—24no—
141C2×C4×C71—42no—
142C2×C4×C71—44no—
143C2×C4×C71—42no—
144C2×C4×C71C4×D412yes—
145C2×C4×C71C4×D412yes—
146C2×C4×C71C2×M1612yes—
147C2×C4×C71#13112yes—
148C2×C4×C71C22×D411yescharacteristic
149C2×C4×C71C2×#4112yes—
150C2×C4×C71C2×#4112yes—
151C2×C4×C71C2×#4111yescharacteristic
152C71⋊C8—24no—
153C71⋊C8—22no—
154C71⋊C8—42no—
155C71⋊C8—42no—
156C71⋊C8#12414yescomplemented
157C22×C71—24no—
158C22×C71—28no—
159C22×C71—42no—
160C22×C71—44no—
161C22×C71—42no—
162C22×C71C2×C4×D411yescharacteristic
163C22×C71C4×#4112yes—
164C22×C71C22×M1611yescharacteristic
165C22×C71C2×#13111yescharacteristic
166C22×C71#56812yes—
167C4×C71—22no—
168C4×C71—42no—
169C4×C71—42no—
170C4×C71—42no—
171C4×C71—42no—
172C4×C71—42no—
173C4×C71#43514yes—
174C4×C71C2×#12411yescharacteristic
175C4×C71#52912yes—
176C4×C71#53211yescharacteristic
177#21633—711nomaximal, Sylow
178C2×C71—44no—
179C2×C71C4×#12411yescharacteristic
180C2×C71C2×#43511yescharacteristic
181C2×C71#293211yescharacteristic
182C2×C71#296912yes—
183C2×C71#297012yes—
184C2×#446—711no—
185C2×C8×D4—711no—
186C4×#124D7111yescharacteristic
187#1984—711no—
188#1954—711no—
189#1924—714no—
190#1831—711no—
191#1752—714no—
192C4×#97—711no—
193C71#2163311yescharacteristic, complemented, Sylow
194C23×C8—712no—
195C2×#124Dic7111yescharacteristic
196C2×C4×Q8C2×D7111yescharacteristic
197C2×C4×D4C2×D7111yescharacteristic
198#446—1424no—
199C8×D4—1424no—
200C2×C4⋊C8—711no—
201C2×C4⋊C8—711no—
202C2×C4⋊C8—712no—
203C2×C22⋊C8—712no—
204C2×C22⋊C8—714no—
205C2×#97—712no—
206C2×#97—712no—
207C2×#97—712no—
208C2×C4×C8—711no—
209C2×C4×C8—711no—
210#391Dic7111yescharacteristic
211#390Dic7111yescharacteristic
212#388Dic7114yes—
213#382Dic7111yescharacteristic
214C4×C22⋊C4C2×D7114yes—
215C43C2×D7111yescharacteristic
216#336—714no—
217#336—714no—
218C22×Q8C2×Dic7111yescharacteristic
219C22×D4C2×Dic7111yescharacteristic
220C23×C4C22×D7112yes—
221C22×C8—711no—
222C22×C8—712no—
223C22×C8—711no—
224C22×C8—712no—
225C22×C8—712no—
226C22×C8—1424no—
227C22×C8—1428no—
228#124—24no—
229C4×Q8—22no—
230C4×Q8—22no—
231C4×D4—22no—
232C4×D4—22no—
233C2×C4⋊C4C2×Dic7111yescharacteristic
234C2×C4⋊C4C2×Dic7111yescharacteristic
235C2×C4⋊C4C2×Dic7112yes—
236C2×C22⋊C4C2×Dic7112yes—
237C2×C22⋊C4C2×Dic7114yes—
238C2×C22⋊C4C2×Dic7114yes—
239C2×C42C2×Dic7111yescharacteristic
240C2×C42C2×Dic7111yescharacteristic
241C2×C42C2×Dic7112yes—
242C2×C42C22×D7112yes—
243C2×C42C22×D7112yes—
244C2×C42C22×D7111yescharacteristic
245C4⋊C8—1424no—
246C4⋊C8—1422no—
247C4⋊C8—1422no—
248C22⋊C8—1428no—
249C22⋊C8—1424no—
250#97—1424no—
251#97—1424no—
252#97—1424no—
253C4×C8—714no—
254C4×C8—1422no—
255#95C2×Dic7114yes—
256#95C2×Dic7114yes—
257C24C22×Dic7112yes—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4#1176584914yes—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4C22×Dic7112yes—
269C22×C4C22×Dic7112yes—
270C22×C4C22×Dic7112yes—
271C22×C4C22×Dic7112yes—
272C22×C4C22×Dic7112yes—
273C22×C4C22×Dic7111yescharacteristic
274C22×C4C22×Dic7111yescharacteristic
275C22×C4C23×D7111yescharacteristic
276C2×C8—2848no—
277C2×C8—712no—
278C2×C8—714no—
279C2×C8—712no—
280C2×C8—714no—
281C2×C8—1422no—
282C2×C8—1422no—
283C2×C8—1424no—
284C2×C8—1424no—
285C2×C8—1422no—
286C2×C8—1422no—
287C2×C8—1422no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—24no—
291C22⋊C4—28no—
292C22⋊C4—24no—
293C22⋊C4—44no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—42no—
301C42D4×D7112yes—
302C42#1176586412yes—
303C23—22no—
304C23—24no—
305C23—22no—
306C23—24no—
307C23C2×#1176584914yes—
308C23C23×Dic7111yescharacteristic
309Q8—44no—
310D4—28no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—24no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—24no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4D4×Dic7112yes—
329C2×C4D4×Dic7112yes—
330C2×C4C2×#1176584912yes—
331C2×C4#1455613512yes—
332C2×C4C2×D4×D7111yescharacteristic
333C2×C4#41×D7112yes—
334C2×C4#41×D7112yes—
335C2×C4C2×#1176586411yescharacteristic
336C8—2842no—
337C8—2842no—
338C8—714no—
339C8—1424no—
340C8—1422no—
341C22—24no—
342C22—28no—
343C22—42no—
344C22—42no—
345C22—44no—
346C22C2×D4×Dic7111yescharacteristic
347C22#41×Dic7112yes—
348C22C22×#1176584911yescharacteristic
349C22C2×#1455613511yescharacteristic
350C22#1798676612yes—
351C4—22no—
352C4—42no—
353C4—42no—
354C4—42no—
355C4—42no—
356C4—42no—
357C4#1798607114yes—
358C4#124×D7111yescharacteristic
359C4#1798652812yes—
360C4#1798654811yescharacteristic
361C2—44no—
362C2#124×Dic7111yescharacteristic
363C2C2×#1798607111yescharacteristic
364C2#2259705811yescharacteristic
365C2#2259734512yes—
366C2#2259735412yes—
subgroupG/H
center, FrattiniC22×C4C23×D71
commutator, socleC23×C71C23×C4
FittingC4×#124×C71C2
prime radicalC22×D4×C71C2×C4
cosocleC22×C4×C71C24
hypercenterC4×#124D71
hypocenterC71#21633
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C22×C4 ◁ C4×#124

Complements

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

NH ≅ G/Ntype
1. C2×#17986130C2semidirect
3. #22589277C2semidirect
5. #22588955C2semidirect
6. #22588457C2semidirect
7. #22588118C2semidirect
8. C4×#14556039C2semidirect
156. C71⋊C8#124semidirect
193. C71#21633semidirect

Maximal subgroups

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

subgroupindex
C2×#179861302
C2×D4×C71⋊C82
#225892772
#225891352
#225889552
#225884572
#225881182
C4×#145560392
C4×#124×C712
#2163371