Finite Group Databasebeta

← #31889055

Subgroups — #31889055

Order 18176.

788 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22588439C212yesmaximal, complemented
2C4×#14556043C211yesmaximal, characteristic
3#22586860C212yesmaximal, complemented
4#22586678C211yesmaximal, characteristic
5C2×#552×C71C211yesmaximal, characteristic
6C2×#14556043—21no—
7#17985963C2212yes—
8C4×C22⋊Dic71C2212yes—
9#17985766—21no—
10#17985758—24no—
11#17985736—24no—
12#17985725—21no—
13#17985723—21no—
14#552×C71—24no—
15#552×C71—24no—
16C2×#126×C71—21no—
17C2×#125×C71—21no—
18C2×#125×C71C411yescharacteristic, complemented
19C2×#123×C71—21no—
20C2×C22⋊Q8×C71C411yescharacteristic, complemented
21C2×C22⋊Q8×C71C2211yescharacteristic
22C2×C4×Q8×C71C411yescharacteristic, complemented
23C2×C4×D4×C71C2211yescharacteristic
24C2×#117×C71—21no—
25C2×#117×C71C2211yescharacteristic
26C22×C4⋊C4×C71C411yescharacteristic, complemented
27C2×#11765849—22no—
28C2×C22⋊Dic71—22no—
29C2×C4×Dic71—22no—
30#14556043—24no—
31C22⋊(C71⋊C8)—22no—
32C22⋊(C71⋊C8)—22no—
33C71⋊#95—22no—
34C22×Q8×C71C2×C411yescharacteristic
35C22×D4×C71D411yescharacteristic
36C23×C4×C71—21no—
37C23×C4×C71C2×C411yescharacteristic
38C23×C4×C71C2311yescharacteristic
39#126×C71—28no—
40#125×C71—28no—
41#125×C71—22no—
42#125×C71—22no—
43#123×C71—28no—
44C22⋊Q8×C71—22no—
45C22⋊Q8×C71—22no—
46C22⋊Q8×C71—22no—
47C22⋊Q8×C71D414yes—
48C4×Q8×C71—22no—
49C4×Q8×C71—22no—
50C4×D4×C71—22no—
51C4×D4×C71—22no—
52#117×C71—28no—
53#117×C71—22no—
54#117×C71D414yes—
55C2×C4⋊C4×C71—21no—
56C2×C4⋊C4×C71—21no—
57C2×C4⋊C4×C71—21no—
58C2×C4⋊C4×C71—21no—
59C2×C4⋊C4×C71—21no—
60C2×C4⋊C4×C71—21no—
61C2×C4⋊C4×C71—22no—
62C2×C4⋊C4×C71—22no—
63C2×C4⋊C4×C71C2×C411yescharacteristic
64C2×C4⋊C4×C71C2×C411yescharacteristic
65C2×C4⋊C4×C71C2×C411yescharacteristic
66C2×C4⋊C4×C71D411yescharacteristic
67C2×C22⋊C4×C71—21no—
68C2×C22⋊C4×C71—21no—
69C2×C22⋊C4×C71—21no—
70C2×C22⋊C4×C71D411yescharacteristic
71C2×C22⋊C4×C71D411yescharacteristic
72C2×C42×C71—21no—
73C2×C42×C71—21no—
74C2×C42×C71C2×C411yescharacteristic
75C22×Dic71—42no—
76#11765849—44no—
77C2×C71⋊C8—42no—
78C22⋊Dic71—24no—
79C22⋊Dic71—42no—
80C4×Dic71—44no—
81C24×C71C22⋊C411yescharacteristic
82C24×C71C2×D411yescharacteristic
83C2×Q8×C71—21no—
84C2×Q8×C71—42no—
85C2×Q8×C71C22⋊C412yes—
86C2×D4×C71—22no—
87C2×D4×C71—22no—
88C2×D4×C71—41no—
89C22×C4×C71—24no—
90C22×C4×C71—21no—
91C22×C4×C71—21no—
92C22×C4×C71—21no—
93C22×C4×C71—21no—
94C22×C4×C71—21no—
95C22×C4×C71—21no—
96C22×C4×C71—42no—
97C22×C4×C71—42no—
98C22×C4×C71—42no—
99C22×C4×C71—42no—
100C22×C4×C71C22⋊C411yescharacteristic
101C22×C4×C71C22⋊C411yescharacteristic
102C22×C4×C71C22⋊C411yescharacteristic
103C22×C4×C71C22×C411yescharacteristic
104C22×C4×C71C2×D412yes—
105C22×C4×C71C2×D412yes—
106C22×C4×C71C2×D411yescharacteristic
107C4⋊C4×C71—24no—
108C4⋊C4×C71—24no—
109C4⋊C4×C71—24no—
110C4⋊C4×C71—24no—
111C4⋊C4×C71—24no—
112C4⋊C4×C71—21no—
113C4⋊C4×C71—21no—
114C4⋊C4×C71—21no—
115C4⋊C4×C71—24no—
116C4⋊C4×C71—22no—
117C4⋊C4×C71C22⋊C412yes—
118C4⋊C4×C71C22⋊C412yes—
119C4⋊C4×C71C22⋊C412yes—
120C22⋊C4×C71—24no—
121C22⋊C4×C71—24no—
122C22⋊C4×C71—24no—
123C22⋊C4×C71—22no—
124C22⋊C4×C71—22no—
125C42×C71—24no—
126C42×C71—21no—
127C42×C71—21no—
128C42×C71—24no—
129C23×C71—21no—
130C23×C71—21no—
131C23×C71—21no—
132C23×C71—21no—
133C23×C71—41no—
134C23×C71—41no—
135C23×C71—41no—
136C23×C71—41no—
137C23×C71#10412yes—
138C23×C71C2×C22⋊C411yescharacteristic
139C23×C71#12012yes—
140Q8×C71—42no—
141Q8×C71—42no—
142D4×C71—44no—
143C2×C4×C71—22no—
144C2×C4×C71—24no—
145C2×C4×C71—21no—
146C2×C4×C71—22no—
147C2×C4×C71—22no—
148C2×C4×C71—22no—
149C2×C4×C71—21no—
150C2×C4×C71—21no—
151C2×C4×C71—22no—
152C2×C4×C71—21no—
153C2×C4×C71—21no—
154C2×C4×C71—41no—
155C2×C4×C71—42no—
156C2×C4×C71—42no—
157C2×C4×C71—42no—
158C2×C4×C71—48no—
159C2×C4×C71—42no—
160C2×C4×C71—41no—
161C2×C4×C71—41no—
162C2×C4×C71—41no—
163C2×C4×C71—41no—
164C2×C4×C71—42no—
165C2×C4×C71—41no—
166C2×C4×C71—41no—
167C2×C4×C71C2×C22⋊C411yescharacteristic
168C2×C4×C71C2×C22⋊C411yescharacteristic
169C2×C4×C71#12012yes—
170C2×Dic71—44no—
171C2×Dic71—82no—
172C2×Dic71—82no—
173C71⋊C8—82no—
174C22×C71—21no—
175C22×C71—21no—
176C22×C71—22no—
177C22×C71—22no—
178C22×C71—22no—
179C22×C71—22no—
180C22×C71—42no—
181C22×C71—42no—
182C22×C71—41no—
183C22×C71—41no—
184C22×C71—81no—
185C22×C71—81no—
186C22×C71#37911yescharacteristic
187C22×C71C2×#10411yescharacteristic
188C22×C71#42111yescharacteristic
189C4×C71—22no—
190C4×C71—21no—
191C4×C71—41no—
192C4×C71—42no—
193C4×C71—41no—
194C4×C71—42no—
195C4×C71—41no—
196C4×C71—42no—
197C4×C71—42no—
198C4×C71#41012yes—
199Dic71—84no—
200#8363—711nomaximal, Sylow
201C2×C71—21no—
202C2×C71—21no—
203C2×C71—42no—
204C2×C71—42no—
205C2×C71#178211yescharacteristic
206C2×C71#178911yescharacteristic
207C2×C71#179611yescharacteristic
208C2×#552D7111yescharacteristic
209#1826—712no—
210C4×#99—711no—
211#1508—712no—
212#1465—711no—
213C71#836311yescharacteristic, complemented, Sylow
214#552—24no—
215#552—24no—
216C2×#126—21no—
217C2×#125—21no—
218C2×#125Dic7111yescharacteristic, complemented
219C2×#123—21no—
220C2×C22⋊Q8Dic7111yescharacteristic, complemented
221C2×C22⋊Q8C2×D7111yescharacteristic
222C2×C4×Q8Dic7111yescharacteristic, complemented
223C2×C4×D4C2×D7111yescharacteristic
224C2×#117—21no—
225C2×#117C2×D7111yescharacteristic
226C22×C4⋊C4Dic7111yescharacteristic, complemented
227C2×#99—1421no—
228#407—712no—
229C4×C22⋊C4—712no—
230#342—1421no—
231#339—1424no—
232#328—1424no—
233#324—1421no—
234#323—1421no—
235C22×Q8C2×Dic7111yescharacteristic
236C22×D4C22⋊D7111yescharacteristic
237C23×C4—21no—
238C23×C4C2×Dic7111yescharacteristic
239C23×C4C22×D7111yescharacteristic
240C2×M16—1422no—
241#126—28no—
242#125—28no—
243#125—22no—
244#125—22no—
245#123—28no—
246C22⋊Q8—22no—
247C22⋊Q8—22no—
248C22⋊Q8—22no—
249C22⋊Q8C22⋊D7114yes—
250C4×Q8—22no—
251C4×Q8—22no—
252C4×D4—22no—
253C4×D4—22no—
254#117—28no—
255#117—22no—
256#117C22⋊D7114yes—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4C2×Dic7111yescharacteristic
266C2×C4⋊C4C2×Dic7111yescharacteristic
267C2×C4⋊C4C2×Dic7111yescharacteristic
268C2×C4⋊C4C22⋊D7111yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—1422no—
273C2×C22⋊C4C22⋊D7111yescharacteristic
274C2×C22⋊C4C22⋊D7111yescharacteristic
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—1422no—
278C2×C42C2×Dic7111yescharacteristic
279#99—1424no—
280C22⋊C8—1422no—
281C22⋊C8—1422no—
282#95—1422no—
283C24C22⋊Dic7111yescharacteristic
284C24C2×C22⋊D7111yescharacteristic
285C2×Q8—21no—
286C2×Q8—42no—
287C2×Q8C22⋊Dic7112yes—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—41no—
291C22×C4—24no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—2842no—
303C22×C4C22⋊Dic7111yescharacteristic
304C22×C4C22⋊Dic7111yescharacteristic
305C22×C4C22⋊Dic7111yescharacteristic
306C22×C4C22×Dic7111yescharacteristic
307C22×C4C2×C22⋊D7112yes—
308C22×C4C2×C22⋊D7112yes—
309C22×C4C2×C22⋊D7111yescharacteristic
310M16—2844no—
311C2×C8—2842no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—24no—
321C4⋊C4—22no—
322C4⋊C4C22⋊Dic7112yes—
323C4⋊C4C22⋊Dic7112yes—
324C4⋊C4C22⋊Dic7112yes—
325C22⋊C4—24no—
326C22⋊C4—24no—
327C22⋊C4—24no—
328C22⋊C4—22no—
329C22⋊C4—22no—
330C22⋊C4—2842no—
331C22⋊C4—1424no—
332C42—24no—
333C42—21no—
334C42—21no—
335C42—24no—
336C42—2844no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23#1455605512yes—
346C23C2×C22⋊Dic7111yescharacteristic
347C23#1455609712yes—
348Q8—42no—
349Q8—42no—
350D4—44no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—48no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—2844no—
376C2×C4—5682no—
377C2×C4—5682no—
378C2×C4C2×C22⋊Dic7111yescharacteristic
379C2×C4C2×C22⋊Dic7111yescharacteristic
380C2×C4#1455609712yes—
381C8—5682no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—42no—
389C22—42no—
390C22—41no—
391C22—41no—
392C22—81no—
393C22—81no—
394C22#1798585511yescharacteristic
395C22C2×#1455605511yescharacteristic
396C22#1798601611yescharacteristic
397C4—22no—
398C4—21no—
399C4—41no—
400C4—42no—
401C4—41no—
402C4—42no—
403C4—42no—
404C4—42no—
405C4—41no—
406C4—5684no—
407C4#1798597312yes—
408C2—21no—
409C2—21no—
410C2—42no—
411C2—42no—
412C2#2258825811yescharacteristic
413C2#2258828411yescharacteristic
414C2#2258831511yescharacteristic
subgroupG/H
centerC22#17985855
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC2×#552×C71C2
socleC22×C71#379
prime radicalC22×D4×C71D4
cosocleC23×C4×C71C23
hypercenterC2×#552D71
hypocenterC71#8363
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#552

Complements

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

NH ≅ G/Ntype
1. #22588439C2semidirect
3. #22586860C2semidirect
18. C2×#125×C71C4semidirect
20. C2×C22⋊Q8×C71C4semidirect
22. C2×C4×Q8×C71C4semidirect
26. C22×C4⋊C4×C71C4semidirect
213. C71#8363semidirect
218. C2×#125Dic71semidirect
220. C2×C22⋊Q8Dic71semidirect
222. C2×C4×Q8Dic71semidirect
226. C22×C4⋊C4Dic71semidirect

Maximal subgroups

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

subgroupindex
#225884392
C4×#145560432
#225868602
#225866782
C2×#552×C712
#836371