Finite Group Databasebeta

← #31885562

Subgroups — #31885562

Order 18176.

504 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22588408C211yesmaximal, characteristic
2#22588320C211yesmaximal, characteristic
3#22586166C212yesmaximal, complemented
4#22586163C212yesmaximal, complemented
5C4×C22⋊Q8×C71C211yesmaximal, characteristic
6C2×C4⋊(C71⋊C8)—21no—
7C2×C22⋊(C71⋊C8)—21no—
8C2×C22⋊(C71⋊C8)—21no—
9#17985754—21no—
10#17985754C2212yes—
11#17985754C2212yes—
12C2×C22⋊Q8×C71C411yescharacteristic
13C2×C4×Q8×C71C411yescharacteristic
14C22×C42×C71C2211yescharacteristic
15#391×C71C2211yescharacteristic
16#389×C71C2211yescharacteristic
17#387×C71—21no—
18#385×C71—21no—
19#380×C71C411yescharacteristic
20C4×C4⋊C4×C71—21no—
21C4×C4⋊C4×C71C411yescharacteristic
22C4×C22⋊C4×C71—21no—
23C22×C71⋊C8—22no—
24C22×C71⋊C8—22no—
25C4⋊(C71⋊C8)—42no—
26C22⋊(C71⋊C8)—42no—
27C22⋊(C71⋊C8)—42no—
28C22×Q8×C71C2×C411yescharacteristic
29C23×C4×C71C2×C411yescharacteristic
30C23×C4×C71D411yescharacteristic
31C23×C4×C71D411yescharacteristic
32C22⋊Q8×C71—22no—
33C22⋊Q8×C71C814yescomplemented
34C4×Q8×C71—42no—
35C2×C4⋊C4×C71—21no—
36C2×C4⋊C4×C71—21no—
37C2×C4⋊C4×C71—21no—
38C2×C4⋊C4×C71C2×C411yescharacteristic
39C2×C4⋊C4×C71C2×C411yescharacteristic
40C2×C4⋊C4×C71C2×C411yescharacteristic
41C2×C22⋊C4×C71—21no—
42C2×C22⋊C4×C71—21no—
43C2×C42×C71—21no—
44C2×C42×C71—21no—
45C2×C42×C71—42no—
46C2×C42×C71C2×C411yescharacteristic
47C2×C42×C71D411yescharacteristic
48C2×C42×C71D411yescharacteristic
49C2×C42×C71C2311yescharacteristic
50#95×C71—21no—
51#95×C71—21no—
52#95×C71—21no—
53#95×C71—21no—
54#95×C71—21no—
55C2×C71⋊C8—42no—
56C2×C71⋊C8—42no—
57C2×C71⋊C8—42no—
58C2×C71⋊C8—42no—
59C2×C71⋊C8—42no—
60C2×C71⋊C8—42no—
61C24×C71C22⋊C411yescharacteristic
62C2×Q8×C71—21no—
63C2×Q8×C71—41no—
64C2×Q8×C71—41no—
65C2×Q8×C71C2×C812yes—
66C22×C4×C71—21no—
67C22×C4×C71—21no—
68C22×C4×C71—22no—
69C22×C4×C71—21no—
70C22×C4×C71—21no—
71C22×C4×C71—22no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—41no—
75C22×C4×C71—41no—
76C22×C4×C71—41no—
77C22×C4×C71—41no—
78C22×C4×C71—41no—
79C22×C4×C71C22⋊C411yescharacteristic
80C22×C4×C71C22⋊C411yescharacteristic
81C22×C4×C71C22⋊C411yescharacteristic
82C22×C4×C71C2×C812yes—
83C22×C4×C71M1612yes—
84C22×C4×C71C22×C411yescharacteristic
85C22×C4×C71C2×D411yescharacteristic
86C22×C4×C71C2×D411yescharacteristic
87C4⋊C4×C71—24no—
88C4⋊C4×C71—21no—
89C4⋊C4×C71—41no—
90C4⋊C4×C71—41no—
91C4⋊C4×C71—42no—
92C4⋊C4×C71—41no—
93C4⋊C4×C71—41no—
94C4⋊C4×C71C2×C812yes—
95C22⋊C4×C71—24no—
96C22⋊C4×C71—42no—
97C42×C71—22no—
98C42×C71—22no—
99C42×C71—41no—
100C42×C71—41no—
101C42×C71—42no—
102C42×C71—42no—
103C23×C71—21no—
104C23×C71—21no—
105C23×C71—21no—
106C23×C71—21no—
107C23×C71—21no—
108C23×C71—21no—
109C23×C71C22⋊C812yes—
110C23×C71C2×C22⋊C411yescharacteristic
111Q8×C71—42no—
112Q8×C71—42no—
113C2×C4×C71—21no—
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—22no—
124C2×C4×C71—22no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—42no—
129C2×C4×C71—42no—
130C2×C4×C71—44no—
131C2×C4×C71—42no—
132C2×C4×C71—42no—
133C2×C4×C71—41no—
134C2×C4×C71—41no—
135C2×C4×C71—41no—
136C2×C4×C71—41no—
137C2×C4×C71—41no—
138C2×C4×C71—41no—
139C2×C4×C71—41no—
140C2×C4×C71—41no—
141C2×C4×C71—41no—
142C2×C4×C71—41no—
143C2×C4×C71C22⋊C812yes—
144C2×C4×C71#9912yes—
145C2×C4×C71C22×C811yescharacteristic
146C2×C4×C71C2×M1611yescharacteristic
147C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C71#13711yescharacteristic
149C2×C4×C71#13712yes—
150C71⋊C8—82no—
151C71⋊C8—82no—
152C22×C71—22no—
153C22×C71—22no—
154C22×C71—22no—
155C22×C71—41no—
156C22×C71—41no—
157C22×C71—41no—
158C22×C71—41no—
159C22×C71C2×C22⋊C811yescharacteristic
160C22×C71C2×#9911yescharacteristic
161C22×C71#41311yescharacteristic
162C22×C71#41711yescharacteristic
163C22×C71#41911yescharacteristic
164C22×C71#42111yescharacteristic
165C22×C71#42111yescharacteristic
166C4×C71—22no—
167C4×C71—21no—
168C4×C71—21no—
169C4×C71—22no—
170C4×C71—22no—
171C4×C71—21no—
172C4×C71—21no—
173C4×C71—42no—
174C4×C71—42no—
175C4×C71—41no—
176C4×C71—81no—
177C4×C71—81no—
178#7600—711nomaximal, Sylow
179C2×C71—42no—
180C2×C71#145911yescharacteristic
181C2×C71#147711yescharacteristic
182C2×C71#147911yescharacteristic
183C2×C71#151111yescharacteristic
184C2×C71#151311yescharacteristic
185C2×C71#153511yescharacteristic
186C2×C71#155911yescharacteristic
187C4×C22⋊Q8D7111yescharacteristic
188#1817—711no—
189#1797—711no—
190#1272—712no—
191#1271—712no—
192C71#760011yescharacteristic, complemented, Sylow
193C2×C22⋊Q8Dic7111yescharacteristic
194C2×C4×Q8Dic7111yescharacteristic
195C22×C42C2×D7111yescharacteristic
196C2×C4⋊C8—1421no—
197C2×C22⋊C8—1421no—
198C2×C22⋊C8—1421no—
199#391C2×D7111yescharacteristic
200#389C2×D7111yescharacteristic
201#387—21no—
202#385—21no—
203#380Dic7111yescharacteristic
204C4×C4⋊C4—21no—
205C4×C4⋊C4Dic7111yescharacteristic
206C4×C22⋊C4—21no—
207#336—712no—
208#336—712no—
209#336—1421no—
210C22×Q8C2×Dic7111yescharacteristic
211C23×C4C2×Dic7111yescharacteristic
212C23×C4C22⋊D7111yescharacteristic
213C23×C4C22⋊D7111yescharacteristic
214C22×C8—1422no—
215C22×C8—1422no—
216C22⋊Q8—22no—
217C22⋊Q8C71⋊C814yescomplemented
218C4×Q8—42no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4C2×Dic7111yescharacteristic
223C2×C4⋊C4C2×Dic7111yescharacteristic
224C2×C4⋊C4C2×Dic7111yescharacteristic
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—42no—
230C2×C42C2×Dic7111yescharacteristic
231C2×C42C22⋊D7111yescharacteristic
232C2×C42C22⋊D7111yescharacteristic
233C2×C42C22×D7111yescharacteristic
234C4⋊C8—2842no—
235C22⋊C8—2842no—
236C22⋊C8—2842no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—21no—
242C24C22⋊Dic7111yescharacteristic
243C2×Q8—21no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8C2×C71⋊C812yes—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4C22⋊Dic7111yescharacteristic
261C22×C4C22⋊Dic7111yescharacteristic
262C22×C4C22⋊Dic7111yescharacteristic
263C22×C4C2×C71⋊C812yes—
264C22×C4#1176584912yes—
265C22×C4C22×Dic7111yescharacteristic
266C22×C4C2×C22⋊D7111yescharacteristic
267C22×C4C2×C22⋊D7111yescharacteristic
268C2×C8—2842no—
269C2×C8—2842no—
270C2×C8—2842no—
271C2×C8—2842no—
272C2×C8—2842no—
273C2×C8—2842no—
274C4⋊C4—24no—
275C4⋊C4—21no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—42no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4C2×C71⋊C812yes—
282C22⋊C4—24no—
283C22⋊C4—42no—
284C42—22no—
285C42—22no—
286C42—41no—
287C42—41no—
288C42—42no—
289C42—42no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23C22⋊(C71⋊C8)12yes—
297C23C2×C22⋊Dic7111yescharacteristic
298Q8—42no—
299Q8—42no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4C22⋊(C71⋊C8)12yes—
331C2×C4#1455604312yes—
332C2×C4C22×C71⋊C811yescharacteristic
333C2×C4C2×#1176584911yescharacteristic
334C2×C4#1455615311yescharacteristic
335C2×C4#1455615811yescharacteristic
336C2×C4#1455615912yes—
337C8—5682no—
338C8—5682no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22C2×C22⋊(C71⋊C8)11yescharacteristic
347C22C2×#1455604311yescharacteristic
348C22#1798598311yescharacteristic
349C22#1798600011yescharacteristic
350C22#1798600811yescharacteristic
351C22#1798601611yescharacteristic
352C22#1798601611yescharacteristic
353C4—21no—
354C4—21no—
355C4—22no—
356C4—21no—
357C4—21no—
358C4—22no—
359C4—22no—
360C4—41no—
361C4—42no—
362C4—42no—
363C4—81no—
364C4—81no—
365C2—42no—
366C2#2258665811yescharacteristic
367C2#2258672711yescharacteristic
368C2#2258673511yescharacteristic
369C2#2258687311yescharacteristic
370C2#2258688311yescharacteristic
371C2#2258697111yescharacteristic
372C2#2258707011yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC2×C4×C71C22×C8
FrattiniC2×C42C22×D71
FittingC4×C22⋊Q8×C71C2
socleC23×C71C2×C22⋊C4
prime radicalC24×C71C22⋊C4
cosocleC2×C42×C71C23
hypercenterC4×C22⋊Q8D71
hypocenterC71#7600
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #22586166C2semidirect
4. #22586163C2semidirect
33. C22⋊Q8×C71C8semidirect
192. C71#7600semidirect
217. C22⋊Q8C71⋊C8semidirect

Maximal subgroups

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

subgroupindex
#225884082
#225883202
#225861662
#225861632
C4×C22⋊Q8×C712
#760071