Finite Group Databasebeta

← #31907369

Subgroups — #31907369

Order 18176.

422 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C8×D4×D71C211yesmaximal, characteristic
2#22590142C211yesmaximal, characteristic, complemented
3#22590137C211yesmaximal, characteristic, complemented
4#22586296C211yesmaximal, characteristic, complemented
5D4⋊(C71⋊C16)C211yesmaximal, characteristic, complemented
6#22586294C211yesmaximal, characteristic, complemented
7C71×D4⋊C16C211yesmaximal, characteristic, complemented
8C22×C8×D71—21no—
9C4×D4×D71C411yescharacteristic
10C8×D284C2211yescharacteristic
11D4×C71⋊C8C411yescharacteristic
12#17986064C411yescharacteristic
13C8×C22⋊D71—21no—
14(C22⋊D71)⋊C8—21no—
15C4⋊C8×D71C411yescharacteristic
16C22⋊C8×D71—21no—
17C4×C8×D71C2211yescharacteristic
18C4⋊(C71⋊C16)C2211yescharacteristic, complemented
19Dic71⋊C16—21no—
20(C2×D71)⋊C16—21no—
21(C2×D71)⋊C16—21no—
22#17985774—21no—
23C4×C71⋊C16C2211yescharacteristic, complemented
24C8×D4×C71C2211yescharacteristic
25C4⋊C16×C71C2211yescharacteristic, complemented
26C4×C16×C71C2211yescharacteristic, complemented
27C2×D4×D71C811yescharacteristic
28C22×C4×D71—21no—
29C22×C71⋊C8—21no—
30C2×C8×D71—21no—
31C2×C8×D71—21no—
32C2×C8×D71—21no—
33C2×C8×D71—42no—
34C2×C8×D71—42no—
35C2×C8×D71D411yescharacteristic
36C2×C8×D71D411yescharacteristic
37C4×D284C2×C411yescharacteristic
38D4×Dic71C811yescharacteristic
39#14556090C811yescharacteristic
40C4×C22⋊D71—21no—
41(C22⋊D71)⋊C4—21no—
42C4⋊C4×D71C811yescharacteristic
43C22⋊C4×D71—21no—
44C42×D71C2×C411yescharacteristic
45C2×C71⋊C16—21no—
46C2×C71⋊C16—21no—
47C4⋊(C71⋊C8)C2×C411yescharacteristic
48Dic71⋊C8—21no—
49C22⋊(C71⋊C8)—21no—
50(C2×D71)⋊C8—21no—
51(C2×D71)⋊C8—21no—
52C4×C71⋊C8C2×C411yescharacteristic
53C8×Dic71D411yescharacteristic
54C8×Dic71D411yescharacteristic
55C22×C8×C71—21no—
56C4×D4×C71C2×C411yescharacteristic
57C2×C16×C71—21no—
58C2×C16×C71—21no—
59C4⋊C8×C71C2×C411yescharacteristic
60C22⋊C8×C71—21no—
61C4×C8×C71C2311yescharacteristic
62C23×D71—21no—
63C2×D284C2×C811yescharacteristic
64C2×C22⋊D71—21no—
65D4×D71—22no—
66D4×D71—22no—
67C22×Dic71—21no—
68C2×C4×D71—21no—
69C2×C4×D71—21no—
70C2×C4×D71—21no—
71C2×C4×D71—42no—
72C2×C4×D71—42no—
73C2×C4×D71C22⋊C411yescharacteristic
74C2×C4×D71C2×C811yescharacteristic
75C2×C4×D71M1611yescharacteristic
76C2×C71⋊C8—21no—
77C2×C71⋊C8—42no—
78C2×C71⋊C8C22⋊C411yescharacteristic
79C2×C71⋊C8C22⋊C411yescharacteristic
80C8×D71—21no—
81C8×D71—21no—
82C8×D71—41no—
83C8×D71—42no—
84C8×D71—42no—
85C8×D71—41no—
86C8×D71—41no—
87C4⋊Dic71C2×C811yescharacteristic
88Dic71⋊C4—21no—
89C22⋊Dic71—21no—
90(C2×D71)⋊C4—21no—
91(C2×D71)⋊C4—21no—
92C4×Dic71C22⋊C411yescharacteristic
93C4×Dic71C2×C811yescharacteristic
94C4×Dic71M1611yescharacteristic
95C71⋊C16—41no—
96C71⋊C16—41no—
97C2×D4×C71C2×C811yescharacteristic
98C22×C4×C71—21no—
99C2×C8×C71—21no—
100C2×C8×C71—41no—
101C2×C8×C71—41no—
102C2×C8×C71C2×D411yescharacteristic
103C2×C8×C71C2×D411yescharacteristic
104C4⋊C4×C71C2×C811yescharacteristic
105C22⋊C4×C71—21no—
106C42×C71C22×C411yescharacteristic
107C16×C71—41no—
108C16×C71—41no—
109C23×C71—21no—
110C22×D71—24no—
111C22×D71—21no—
112C22×D71—42no—
113C22×D71—42no—
114C22×D71C22⋊C811yescharacteristic
115D4×C71—21no—
116D4×C71M3212yes—
117C2×C4×C71—21no—
118C2×C4×C71—41no—
119C2×C4×C71—41no—
120C2×C4×C71C2×C22⋊C411yescharacteristic
121C2×C4×C71C22×C811yescharacteristic
122C2×C4×C71C2×M1611yescharacteristic
123C22⋊D71—24no—
124C2×Dic71—21no—
125C2×Dic71—42no—
126C2×Dic71C22⋊C811yescharacteristic
127C2×Dic71C22⋊C811yescharacteristic
128C2×Dic71C22⋊C811yescharacteristic
129D284—21no—
130D284M3212yes—
131C4×D71—22no—
132C4×D71—21no—
133C4×D71—21no—
134C4×D71—21no—
135C4×D71—21no—
136C4×D71—42no—
137C4×D71—42no—
138C4×D71—41no—
139C4×D71—41no—
140C8×C71—21no—
141C8×C71—41no—
142C8×C71AGL(1,ℤ/8)11yescharacteristic
143C8×C71AGL(1,ℤ/8)11yescharacteristic
144C71⋊C8—21no—
145C71⋊C8—21no—
146C71⋊C8—41no—
147C22×C71—22no—
148C22×C71—41no—
149C22×C71—41no—
150C22×C71C2×C22⋊C811yescharacteristic
151C2×D71—21no—
152C2×D71—21no—
153C2×D71—22no—
154C2×D71—48no—
155C2×D71—41no—
156C2×D71—41no—
157C2×D71#35012yes—
158C4×C71—21no—
159C4×C71—41no—
160C4×C71#41811yescharacteristic
161C4×C71#42111yescharacteristic
162C4×C71C2×M3211yescharacteristic
163C4×C71C2×M3211yescharacteristic
164Dic71—21no—
165Dic71—21no—
166Dic71—21no—
167Dic71—41no—
168#11631—711nomaximal, Sylow
169C2×C71—42no—
170C2×C71#147811yescharacteristic
171C2×C71#211211yescharacteristic
172C2×C71C2×#35011yescharacteristic
173D71—22no—
174D71—42no—
175C2×C8×D4—711no—
176#2163—711no—
177#2162—711no—
178D4⋊C16—711no—
179D4⋊C16—711no—
180D4⋊C16—711no—
181D4⋊C16D7111yescharacteristic, complemented
182C71#1163111yescharacteristic, complemented, Sylow
183C23×C8—1421no—
184C2×C4×D4—711no—
185C8×D4—711no—
186C8×D4—711no—
187C8×D4—711no—
188C8×D4—1421no—
189C8×D4—1421no—
190C8×D4C2×D7111yescharacteristic
191C2×C4⋊C8—711no—
192C2×C22⋊C8—1421no—
193C2×C4×C8—711no—
194C4⋊C16—711no—
195C4⋊C16—1421no—
196C4⋊C16C2×D7111yescharacteristic, complemented
197C22⋊C16—1421no—
198C22⋊C16—1421no—
199#346—1421no—
200C4×C16—711no—
201C4×C16C2×D7111yescharacteristic, complemented
202C22×D4—711no—
203C23×C4—1421no—
204C22×C8—21no—
205C22×C8—2842no—
206C22×C8—2842no—
207C22×C8—711no—
208C22×C8—711no—
209C22×C8—1421no—
210C22×C8—1421no—
211C22×C8—1421no—
212C22×C8—1421no—
213C4×D4—711no—
214C4×D4—711no—
215C4×D4—711no—
216C4×D4—1421no—
217C4×D4—1421no—
218C4×D4C4×D7111yescharacteristic
219C2×C4⋊C4—711no—
220C2×C22⋊C4—1421no—
221C2×C42—711no—
222C2×C16—21no—
223C2×C16—21no—
224C2×C16—1421no—
225C2×C16—1421no—
226C4⋊C8—711no—
227C4⋊C8—1421no—
228C4⋊C8C4×D7111yescharacteristic
229C22⋊C8—21no—
230C22⋊C8—1421no—
231C22⋊C8—1421no—
232C22⋊C8—1421no—
233C4×C8—711no—
234C4×C8—711no—
235C4×C8—711no—
236C4×C8C22×D7111yescharacteristic
237C24—1421no—
238C2×D4—711no—
239C2×D4—1421no—
240C2×D4—1422no—
241C2×D4—1422no—
242C2×D4C8×D7111yescharacteristic
243C22×C4—21no—
244C22×C4—2842no—
245C22×C4—2842no—
246C22×C4—711no—
247C22×C4—711no—
248C22×C4—711no—
249C22×C4—1421no—
250C22×C4—1421no—
251C22×C4—1421no—
252C22×C4—1421no—
253C2×C8—21no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—2841no—
257C2×C8—2842no—
258C2×C8—2842no—
259C2×C8—2842no—
260C2×C8—2841no—
261C2×C8—2841no—
262C2×C8—711no—
263C2×C8—711no—
264C2×C8—1421no—
265C2×C8—1421no—
266C2×C8—1421no—
267C2×C8D4×D7111yescharacteristic
268C2×C8D4×D7111yescharacteristic
269C4⋊C4—711no—
270C4⋊C4—1421no—
271C4⋊C4C8×D7111yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—1421no—
274C22⋊C4—1421no—
275C22⋊C4—1421no—
276C42—711no—
277C42—711no—
278C42—711no—
279C42C2×C4×D7111yescharacteristic
280C16—41no—
281C16—41no—
282C16—2841no—
283C16—2841no—
284C23—21no—
285C23—2842no—
286C23—2842no—
287C23—711no—
288C23—1424no—
289C23—1421no—
290D4—21no—
291D4—712no—
292D4—1424no—
293D4—1421no—
294D4#1455606712yes—
295C2×C4—21no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—2842no—
299C2×C4—2842no—
300C2×C4—2842no—
301C2×C4—2841no—
302C2×C4—2841no—
303C2×C4—711no—
304C2×C4—711no—
305C2×C4—711no—
306C2×C4—1422no—
307C2×C4—1421no—
308C2×C4—1421no—
309C2×C4—1421no—
310C2×C4—1421no—
311C2×C4—1421no—
312C2×C4C22⋊C4×D7111yescharacteristic
313C2×C4C2×C8×D7111yescharacteristic
314C2×C4M16×D7111yescharacteristic
315C8—21no—
316C8—41no—
317C8—2841no—
318C8—1421no—
319C8—1421no—
320C8#1455615211yescharacteristic
321C8#1455615511yescharacteristic
322C22—22no—
323C22—41no—
324C22—41no—
325C22—2848no—
326C22—2841no—
327C22—2841no—
328C22—712no—
329C22—1422no—
330C22—1421no—
331C22—1421no—
332C22C22⋊C8×D7111yescharacteristic
333C4—21no—
334C4—41no—
335C4—2841no—
336C4—1421no—
337C4—1421no—
338C4—1421no—
339C4#1798600211yescharacteristic
340C4#1798601411yescharacteristic
341C4M32×D7111yescharacteristic
342C4C2×#1455606711yescharacteristic
343C2—42no—
344C2—2842no—
345C2—1422no—
346C2#2258672911yescharacteristic
347C2#2258994911yescharacteristic
348C2#350×D7111yescharacteristic
subgroupG/H
centerC2×C8D4×D71
commutatorC2×C4×C71C22×C8
FrattiniC4×C8C22×D71
FittingC71×D4⋊C16C2
socleC22×C71C2×C22⋊C8
prime radicalC2×D4×D71C8
cosocleC4×C8×C71C23
hypercenterD4⋊C16D71
hypocenterC71#11631
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C8 ◁ C4×C8 ◁ D4⋊C16

Complements

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

NH ≅ G/Ntype
2. #22590142C2semidirect
3. #22590137C2semidirect
4. #22586296C2semidirect
5. D4⋊(C71⋊C16)C2semidirect
6. #22586294C2semidirect
7. C71×D4⋊C16C2semidirect
18. C4⋊(C71⋊C16)C22semidirect
23. C4×C71⋊C16C22semidirect
25. C4⋊C16×C71C22semidirect
26. C4×C16×C71C22semidirect
181. D4⋊C16D71semidirect
182. C71#11631semidirect
196. C4⋊C16C2×D71semidirect
201. C4×C16C2×D71semidirect

Maximal subgroups

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

subgroupindex
C8×D4×D712
#225901422
#225901372
#225862962
D4⋊(C71⋊C16)2
#225862942
C71×D4⋊C162
#1163171