Finite Group Databasebeta

← #11993647

Subgroups — #11993647

Order 1280.

376 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#681C211yesmaximal, characteristic
2C5×#1487C211yesmaximal, characteristic
3C2×Q8⋊#168C211yesmaximal, characteristic
4#10601313C211yesmaximal, characteristic
5#10601283C211yesmaximal, characteristic
6#10598236C211yesmaximal, characteristic
7#10598230C211yesmaximal, characteristic
8C2×C4×C5×Q8C2211yescharacteristic
9C5×#431C2211yescharacteristic
10C5×#423C2211yescharacteristic
11C2×#3954—21no—
12#65313—22no—
13#65313—21no—
14Q8×#168—21no—
15#65213—21no—
16#65213C2211yescharacteristic
17C2×C4⋊#168C2211yescharacteristic
18C4×#681—21no—
19C4×#681C2211yescharacteristic
20C2×C4×#168C2211yescharacteristic
21C5×Q8⋊C8—21no—
22C5×Q8⋊C8—21no—
23#64778—21no—
24Q8⋊#168C411yescharacteristic
25Q8⋊#168C411yescharacteristic
26Q8⋊#168C411yescharacteristic
27Q8⋊#168C411yescharacteristic
28#64700—22no—
29#8909—51nomaximal, Sylow
30C22×C5×Q8Q811yescharacteristic
31C2×C5×M16—21no—
32C4×C5×Q8—21no—
33C4×C5×Q8—21no—
34C4×C5×Q8C2×C411yescharacteristic
35C4×C5×Q8C2×C411yescharacteristic
36C4×C5×Q8D411yescharacteristic
37C4×C5×Q8D411yescharacteristic
38C2×C5×C4⋊C4—21no—
39C2×C5×C4⋊C4Q811yescharacteristic
40C2×C42×C5—21no—
41C2×C42×C5C2311yescharacteristic
42C2×#681—21no—
43C2×#681D411yescharacteristic
44C2×#681D411yescharacteristic
45C22×#168—21no—
46C22×#168—21no—
47C5×C4⋊C8—21no—
48C5×C4⋊C8—21no—
49C5×C4⋊C8—21no—
50C5×C22⋊C8—21no—
51C4×C5×C8—21no—
52#3954—42no—
53C4⋊#168—22no—
54C4⋊#168—22no—
55C4⋊#168—21no—
56C4⋊#168—21no—
57C4⋊#168C2×C411yescharacteristic
58C4⋊#168C2×C411yescharacteristic
59C4⋊#168D412yes—
60#3935—22no—
61#3935—21no—
62C4×#168—22no—
63C4×#168—21no—
64C4×#168—21no—
65C4×#168C2×C411yescharacteristic
66C4×#168C2×C411yescharacteristic
67Q8×M16—51no—
68#1950—51no—
69#1566—51no—
70#1487D511yescharacteristic
71C2×Q8⋊C8—51no—
72#1301—51no—
73#1278—51no—
74C2×C5×Q8—21no—
75C2×C5×Q8—22no—
76C2×C5×Q8—41no—
77C2×C5×Q8C4⋊C411yescharacteristic
78C2×C5×Q8C4⋊C411yescharacteristic
79C22×C4×C5—21no—
80C22×C4×C5—21no—
81C22×C4×C5C2×D411yescharacteristic
82C22×C4×C5C2×Q811yescharacteristic
83C22×C4×C5#4111yescharacteristic
84C5×M16—41no—
85C5×M16—41no—
86C2×C5×C8—41no—
87C2×C5×C8—41no—
88C5×C4⋊C4—21no—
89C5×C4⋊C4—21no—
90C5×C4⋊C4—21no—
91C5×C4⋊C4—21no—
92C5×C4⋊C4—21no—
93C5×C4⋊C4C4⋊C411yescharacteristic
94C5×C4⋊C4C4⋊C411yescharacteristic
95C42×C5—21no—
96C42×C5—21no—
97C42×C5—21no—
98C42×C5—21no—
99C42×C5C22×C411yescharacteristic
100C42×C5C2×D411yescharacteristic
101C42×C5C2×D411yescharacteristic
102C42×C5#4111yescharacteristic
103#681—21no—
104#681—21no—
105#681—21no—
106#681—41no—
107#681—41no—
108#681SD1612yes—
109C2×#168—21no—
110C2×#168—21no—
111C2×#168—21no—
112C2×#168—21no—
113C2×#168—21no—
114C2×#168—21no—
115C2×#168—22no—
116C2×#168—42no—
117C2×#168—41no—
118C2×#168—41no—
119C2×C4×Q8C2×D511yescharacteristic
120#446—102no—
121#446—101no—
122C8×Q8—101no—
123#431C2×D511yescharacteristic
124C2×#108—101no—
125#423—51no—
126#423—101no—
127#423C2×D511yescharacteristic
128C2×C4⋊C8—51no—
129C4×M16—51no—
130C4×M16—101no—
131C2×C4×C8—51no—
132#341—101no—
133#334—102no—
134Q8⋊C8—21no—
135Q8⋊C8—21no—
136Q8⋊C8—51no—
137Q8⋊C8—51no—
138Q8⋊C8—51no—
139Q8⋊C8—51no—
140C23×C5C22⋊Q811yescharacteristic
141C5×Q8—21no—
142C5×Q8—21no—
143C5×Q8—21no—
144C5×Q8—42no—
145C5×Q8#10812yescomplemented
146C2×C4×C5—21no—
147C2×C4×C5—21no—
148C2×C4×C5—21no—
149C2×C4×C5—21no—
150C2×C4×C5—21no—
151C2×C4×C5—21no—
152C2×C4×C5—41no—
153C2×C4×C5—42no—
154C2×C4×C5—81no—
155C2×C4×C5C2×C4⋊C411yescharacteristic
156C2×C4×C5C4×D411yescharacteristic
157C2×C4×C5C4×D411yescharacteristic
158C2×C4×C5#12011yescharacteristic
159C2×C4×C5C22⋊Q811yescharacteristic
160C2×C4×C5#12311yescharacteristic
161C2×C4×C5C2×SD1611yescharacteristic
162C2×C4×C5#13511yescharacteristic
163C2×C4×C5#13711yescharacteristic
164C2×C4×C5#13711yescharacteristic
165C5×C8—81no—
166C5×C8—81no—
167#168—22no—
168#168—22no—
169#168—41no—
170#168—41no—
171#168—41no—
172#168—41no—
173#168—42no—
174C22×Q8Dic1011yescharacteristic
175C2×M16—21no—
176C2×M16—51no—
177C2×M16—51no—
178C2×M16—101no—
179C22×C8—101no—
180C22×C8—101no—
181C4×Q8—21no—
182C4×Q8—21no—
183C4×Q8C4×D511yescharacteristic
184C4×Q8C4×D511yescharacteristic
185C4×Q8C22⋊D511yescharacteristic
186C4×Q8C22⋊D511yescharacteristic
187C2×C4⋊C4—21no—
188C2×C4⋊C4Dic1011yescharacteristic
189C2×C42—21no—
190C2×C42C22×D511yescharacteristic
191#108—202no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—21no—
195C4⋊C8—51no—
196C4⋊C8—51no—
197C4⋊C8—52no—
198C4⋊C8—102no—
199C4⋊C8—102no—
200C4⋊C8—101no—
201C4⋊C8—101no—
202C22⋊C8—21no—
203#97—102no—
204#97—101no—
205C4×C8—21no—
206C4×C8—51no—
207C4×C8—51no—
208C4×C8—102no—
209C4×C8—101no—
210C4×C8—101no—
211C22×C5—21no—
212C22×C5—21no—
213C22×C5#38511yescharacteristic
214C22×C5#47511yescharacteristic
215C22×C5#47911yescharacteristic
216C4×C5—21no—
217C4×C5—22no—
218C4×C5—21no—
219C4×C5—21no—
220C4×C5—41no—
221C4×C5—41no—
222C4×C5—41no—
223C4×C5C2×#10811yescharacteristic
224C4×C5#43011yescharacteristic
225C4×C5C4×SD1611yescharacteristic
226C4×C5#44111yescharacteristic
227C4×C5#44811yescharacteristic
228C4×C5#45211yescharacteristic
229C2×Q8—21no—
230C2×Q8—22no—
231C2×Q8—41no—
232C2×Q8#68311yescharacteristic
233C2×Q8#68311yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4C2×Dic1011yescharacteristic
237C22×C4D4×D511yescharacteristic
238C22×C4#71111yescharacteristic
239M16—41no—
240M16—41no—
241M16—52no—
242M16—101no—
243M16—101no—
244M16—101no—
245M16—201no—
246M16—201no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—101no—
250C2×C8—101no—
251C2×C8—101no—
252C2×C8—101no—
253C2×C8—101no—
254C2×C8—102no—
255C2×C8—101no—
256C2×C8—202no—
257C2×C8—201no—
258C2×C8—201no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4#68311yescharacteristic
265C4⋊C4#68311yescharacteristic
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42C2×C4×D511yescharacteristic
271C42D4×D511yescharacteristic
272C42#71111yescharacteristic
273C42C2×C22⋊D511yescharacteristic
274C2×C5—21no—
275C2×C5—21no—
276C2×C5#185511yescharacteristic
277C2×C5#186111yescharacteristic
278C2×C5#186511yescharacteristic
279C23#402411yescharacteristic
280Q8—21no—
281Q8—21no—
282Q8—21no—
283Q8—42no—
284Q8#395412yescomplemented
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—81no—
294C2×C4#402411yescharacteristic
295C2×C4#402711yescharacteristic
296C2×C4#402711yescharacteristic
297C2×C4D5×SD1611yescharacteristic
298C2×C4#406111yescharacteristic
299C2×C4#406411yescharacteristic
300C2×C4#406511yescharacteristic
301C2×C4C2×#68311yescharacteristic
302C2×C4#408111yescharacteristic
303C2×C4#408311yescharacteristic
304C8—81no—
305C8—81no—
306C8—102no—
307C8—102no—
308C8—201no—
309C8—201no—
310C8—201no—
311C8—201no—
312C8—202no—
313C5#890911yescharacteristic, complemented, Sylow
314C22—21no—
315C22—21no—
316C22#6508011yescharacteristic
317C22#6508511yescharacteristic
318C22#6523511yescharacteristic
319C4—21no—
320C4—22no—
321C4—21no—
322C4—21no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4#6507711yescharacteristic
327C4#6507811yescharacteristic
328C4C2×#395411yescharacteristic
329C4#6541311yescharacteristic
330C4#6545911yescharacteristic
331C4#6547611yescharacteristic
332C2—21no—
333C2—21no—
334C2#1060077711yescharacteristic
335C2#1060103011yescharacteristic
336C2#1060119111yescharacteristic
subgroupG/H
centerC2×C4#4083
commutatorC42×C5C22×C4
FrattiniC2×C42C22×D5
FittingC5×#1487C2
socleC22×C5#385
prime radicalC23×C5C22⋊Q8
cosocleC2×C42×C5C23
hypercenter#1487D5
hypocenterC5#8909
LCSG ▷ C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1487

Complements

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

NH ≅ G/Ntype
145. C5×Q8#108semidirect
284. Q8#3954semidirect
313. C5#8909semidirect

Maximal subgroups

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

subgroupindex
Q8×#6812
C5×#14872
C2×Q8⋊#1682
#106013132
#106012832
#105982362
#105982302
#89095