Finite Group Databasebeta

← #29600283

Subgroups — #29600283

Order 15616.

470 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C2×#16839944C211yesmaximal, characteristic
2#21444545C211yesmaximal, characteristic, complemented
3#21443972C214yesmaximal, complemented
4C2×C61×#356C211yesmaximal, characteristic, complemented
5C2×#14519825C411yescharacteristic
6C2×#14519808—21no—
7#16839944—21no—
8#16839944—21no—
9#16839944C412yes—
10#16839944C2212yes—
11#16839819—21no—
12C2×#14519705—21no—
13C2×#14519696C411yescharacteristic
14#16839511—21no—
15#16839511C2212yescomplemented
16C2×C4⋊Q8×C61C2211yescharacteristic
17C2×#101×C61—21no—
18C61×#356—21no—
19C61×#356C2212yescomplemented
20#14519825—21no—
21#14519825—42no—
22#14519825—41no—
23#14519825C2×C412yes—
24C2×#11758226—41no—
25C2×#11758222—21no—
26C2×#11758222—41no—
27C2×Q8×D61—41no—
28#14519808—22no—
29#14519808—22no—
30#14519808—42no—
31C2×#11758202—41no—
32C2×#11758201—41no—
33C2×#11758194—21no—
34C4⋊#92418—22no—
35#14519705—22no—
36#14519705—22no—
37#14519696—21no—
38#14519696C2×C412yes—
39C22⋊#92418—41no—
40#14519688—41no—
41C22×Q8×C61C2×C411yescharacteristic
42C22×Q8×C61D411yescharacteristic
43C2×M16×C61—41no—
44C4⋊Q8×C61—41no—
45C4⋊Q8×C61C2×C411yescharacteristic
46C4⋊Q8×C61D411yescharacteristic
47C4⋊Q8×C61D411yescharacteristic
48C4⋊Q8×C61C2311yescharacteristic
49C2×C4⋊C4×C61—41no—
50C2×C42×C61D411yescharacteristic
51#101×C61—22no—
52#101×C61—41no—
53C2×Q8×C61—21no—
54C2×Q8×C61—21no—
55C2×Q8×C61—41no—
56C2×Q8×C61—41no—
57C2×Q8×C61—41no—
58C2×Q8×C61—41no—
59C2×Q8×C61C22⋊C411yescharacteristic
60C2×Q8×C61C22⋊C411yescharacteristic
61C2×Q8×C61C22×C411yescharacteristic
62C2×Q8×C61C2×D411yescharacteristic
63C22×C4×C61—21no—
64C22×C4×C61—21no—
65C22×C4×C61—21no—
66C22×C4×C61C22⋊C411yescharacteristic
67C2×C22⋊D61—41no—
68#11758226—42no—
69#11758226—42no—
70#11758226—44no—
71Q8×D61—42no—
72Q8×D61—42no—
73Q8×D61—44no—
74#11758222—22no—
75#11758222—22no—
76#11758222—41no—
77#11758222—42no—
78#11758222—42no—
79#11758222—41no—
80#11758222—42no—
81#11758222—42no—
82C2×D244—21no—
83C2×D244—41no—
84C2×C4×D61—41no—
85C2×C4×D61—42no—
86C2×Dic122—21no—
87C2×Dic122—41no—
88M16×C61—42no—
89M16×C61—81no—
90C2×C8×C61—81no—
91C4⋊C4×C61—41no—
92C4⋊C4×C61—41no—
93C4⋊C4×C61—41no—
94C4⋊C4×C61—41no—
95C42×C61—21no—
96C42×C61C22⋊C411yescharacteristic
97C42×C61C2×D411yescharacteristic
98#11758202—42no—
99#11758202—42no—
100#11758201—42no—
101#11758201—42no—
102#11758194—22no—
103#11758194—41no—
104C2×#92418—41no—
105C2×#92418—42no—
106C23×C61#9911yescharacteristic
107C22×D61—41no—
108Q8×C61—42no—
109Q8×C61—41no—
110Q8×C61—41no—
111Q8×C61—41no—
112Q8×C61—41no—
113Q8×C61—81no—
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—21no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—21no—
122C2×C4×C61—42no—
123C2×C4×C61—41no—
124C2×C4×C61—41no—
125C2×C4×C61—41no—
126C2×C4×C61—41no—
127C2×C4×C61#9911yescharacteristic
128C2×C4×C61C2×C22⋊C411yescharacteristic
129C22⋊D61—42no—
130C22⋊D61—42no—
131C2×Dic61—41no—
132D244—22no—
133D244—41no—
134D244—42no—
135D244—42no—
136C4×D61—42no—
137C4×D61—42no—
138C4×D61—44no—
139C4×D61—44no—
140Dic122—22no—
141Dic122—41no—
142Dic122—42no—
143Dic122—42no—
144C8×C61—82no—
145#92418—42no—
146#92418—82no—
147#13573—611nomaximal, Sylow
148C22×C61—41no—
149C22×C61AΣL(1,𝔽16)11yescharacteristic
150C22×C61AΣL(1,𝔽16)11yescharacteristic
151C22×C61C2×#9911yescharacteristic
152C2×D61—42no—
153C2×D61—82no—
154C4×C61—22no—
155C4×C61—21no—
156C4×C61—21no—
157C4×C61—41no—
158C4×C61—41no—
159C4×C61—41no—
160C4×C61—41no—
161C4×C61—41no—
162Dic61—42no—
163C2×#456—611no—
164C2×#356D6111yescharacteristic, complemented
165#1558—611no—
166#1405—614no—
167C2×C61—21no—
168C2×C61—21no—
169C2×C61—21no—
170C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
171D61—82no—
172C2×2−1+4—611no—
173C2×#137—1221no—
174C2×C4⋊Q8C2×D6111yescharacteristic
175#456—612no—
176#456—612no—
177#456—1221no—
178#456—1221no—
179#432—1221no—
180C2×#104—1221no—
181C2×#101—21no—
182C2×#101—611no—
183#356—21no—
184#356C2×D6112yescomplemented
185#333—612no—
186#333—1221no—
187C61#1357311yescharacteristic, complemented, Sylow
1882−1+4—612no—
1892−1+4—1221no—
1902−1+4—2441no—
1912−1+4—2442no—
192C2×#41—1221no—
193C2×#41—2441no—
194C2×#41—2441no—
195C22×Q8—2441no—
196C22×Q8C4×D6111yescharacteristic
197C22×Q8C22⋊D6111yescharacteristic
198#137—1222no—
199#137—1222no—
200#137—2442no—
201C2×Q16—2441no—
202C2×SD16—2441no—
203C2×M16—41no—
204C2×M16—1221no—
205C4⋊Q8—41no—
206C4⋊Q8C4×D6111yescharacteristic
207C4⋊Q8D24411yescharacteristic
208C4⋊Q8C22⋊D6111yescharacteristic
209C4⋊Q8C22×D6111yescharacteristic
210C2×C4⋊C4—41no—
211C2×C42D24411yescharacteristic
212C4⋊C8—1222no—
213#104—1222no—
214#104—1222no—
215#101—22no—
216#101—41no—
217#101—612no—
218#101—1221no—
219C22⋊C8—2441no—
220#97—2441no—
221#41—1222no—
222#41—1222no—
223#41—2441no—
224#41—2442no—
225#41—2442no—
226#41—2442no—
227#41—2442no—
228#41—2442no—
229#41—2442no—
230#41—2444no—
231#41—2441no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—1221no—
239C2×Q8—2441no—
240C2×Q8—2442no—
241C2×Q8—2442no—
242C2×Q8—2444no—
243C2×Q8#1175819811yescharacteristic
244C2×Q8#1175819811yescharacteristic
245C2×Q8C2×C4×D6111yescharacteristic
246C2×Q8C2×C22⋊D6111yescharacteristic
247C2×D4—1221no—
248C2×D4—2441no—
249C2×D4—2441no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—2441no—
254C22×C4—2442no—
255C22×C4#1175819811yescharacteristic
256Q16—2442no—
257Q16—2442no—
258SD16—2442no—
259SD16—2442no—
260M16—42no—
261M16—81no—
262M16—1222no—
263M16—2441no—
264C2×C8—81no—
265C2×C8—2442no—
266C2×C8—2441no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C42—21no—
272C42#1175819811yescharacteristic
273C42C2×D24411yescharacteristic
274C23—2441no—
275C23#1451969111yescharacteristic
276Q8—41no—
277Q8—41no—
278Q8—41no—
279Q8—41no—
280Q8—42no—
281Q8—81no—
282Q8—1222no—
283Q8—2441no—
284Q8—2442no—
285Q8—2442no—
286D4—1222no—
287D4—2441no—
288D4—2442no—
289D4—2442no—
290D4—2442no—
291D4—2442no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—2442no—
306C2×C4—2442no—
307C2×C4—2444no—
308C2×C4—2444no—
309C2×C4—2441no—
310C2×C4#1451969111yescharacteristic
311C2×C4C2×#1175819811yescharacteristic
312C8—82no—
313C8—4882no—
314C8—2442no—
315C22—41no—
316C22—4882no—
317C22—2442no—
318C22#1683954811yescharacteristic
319C22#1683954811yescharacteristic
320C22C2×#1451969111yescharacteristic
321C4—22no—
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—2442no—
330C2—21no—
331C2—21no—
332C2—21no—
333C2—4882no—
334C2C2×#1683954811yescharacteristic
subgroupG/H
centerC2C2×#16839548
commutatorC2×Q8×C61C22×C4
FrattiniC4⋊Q8C22×D61
FittingC2×C61×#356C2
socleC2×C61C2×AΣL(1,𝔽16)
prime radicalC2×#14519825C4
cosocleC4⋊Q8×C61C23
hypercenterC2×#356D61
hypocenterC61#13573
LCSG ▷ C2×Q8×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ C2×#356

Complements

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

NH ≅ G/Ntype
2. #21444545C2semidirect
3. #21443972C2semidirect
4. C2×C61×#356C2semidirect
15. #16839511C22semidirect
19. C61×#356C22semidirect
164. C2×#356D61semidirect
184. #356C2×D61semidirect
187. C61#13573semidirect

Maximal subgroups

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

subgroupindex
C2×#168399442
#214445452
#214439722
C2×C61×#3562
#1357361