Finite Group Databasebeta

← #30607451

Subgroups — #30607451

Order 15616.

396 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 86 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×#16840802C211yesmaximal, characteristic
2C2×#16840687C211yesmaximal, characteristic
3C2×#16840681C211yesmaximal, characteristic
4#21461928C211yesmaximal, characteristic
5#21460890C211yesmaximal, characteristic
6#21460859C211yesmaximal, characteristic
7#21447537C211yesmaximal, characteristic
8C2×Q8×#6923—21no—
9C2×#14519852C2211yescharacteristic
10#16840802—22no—
11#16840802—22no—
12C2×#14519847C2211yescharacteristic
13C2×#14519845C2211yescharacteristic
14#16840769—21no—
15C2×#14519840—21no—
16C2×#14519831C2211yescharacteristic
17#16840687—22no—
18#16840681—22no—
19#16840632—21no—
20C2×Q16×D61C2211yescharacteristic
21C2×#14519716C2211yescharacteristic
22#16839799C2211yescharacteristic
23#16839567C411yescharacteristic, complemented
24#16839566C411yescharacteristic, complemented
25#16839565C411yescharacteristic, complemented
26C61×#358C411yescharacteristic, complemented
27C2×#11758213—21no—
28C2×D61⋊C8D411yescharacteristic
29Q8×#6923—24no—
30Q8×#6923—42no—
31C2×C4⋊#6923—21no—
32C2×C4⋊#6923—41no—
33C2×C4×#6923—41no—
34C2×C4×#6923D411yescharacteristic
35#14519852—21no—
36#14519852—21no—
37C2×#11758187—21no—
38#14519847—21no—
39#14519847—21no—
40#14519845—21no—
41#14519845D412yes—
42#14519844—41no—
43C4⋊#92420—41no—
44#14519840—24no—
45#14519831—21no—
46#14519831—21no—
47C2×Q8×D61—21no—
48C2×Dic244C2×C411yescharacteristic
49C2×#11758202—21no—
50Q16×D61—21no—
51Q16×D61—21no—
52Q16×D61D412yes—
53Q16×D61Q812yes—
54C2×C8×D61C2311yescharacteristic
55#14519735—21no—
56#14519716—21no—
57#14519716—21no—
58C2×#11758185C2×C411yescharacteristic
59#14519711C2×C411yescharacteristic
60#14519708—21no—
61C2×Q16×C61C2×C411yescharacteristic
62C2×C16×C61C2×C411yescharacteristic
63#107×C61C2×C411yescharacteristic
64C22×#6923—21no—
65C22×#6923—41no—
66C2×Q8×C61—21no—
67Q8×D61—22no—
68Q8×D61—22no—
69Q8×D61—41no—
70Q8×D61—41no—
71C2×C4×D61—41no—
72C2×C4×D61C2×D411yescharacteristic
73C2×Dic122—21no—
74C2×Dic122—41no—
75C2×#92420—21no—
76C2×#92420—41no—
77C4⋊#6923—22no—
78C4⋊#6923—44no—
79C4⋊#6923—41no—
80C4×#6923—21no—
81C4×#6923—44no—
82C4×#6923SD1612yes—
83#11758213—41no—
84#11758213—41no—
85D61⋊C8—21no—
86D61⋊C8—21no—
87Q16×C61—21no—
88Q16×C61C4⋊C412yes—
89C2×C8×C61C22×C411yescharacteristic
90C4⋊C4×C61—21no—
91#11758202—22no—
92#11758202—22no—
93#11758198—41no—
94C4⋊Dic61—21no—
95C4×Dic61—41no—
96C2×#92418#4111yescharacteristic
97Dic244—21no—
98Dic244C4⋊C412yes—
99C8×D61C2×D411yescharacteristic
100C8×D61C2×D411yescharacteristic
101C8×D61C2×Q811yescharacteristic
102C8×D61#4111yescharacteristic
103#11758187—41no—
104C16×C61—21no—
105#11758185—21no—
106C22×D61AGL(1,ℤ/8)11yescharacteristic
107C2×#6923—22no—
108C2×#6923—41no—
109C2×#6923—41no—
110C2×#6923—42no—
111C2×#6923—81no—
112C2×#6923—81no—
113Q8×C61—22no—
114Q8×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61C4×D411yescharacteristic
118C2×Dic61—41no—
119C2×Dic61—41no—
120C2×Dic61AGL(1,ℤ/8)11yescharacteristic
121C4×D61—42no—
122C4×D61—42no—
123C4×D61#12011yescharacteristic
124C4×D61C22⋊Q811yescharacteristic
125C4×D61C2×SD1611yescharacteristic
126C4×D61#13511yescharacteristic
127Dic122—22no—
128Dic122—41no—
129Dic122—42no—
130Dic122—42no—
131#92420—41no—
132#92420—81no—
133C8×C61C2×C4⋊C411yescharacteristic
134C8×C61C4×D411yescharacteristic
135#92418C22⋊Q811yescharacteristic
136#92418#12311yescharacteristic
137#12957—611nomaximal, Sylow
138C22×C61#44211yescharacteristic
139C2×D61—21no—
140C2×D61—21no—
141C2×D61#45011yescharacteristic
142C2×D61#47811yescharacteristic
143#6923—42no—
144#6923—82no—
145C4×C61—42no—
146C4×C61—81no—
147C4×C61#38511yescharacteristic
148C4×C61C4×SD1611yescharacteristic
149Dic61—42no—
150Dic61—81no—
151Dic61#44911yescharacteristic
152Dic61#47611yescharacteristic
153C2×#441—611no—
154C2×#365—611no—
155C2×#362—611no—
156#2141—611no—
157#1946—611no—
158#1386—611no—
159#1375—611no—
160C2×C61#186411yescharacteristic
161C2×C61#218011yescharacteristic
162C2×C61#218011yescharacteristic
163D61—21no—
164D61—21no—
165C22×Q16—611no—
166C2×C4×Q8—1221no—
167C2×M32—611no—
168C2×M32—611no—
169#441—1222no—
170#441—1222no—
171C2×#108—611no—
172#427—611no—
173C2×#106—611no—
174#424—1221no—
175C2×#103—1221no—
176C2×#97—611no—
177#365—1222no—
178#362—1222no—
179#358—611no—
180#358—611no—
181#358—611no—
182#358#692311yescharacteristic, complemented
183#339—1221no—
184C61#1295711yescharacteristic, complemented, Sylow
185C22×Q8—1221no—
186C2×Q16—612no—
187C2×Q16—612no—
188C2×Q16—611no—
189C2×Q16—1221no—
190C2×Q16—1221no—
191C2×Q16—1221no—
192C2×Q16C2×#692311yescharacteristic
193C2×M16—1221no—
194C22×C8—611no—
195C22×C8—611no—
196C4×Q8—1224no—
197C4×Q8—2442no—
198#117—1221no—
199C2×C4⋊C4—1221no—
200C2×C4⋊C4—2441no—
201C2×C42—611no—
202C2×C42—2441no—
203M32—1221no—
204M32—1221no—
205M32—1221no—
206M32—1221no—
207C2×C16—611no—
208C2×C16—1221no—
209C2×C16C2×#692311yescharacteristic
210#108—1221no—
211#108—1221no—
212#107—611no—
213#107C2×#692311yescharacteristic
214#106—612no—
215#106—1221no—
216#106—1221no—
217C4⋊C8—2441no—
218C4⋊C8—2441no—
219#103—1224no—
220#97—1221no—
221#97—1221no—
222C2×Q8—21no—
223C2×Q8—1222no—
224C2×Q8—1222no—
225C2×Q8—1221no—
226C2×Q8—2441no—
227C2×Q8—2441no—
228C2×Q8—2441no—
229C22×C4—611no—
230C22×C4—1221no—
231C22×C4—2441no—
232C22×C4—2441no—
233Q16—21no—
234Q16—612no—
235Q16—1222no—
236Q16—1222no—
237Q16—1221no—
238Q16C4⋊#692312yes—
239M16—2441no—
240M16—2441no—
241C2×C8—611no—
242C2×C8—611no—
243C2×C8—611no—
244C2×C8—611no—
245C2×C8—611no—
246C2×C8—1221no—
247C2×C8—1221no—
248C2×C8—1221no—
249C2×C8—2441no—
250C2×C8C22×#692311yescharacteristic
251C4⋊C4—21no—
252C4⋊C4—1222no—
253C4⋊C4—1221no—
254C4⋊C4—2441no—
255C4⋊C4—2444no—
256C22⋊C4—2441no—
257C42—612no—
258C42—1221no—
259C42—2441no—
260C42—2444no—
261C16—21no—
262C16—1221no—
263C16—2441no—
264C23—611no—
265Q8—22no—
266Q8—41no—
267Q8—1222no—
268Q8—2442no—
269Q8—2442no—
270Q8—2441no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—611no—
274C2×C4—611no—
275C2×C4—611no—
276C2×C4—611no—
277C2×C4—611no—
278C2×C4—1222no—
279C2×C4—4881no—
280C2×C4—4881no—
281C2×C4—2442no—
282C2×C4—2442no—
283C2×C4—2442no—
284C2×C4—2441no—
285C2×C4—2441no—
286C2×C4—2441no—
287C2×C4—2441no—
288C2×C4D4×#692311yescharacteristic
289C8—611no—
290C8—611no—
291C8—4881no—
292C8—2441no—
293C8C2×C4⋊#692311yescharacteristic
294C8D4×#692311yescharacteristic
295C22—611no—
296C22—611no—
297C22—1221no—
298C22—1221no—
299C22#1684080311yescharacteristic
300C4—42no—
301C4—81no—
302C4—611no—
303C4—611no—
304C4—4882no—
305C4—4881no—
306C4—2442no—
307C4—2442no—
308C4#1684070911yescharacteristic
309C4SD16×#692311yescharacteristic
310C2—1221no—
311C2—1221no—
312C2#2146178411yescharacteristic
313C2#2146222411yescharacteristic
314C2#2146222711yescharacteristic
subgroupG/H
centerC22#16840803
commutatorC2×C8×C61C22×C4
FrattiniC2×C8C22×#6923
FittingC61×#358C4
socleC22×C61#442
prime radicalC22×D61AGL(1,ℤ/8)
cosocleC2×C8×D61C23
hypercenter#358#6923
hypocenterC61#12957
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ #358

Complements

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

NH ≅ G/Ntype
23. #16839567C4semidirect
24. #16839566C4semidirect
25. #16839565C4semidirect
26. C61×#358C4semidirect
182. #358#6923semidirect
184. C61#12957semidirect

Maximal subgroups

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

subgroupindex
C2×#168408022
C2×#168406872
C2×#168406812
#214619282
#214608902
#214608592
#214475372
#1295761