Finite Group Databasebeta

← #30594504

Subgroups — #30594504

Order 15616.

372 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 62 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
1#21461793C211yesmaximal, characteristic
2#21461684C211yesmaximal, characteristic
3#21445806C211yesmaximal, characteristic
4C22×D61⋊C8C411yescharacteristic, complemented
5#16840765—21no—
6C2×#14519837—21no—
7C2×#14519834C411yescharacteristic, complemented
8C2×C22⋊#92420—21no—
9#16840642C412yescomplemented
10#16840628—21no—
11C2×#14519737C2211yescharacteristic
12C2×#14519691—21no—
13#16839620—21no—
14#16839529C411yescharacteristic
15C61×#342C411yescharacteristic
16C2×#11758217—22no—
17C2×D61⋊C8—41no—
18C2×D61⋊C8—42no—
19C22×#92420—21no—
20C22×#92420—21no—
21#14519844—41no—
22C4⋊#92420—41no—
23#14519837—41no—
24#14519837—41no—
25#14519834—41no—
26C22⋊#92420—42no—
27C22×C22⋊D61D411yescharacteristic
28C23×Dic61D411yescharacteristic
29C22×C4×D61C2×C411yescharacteristic
30#14519737—41no—
31#14519737—41no—
32C2×#11758196—41no—
33C2×#11758196C2×C411yescharacteristic
34C2×C22⋊Dic61—21no—
35C2×#11758198Q811yescharacteristic
36C2×C4×Dic61D411yescharacteristic
37#14519691—41no—
38#14519691—41no—
39#14519684—41no—
40C2×C22⋊C4×C61—21no—
41C2×C22⋊C4×C61C2×C411yescharacteristic
42C24×C61C22⋊C411yescharacteristic
43C23×D61C4⋊C411yescharacteristic
44C22×C4×C61—41no—
45C22×C4×C61C4211yescharacteristic
46C22×C4×C61C4⋊C411yescharacteristic
47C2×C22⋊D61—41no—
48C2×C22⋊D61—41no—
49C2×C22⋊D61D811yescharacteristic
50C2×C22⋊D61SD1611yescharacteristic
51C2×C22⋊D61SD1611yescharacteristic
52C2×C22⋊D61Q1611yescharacteristic
53C22×Dic61—21no—
54C22×Dic61—21no—
55C22×Dic61—41no—
56C22×Dic61—41no—
57C22×Dic61—41no—
58C22×Dic61C22⋊C411yescharacteristic
59C22×Dic61C22⋊C411yescharacteristic
60C22×Dic61C4⋊C411yescharacteristic
61C2×C4×D61—21no—
62C2×C4×D61—21no—
63C2×C4×D61—41no—
64C2×C4×D61—41no—
65#11758217—42no—
66#11758217—42no—
67C2×#92420—22no—
68C2×#92420—41no—
69C2×#92420—41no—
70C2×#92420—41no—
71C2×#92420—42no—
72C2×#92420—41no—
73C2×#92420—41no—
74D61⋊C8—42no—
75D61⋊C8—42no—
76C22⋊C4×C61—41no—
77C22⋊C4×C61—41no—
78C22⋊C4×C61—41no—
79C22⋊Dic61—41no—
80C22⋊Dic61—41no—
81#11758198—41no—
82#11758196—41no—
83#11758196—81no—
84#11758196—81no—
85C4×Dic61—21no—
86C4×Dic61—21no—
87C23×C61—21no—
88C23×C61—21no—
89C23×C61—41no—
90C23×C61—41no—
91C23×C61#9511yescharacteristic
92C23×C61#10211yescharacteristic
93C23×C61#10311yescharacteristic
94C22×D61—21no—
95C22×D61—21no—
96C22×D61—41no—
97C22×D61—41no—
98C22×D61#10611yescharacteristic
99C22×D61#10711yescharacteristic
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61—41no—
103C2×C4×C61—41no—
104C2×C4×C61—81no—
105C2×C4×C61—81no—
106C2×C4×C61—81no—
107C22⋊D61—41no—
108C22⋊D61—41no—
109C22⋊D61—41no—
110C22⋊D61—41no—
111C2×Dic61—21no—
112C2×Dic61—21no—
113C2×Dic61—21no—
114C2×Dic61—21no—
115C2×Dic61—21no—
116C2×Dic61—41no—
117C2×Dic61—41no—
118C2×Dic61—41no—
119C2×Dic61—81no—
120C2×Dic61—81no—
121C2×Dic61—81no—
122C2×Dic61—81no—
123C2×Dic61#9911yescharacteristic
124C2×Dic61#9911yescharacteristic
125C2×Dic61#10211yescharacteristic
126C2×Dic61#10311yescharacteristic
127C4×D61—42no—
128C4×D61—41no—
129C4×D61—41no—
130#92420—42no—
131#92420—81no—
132#92420—82no—
133#7236—611nomaximal, Sylow
134C22×C61—21no—
135C22×C61—21no—
136C22×C61—21no—
137C22×C61—21no—
138C22×C61—41no—
139C22×C61—41no—
140C22×C61—41no—
141C22×C61—41no—
142C22×C61—81no—
143C22×C61H(3,ℤ/4)11yescharacteristic
144C22×C61#34011yescharacteristic
145C22×C61#34211yescharacteristic
146C2×D61—21no—
147C2×D61—21no—
148C2×D61—42no—
149C2×D61—41no—
150C2×D61—41no—
151C2×D61—41no—
152C2×D61—41no—
153C4×C61—41no—
154C4×C61—41no—
155C4×C61—161no—
156Dic61—21no—
157Dic61—21no—
158Dic61—21no—
159Dic61—41no—
160Dic61—161no—
161Dic61AΣL(1,𝔽16)11yescharacteristic
162Dic61#35211yescharacteristic
163#1869—611no—
164#1800—611no—
165#1799—611no—
166C2×C61—21no—
167C2×C61—21no—
168C2×C61—41no—
169C2×C61—41no—
170C2×C61#128911yescharacteristic
171C2×C61#139011yescharacteristic
172C2×C61#139511yescharacteristic
173D61—41no—
174D61—41no—
175C23×C8—611no—
176C2×C4×D4—611no—
177#423—1221no—
178C2×#100—1221no—
179C2×#99—1221no—
180C2×C22⋊C8—611no—
181C2×C22⋊C8—1221no—
182#380—1221no—
183#343—612no—
184#342—611no—
185#342#692311yescharacteristic
186#336—1221no—
187C61#723611yescharacteristic, complemented, Sylow
188C22×D4—611no—
189C23×C4—611no—
190C23×C4—611no—
191C2×M16—1222no—
192C22×C8—1221no—
193C22×C8—1221no—
194C22×C8—2442no—
195C22×C8—2441no—
196C4×D4—2441no—
197C4×D4—2441no—
198C2×C4⋊C4—611no—
199C2×C4⋊C4—2441no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—611no—
202C2×C22⋊C4—1221no—
203C2×C22⋊C4C2×#692311yescharacteristic
204C2×C42—611no—
205C4⋊C8—2441no—
206C4⋊C8—2441no—
207#100—2441no—
208#100—2441no—
209#99—2441no—
210#99—2441no—
211C22⋊C8—2442no—
212C22⋊C8—2441no—
213#95—2441no—
214C24—611no—
215C24C22⋊#692311yescharacteristic
216C2×D4—611no—
217C2×D4—611no—
218C2×D4—611no—
219C2×D4—611no—
220C2×D4—2441no—
221C2×D4—2441no—
222C22×C4—41no—
223C22×C4—611no—
224C22×C4—611no—
225C22×C4—611no—
226C22×C4—1221no—
227C22×C4—1221no—
228C22×C4—1221no—
229C22×C4—1221no—
230C22×C4—2441no—
231C22×C4—2441no—
232C22×C4—2441no—
233C22×C4—2441no—
234C22×C4—2441no—
235C22×C4C4×#692311yescharacteristic
236C22×C4C4⋊#692311yescharacteristic
237M16—2442no—
238M16—2442no—
239C2×C8—1222no—
240C2×C8—2441no—
241C2×C8—2441no—
242C2×C8—2442no—
243C2×C8—2441no—
244C2×C8—2441no—
245C2×C8—2441no—
246C2×C8—2442no—
247C2×C8—2442no—
248C4⋊C4—4881no—
249C4⋊C4—4881no—
250C4⋊C4—2441no—
251C22⋊C4—41no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—2441no—
255C22⋊C4—2441no—
256C22⋊C4—2441no—
257C42—1221no—
258C42—1221no—
259C23—21no—
260C23—21no—
261C23—41no—
262C23—41no—
263C23—611no—
264C23—611no—
265C23—1221no—
266C23—1221no—
267C23—2441no—
268C23—2441no—
269C23(C2×#6923)⋊C411yescharacteristic
270C23#1451983911yescharacteristic
271C23#1451984011yescharacteristic
272D4—2441no—
273D4—2441no—
274D4—2441no—
275D4—2441no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—611no—
284C2×C4—611no—
285C2×C4—611no—
286C2×C4—611no—
287C2×C4—1221no—
288C2×C4—1221no—
289C2×C4—1221no—
290C2×C4—1221no—
291C2×C4—1221no—
292C2×C4—4881no—
293C2×C4—4881no—
294C2×C4—4881no—
295C2×C4—4881no—
296C2×C4—2441no—
297C2×C4—2441no—
298C2×C4—2442no—
299C2×C4—2441no—
300C2×C4—2441no—
301C2×C4—2441no—
302C8—4881no—
303C8—4882no—
304C8—2442no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—81no—
314C22—1221no—
315C22—1221no—
316C22—2441no—
317C22—2441no—
318C22—2442no—
319C22—2441no—
320C22—2441no—
321C22#1684062911yescharacteristic
322C22#1684063511yescharacteristic
323C22#1684063911yescharacteristic
324C4—41no—
325C4—41no—
326C4—161no—
327C4—611no—
328C4—611no—
329C4—1221no—
330C4—1221no—
331C4—1221no—
332C4—9761no—
333C4—2441no—
334C2—21no—
335C2—21no—
336C2—41no—
337C2—41no—
338C2—2441no—
339C2—2441no—
340C2#2146068111yescharacteristic
341C2#2146090211yescharacteristic
342C2#2146091811yescharacteristic
subgroupG/H
centerC22#16840639
commutatorC22×C4×C61C42
FrattiniC2×C22⋊C4C2×#6923
FittingC61×#342C4
socleC22×C61#342
prime radicalC22×C22⋊D61D4
cosocleC2×#14519737C22
hypercenter#342#6923
hypocenterC61#7236
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #342

Complements

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

NH ≅ G/Ntype
4. C22×D61⋊C8C4semidirect
7. C2×#14519834C4semidirect
9. #16840642C4semidirect
187. C61#7236semidirect

Maximal subgroups

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

subgroupindex
#214617932
#214616842
#214458062
#723661