Finite Group Databasebeta

← #29590837

Subgroups — #29590837

Order 15616.

382 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 79 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#21451679C211yesmaximal, characteristic
2#21444163C211yesmaximal, characteristic, complemented
3#21443848C212yesmaximal, complemented
4#21443821C212yesmaximal
5C61×#1801C211yesmaximal, characteristic, complemented
6C2×#14519765C411yescharacteristic
7#16839819—21no—
8#16839697—21no—
9#16839682C411yescharacteristic
10#16839682C2211yescharacteristic
11#16839679C411yescharacteristic
12#16839679C2211yescharacteristic
13#16839673—21no—
14#16839653—21no—
15#16839653—21no—
16#16839630C411yescharacteristic
17#16839488—21no—
18#16839488C2212yes—
19C2×C4×Q8×C61C2211yescharacteristic
20C61×#423—21no—
21C2×#101×C61—21no—
22C61×#343C2212yes—
23C22×Dic122C2×C411yescharacteristic
24C22×D244C2×C411yescharacteristic
25#14519765—41no—
26#14519765—41no—
27C2×#11758196—21no—
28C2×#11758196—21no—
29C2×#11758198—21no—
30C2×#11758198—21no—
31C2×#11758198—21no—
32C2×#11758198—21no—
33C2×#11758198—21no—
34C2×#11758198—21no—
35C4⋊#92418—41no—
36C22⋊#92418—22no—
37#14519688—41no—
38#14519684—21no—
39#14519684—21no—
40#14519684—21no—
41#14519684—21no—
42#14519684C2×C411yescharacteristic
43#14519684C2×C411yescharacteristic
44C22×Q8×C61D411yescharacteristic
45C2×M16×C61—22no—
46C4×Q8×C61—41no—
47C4×Q8×C61—41no—
48C2×C4⋊C4×C61C2×C411yescharacteristic
49C2×C4⋊C4×C61D411yescharacteristic
50C2×C4⋊C4×C61C2311yescharacteristic
51C2×C42×C61C2×C411yescharacteristic
52C2×C42×C61D411yescharacteristic
53C2×C42×C61D411yescharacteristic
54C4⋊C8×C61—41no—
55C4⋊C8×C61—41no—
56#101×C61—42no—
57C23×D61—21no—
58C2×Q8×C61—21no—
59C2×Q8×C61—21no—
60C2×Q8×C61—41no—
61C2×Q8×C61—41no—
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22⋊C411yescharacteristic
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C22⋊C411yescharacteristic
66C22×C4×C61C22×C411yescharacteristic
67C22×C4×C61C2×D411yescharacteristic
68C22×C4×C61C2×D411yescharacteristic
69C22×Dic61—21no—
70C22×Dic61—21no—
71C22×Dic61—21no—
72C2×D244—41no—
73C2×D244—81no—
74C2×Dic122—41no—
75C2×Dic122—81no—
76M16×C61—42no—
77M16×C61—42no—
78C2×C8×C61—42no—
79C4⋊C4×C61—41no—
80C4⋊C4×C61—41no—
81C4⋊C4×C61—41no—
82C42×C61—21no—
83C42×C61—21no—
84C42×C61—21no—
85C42×C61—21no—
86C42×C61—21no—
87C42×C61—21no—
88#11758198—42no—
89#11758198—42no—
90#11758198—41no—
91#11758198—41no—
92#11758198—41no—
93#11758198—41no—
94#11758198—81no—
95#11758198—81no—
96#11758196—81no—
97#11758196—81no—
98C2×#92418—42no—
99C23×C61C2×C22⋊C411yescharacteristic
100C22×D61—41no—
101C22×D61—41no—
102C22×D61—41no—
103C22×D61—41no—
104C22×D61—41no—
105C22×D61—41no—
106C22×D61—41no—
107Q8×C61—42no—
108Q8×C61—42no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
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—41no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61#9911yescharacteristic
125C2×C4×C61#9911yescharacteristic
126C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
127C2×C4×C61#13711yescharacteristic
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137D244—82no—
138Dic122—82no—
139C8×C61—82no—
140#92418—82no—
141#11833—611nomaximal, Sylow
142C22×C61—21no—
143C22×C61—21no—
144C22×C61C2×#9911yescharacteristic
145C22×C61#41711yescharacteristic
146C22×C61#42111yescharacteristic
147C2×D61—81no—
148C2×D61—81no—
149C2×D61—81no—
150C2×D61—81no—
151C2×D61—81no—
152C2×D61—81no—
153C2×D61—81no—
154C4×C61—21no—
155C4×C61—21no—
156C4×C61—21no—
157C4×C61—21no—
158C4×C61—41no—
159C4×C61—41no—
160C4×C61—41no—
161C4×C61—41no—
162C4×C61—41no—
163Dic61—81no—
164Dic61—81no—
165Dic61—81no—
166#2632—611no—
167#1801D6111yescharacteristic, complemented
168#1467—611no—
169#1354—612no—
170#1345—612no—
171C2×C61—21no—
172C2×C61—21no—
173C2×C61#151311yescharacteristic
174C2×C61#212211yescharacteristic
175C2×C61#213311yescharacteristic
176D61—161no—
177C2×#124—611no—
178C2×C4×Q8C2×D6111yescharacteristic
179#432—1221no—
180#423—21no—
181C2×#101—21no—
182#399—1221no—
183#395—611no—
184#395—611no—
185#394—611no—
186#394—611no—
187#393—1221no—
188#388—1221no—
189#388—1221no—
190#383—611no—
191#343C2×D6112yes—
192#324—612no—
193#324—1221no—
194C61#1183311yescharacteristic, complemented, Sylow
195C22×Q8—611no—
196C22×Q8C22⋊D6111yescharacteristic
197C22×D4—611no—
198C2×M16—22no—
199#124—2441no—
200#124—2441no—
201C4×Q8—41no—
202C4×Q8—41no—
203C2×C4⋊C4—1221no—
204C2×C4⋊C4—1221no—
205C2×C4⋊C4C4×D6111yescharacteristic
206C2×C4⋊C4D24411yescharacteristic
207C2×C4⋊C4C22×D6111yescharacteristic
208C2×C22⋊C4—1221no—
209C2×C22⋊C4—1221no—
210C2×C22⋊C4—1221no—
211C2×C22⋊C4—1221no—
212C2×C22⋊C4—1221no—
213C2×C22⋊C4—1221no—
214C2×C42C4×D6111yescharacteristic
215C2×C42D24411yescharacteristic
216C2×C42C22⋊D6111yescharacteristic
217C4⋊C8—41no—
218C4⋊C8—41no—
219C4⋊C8—2441no—
220#101—42no—
221C22⋊C8—1222no—
222#97—2441no—
223#95—611no—
224#95—611no—
225#95—1221no—
226#95—1221no—
227#95—1221no—
228#95—1221no—
229C24—1221no—
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—4881no—
235C2×Q8—2441no—
236C2×D4—4881no—
237C2×D4—2441no—
238C22×C4—1221no—
239C22×C4—1221no—
240C22×C4—1221no—
241C22×C4#1175819811yescharacteristic
242C22×C4#1175819811yescharacteristic
243C22×C4#1175819811yescharacteristic
244C22×C4#1175819811yescharacteristic
245C22×C4C2×C4×D6111yescharacteristic
246C22×C4C2×D24411yescharacteristic
247C22×C4C2×C22⋊D6111yescharacteristic
248M16—42no—
249M16—42no—
250C2×C8—42no—
251C2×C8—2442no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—4881no—
256C4⋊C4—4881no—
257C22⋊C4—4881no—
258C22⋊C4—4881no—
259C22⋊C4—2442no—
260C22⋊C4—2442no—
261C22⋊C4—2441no—
262C22⋊C4—2441no—
263C22⋊C4—2441no—
264C22⋊C4—2441no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C23—2441no—
272C23—2441no—
273C23—2441no—
274C23—2441no—
275C23—2441no—
276C23—2441no—
277C23—2441no—
278C23C2×#1175819811yescharacteristic
279Q8—42no—
280Q8—42no—
281Q8—4882no—
282D4—4882no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—2441no—
299C2×C4—2441no—
300C2×C4—2441no—
301C2×C4—2441no—
302C2×C4—2441no—
303C2×C4—2441no—
304C2×C4—2441no—
305C2×C4—2441no—
306C2×C4—2441no—
307C2×C4#1451969111yescharacteristic
308C2×C4#1451969111yescharacteristic
309C2×C4#1451980311yescharacteristic
310C2×C4#1451980811yescharacteristic
311C8—82no—
312C8—4882no—
313C22—21no—
314C22—21no—
315C22—4881no—
316C22—4881no—
317C22—4881no—
318C22—4881no—
319C22—4881no—
320C22—4881no—
321C22—4881no—
322C22C2×#1451969111yescharacteristic
323C22#1683976111yescharacteristic
324C22#1683978011yescharacteristic
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—4881no—
335C4—4881no—
336C4—4881no—
337C2—21no—
338C2—21no—
339C2—9761no—
340C2#2144436011yescharacteristic
341C2#2144747011yescharacteristic
342C2#2144750611yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC2×C4⋊C4C22×D61
FittingC61×#1801C2
socleC22×C61C2×#99
prime radicalC22×D244C2×C4
cosocleC2×C4⋊C4×C61C23
hypercenter#1801D61
hypocenterC61#11833
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1801

Complements

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

NH ≅ G/Ntype
2. #21444163C2semidirect
3. #21443848C2semidirect
5. C61×#1801C2semidirect
167. #1801D61semidirect
194. C61#11833semidirect

Maximal subgroups

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

subgroupindex
#214516792
#214441632
#214438482
#214438212
C61×#18012
#1183361