Finite Group Databasebeta

← #29590817

Subgroups — #29590817

Order 15616.

416 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 83 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#21449102C211yesmaximal, characteristic
2#21444163C211yesmaximal, characteristic, complemented
3#21443845C212yesmaximal, complemented
4#21443821C212yesmaximal
5C61×#1801C211yesmaximal, characteristic, complemented
6C2×C4×D244C411yescharacteristic
7#16839819—21no—
8#16839701C411yescharacteristic
9#16839701C2211yescharacteristic
10#16839688—21no—
11#16839679C411yescharacteristic
12#16839679C2211yescharacteristic
13#16839673—21no—
14C4×#11758198—21no—
15C4×#11758198—21no—
16#16839608C411yescharacteristic
17#16839488—21no—
18#16839488C2212yes—
19C2×C4×Q8×C61C2211yescharacteristic
20C61×#423—21no—
21C2×#101×C61—21no—
22C61×#343C2212yes—
23C22×D244C2×C411yescharacteristic
24C22×C4×D61—21no—
25C4×D244—41no—
26C4×D244—41no—
27C2×C4⋊Dic61C2×C411yescharacteristic
28C2×#11758198—21no—
29C2×#11758198—21no—
30C2×#11758198—21no—
31C2×#11758198—21no—
32C2×#11758198—21no—
33C2×C4×Dic61—21no—
34C4⋊#92418—41no—
35C22⋊#92418—22no—
36#14519688—41no—
37#14519684—21no—
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—42no—
74C2×C4×D61—42no—
75C2×C4×D61—42no—
76C2×C4×D61—41no—
77C2×C4×D61—41no—
78M16×C61—42no—
79M16×C61—42no—
80C2×C8×C61—42no—
81C4⋊C4×C61—41no—
82C4⋊C4×C61—41no—
83C4⋊C4×C61—41no—
84C42×C61—21no—
85C42×C61—21no—
86C42×C61—21no—
87C42×C61—21no—
88C42×C61—21no—
89C42×C61—21no—
90#11758198—42no—
91#11758198—42no—
92#11758198—42no—
93#11758198—42no—
94#11758198—41no—
95#11758198—41no—
96C4⋊Dic61—41no—
97C4×Dic61—42no—
98C2×#92418—42no—
99C23×C61C2×C22⋊C411yescharacteristic
100C22×D61—22no—
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—22no—
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—21no—
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—42no—
138D244—42no—
139C4×D61—82no—
140C4×D61—82no—
141C8×C61—82no—
142#92418—82no—
143#11829—611nomaximal, Sylow
144C22×C61—21no—
145C22×C61—21no—
146C22×C61C2×#9911yescharacteristic
147C22×C61#41711yescharacteristic
148C22×C61#42111yescharacteristic
149C2×D61—42no—
150C2×D61—42no—
151C2×D61—42no—
152C2×D61—81no—
153C2×D61—81no—
154C2×D61—81no—
155C2×D61—81no—
156C4×C61—21no—
157C4×C61—21no—
158C4×C61—21no—
159C4×C61—41no—
160C4×C61—41no—
161C4×C61—41no—
162C4×C61—41no—
163C4×C61—41no—
164C4×C61#35512yes—
165Dic61—81no—
166Dic61—81no—
167Dic61—81no—
168#2398—611no—
169#1801D6111yescharacteristic, complemented
170#1467—611no—
171#1353—612no—
172#1345—612no—
173C2×C61—21no—
174C2×C61—21no—
175C2×C61#151311yescharacteristic
176C2×C61#212011yescharacteristic
177C2×C61C2×#35511yescharacteristic
178D61—82no—
179C2×C4×Q8C2×D6111yescharacteristic
180C2×C4×D4—611no—
181#432—1221no—
182#423—21no—
183C2×#101—21no—
184#400—611no—
185#400—611no—
186#397—1221no—
187#394—611no—
188#394—611no—
189#393—1221no—
190C4×C22⋊C4—1221no—
191C4×C22⋊C4—1221no—
192#376—611no—
193#343C2×D6112yes—
194#324—612no—
195#324—1221no—
196C61#1182911yescharacteristic, complemented, Sylow
197C22×Q8C22⋊D6111yescharacteristic
198C22×D4—611no—
199C23×C4—1221no—
200C2×M16—22no—
201C4×Q8—41no—
202C4×Q8—41no—
203C4×D4—2441no—
204C4×D4—2441no—
205C2×C4⋊C4—611no—
206C2×C4⋊C4C4×D6111yescharacteristic
207C2×C4⋊C4D24411yescharacteristic
208C2×C4⋊C4C22×D6111yescharacteristic
209C2×C22⋊C4—1221no—
210C2×C22⋊C4—1221no—
211C2×C22⋊C4—1221no—
212C2×C22⋊C4—1221no—
213C2×C22⋊C4—1221no—
214C2×C42—1221no—
215C2×C42C4×D6111yescharacteristic
216C2×C42D24411yescharacteristic
217C2×C42C22⋊D6111yescharacteristic
218C4⋊C8—41no—
219C4⋊C8—41no—
220C4⋊C8—2441no—
221#101—42no—
222C22⋊C8—1222no—
223#97—2441no—
224#95—611no—
225#95—611no—
226#95—1221no—
227#95—1221no—
228#95—1221no—
229#95—1221no—
230#95—1221no—
231C24—1221no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×D4—2441no—
237C2×D4—2442no—
238C22×C4—1221no—
239C22×C4—1221no—
240C22×C4—1221no—
241C22×C4—2442no—
242C22×C4—2442no—
243C22×C4—2441no—
244C22×C4—2441no—
245C22×C4#1175819811yescharacteristic
246C22×C4#1175819811yescharacteristic
247C22×C4#1175819811yescharacteristic
248C22×C4#1175819811yescharacteristic
249C22×C4C2×C4×D6111yescharacteristic
250C22×C4C2×D24411yescharacteristic
251C22×C4C2×C22⋊D6111yescharacteristic
252M16—42no—
253M16—42no—
254C2×C8—42no—
255C2×C8—2442no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—2441no—
260C22⋊C4—2442no—
261C22⋊C4—2442no—
262C22⋊C4—2442no—
263C22⋊C4—2442no—
264C22⋊C4—2441no—
265C22⋊C4—2441no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—2442no—
273C23—1222no—
274C23—2441no—
275C23—2441no—
276C23—2441no—
277C23—2441no—
278C23—2441no—
279C23—2441no—
280C23C2×#1175819811yescharacteristic
281Q8—42no—
282Q8—42no—
283D4—2442no—
284D4—2442no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—4882no—
301C2×C4—4882no—
302C2×C4—2441no—
303C2×C4—2441no—
304C2×C4—2441no—
305C2×C4—2441no—
306C2×C4—2441no—
307C2×C4—2441no—
308C2×C4—2441no—
309C2×C4—2441no—
310C2×C4—2441no—
311C2×C4#1451969111yescharacteristic
312C2×C4#1451969111yescharacteristic
313C2×C4#1451980311yescharacteristic
314C2×C4#1451980811yescharacteristic
315C8—82no—
316C8—4882no—
317C22—21no—
318C22—21no—
319C22—4881no—
320C22—4881no—
321C22—4881no—
322C22—4881no—
323C22—2442no—
324C22—2442no—
325C22—2442no—
326C22C2×#1451969111yescharacteristic
327C22#1683976111yescharacteristic
328C22#1683978011yescharacteristic
329C4—21no—
330C4—21no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—4881no—
338C4—4881no—
339C4—4881no—
340C4#1683955812yes—
341C2—21no—
342C2—21no—
343C2—4882no—
344C2#2144436011yescharacteristic
345C2#2144746111yescharacteristic
346C2C2×#1683955811yescharacteristic
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#11829
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. #21443845C2semidirect
5. C61×#1801C2semidirect
169. #1801D61semidirect
196. C61#11829semidirect

Maximal subgroups

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

subgroupindex
#214491022
#214441632
#214438452
#214438212
C61×#18012
#1182961