Finite Group Databasebeta

← #29589116

Subgroups — #29589116

Order 15616.

602 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 93 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#21448306C211yesmaximal, characteristic
2#21444132C212yesmaximal, complemented
3#21443752C212yesmaximal, complemented
4#21443751C211yesmaximal, characteristic, complemented
5C61×#1318C211yesmaximal, characteristic, complemented
6C22×#11758224C411yescharacteristic
7C2×#14519735C2211yescharacteristic
8C2×C42×D61C411yescharacteristic
9#16839812—22no—
10#16839668C411yescharacteristic
11#16839667C2211yescharacteristic
12#16839665C411yescharacteristic
13#16839638—22no—
14C22⋊C4×Dic61—22no—
15#16839488C2212yes—
16#16839485—22no—
17C61×#390C2211yescharacteristic
18C61×#324C2212yes—
19C2×#11758224—22no—
20C2×#11758224—41no—
21C2×#11758224—41no—
22C2×#11758224—42no—
23C2×#11758224C814yescomplemented
24C22×Dic122C2×C411yescharacteristic
25C22×C22⋊D61—21no—
26C23×Dic61—21no—
27C22×C4×D61—21no—
28C22×C4×D61C2×C411yescharacteristic
29#14519735—41no—
30#14519735—41no—
31C2×C4⋊Dic61C2×C411yescharacteristic
32C2×C22⋊Dic61—22no—
33C2×#11758198D412yes—
34C2×C4×Dic61—21no—
35C2×C4×Dic61C2×C411yescharacteristic
36C2×C4×Dic61D412yes—
37C42×D61—42no—
38#14519706—42no—
39C22⋊#92418—22no—
40C8×Dic61—42no—
41#14519684—22no—
42#14519684—22no—
43C22×D4×C61C2×C411yescharacteristic
44C2×C4⋊C4×C61C2311yescharacteristic
45C2×C22⋊C4×C61—22no—
46C2×C42×C61C2×C411yescharacteristic
47C22⋊C8×C61—22no—
48C24×C61—21no—
49C23×D61C22⋊C411yescharacteristic
50C2×D4×C61—21no—
51C2×D4×C61—41no—
52C2×D4×C61—41no—
53C2×D4×C61C2×C812yes—
54C22×C4×C61—21no—
55C22×C4×C61C22×C411yescharacteristic
56C22×C4×C61C2×D412yes—
57C2×C22⋊D61—24no—
58C2×C22⋊D61—42no—
59C2×C22⋊D61—42no—
60C22×Dic61—21no—
61C22×Dic61—21no—
62C22×Dic61—24no—
63C22×Dic61—41no—
64C22×Dic61—41no—
65C22×Dic61—42no—
66C22×Dic61C22⋊C412yes—
67C22×Dic61C22⋊C411yescharacteristic
68#11758224—44no—
69#11758224—44no—
70#11758224—44no—
71#11758224—44no—
72C2×C4×D61—21no—
73C2×C4×D61—21no—
74C2×C4×D61—22no—
75C2×C4×D61—42no—
76C2×C4×D61—42no—
77C2×C4×D61—42no—
78C2×C4×D61C2×C812yes—
79C2×C4×D61M1612yes—
80C2×Dic122—21no—
81C2×Dic122—41no—
82C2×Dic122—41no—
83C2×Dic122C2×C812yes—
84C2×C8×C61—42no—
85C4⋊C4×C61—41no—
86C22⋊C4×C61—44no—
87C42×C61—41no—
88C22⋊Dic61—44no—
89#11758198—42no—
90C4⋊Dic61—41no—
91C4×Dic61—22no—
92C4×Dic61—22no—
93C4×Dic61—42no—
94C4×Dic61—41no—
95C2×#92418—42no—
96C23×C61—22no—
97C23×C61—42no—
98C23×C61—42no—
99C23×C61—41no—
100C23×C61—41no—
101C23×C61C2×C22⋊C411yescharacteristic
102C22×D61—21no—
103C22×D61—21no—
104C22×D61—41no—
105C22×D61—41no—
106C22×D61C22⋊C812yes—
107D4×C61—42no—
108D4×C61—42no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—22no—
112C2×C4×C61—22no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—42no—
116C2×C4×C61C22×C811yescharacteristic
117C2×C4×C61C2×M1611yescharacteristic
118C22⋊D61—48no—
119C2×Dic61—22no—
120C2×Dic61—22no—
121C2×Dic61—22no—
122C2×Dic61—41no—
123C2×Dic61—41no—
124C2×Dic61—41no—
125C2×Dic61—41no—
126C2×Dic61—44no—
127C2×Dic61—44no—
128C2×Dic61—42no—
129C2×Dic61C22⋊C812yes—
130C2×Dic61#9911yescharacteristic
131C2×Dic61#9911yescharacteristic
132C2×Dic61#10012yes—
133C4×D61—22no—
134C4×D61—22no—
135C4×D61—22no—
136C4×D61—22no—
137C4×D61—48no—
138Dic122—42no—
139Dic122—42no—
140C8×C61—82no—
141#92418—82no—
142#11431—611nomaximal, Sylow
143C22×C61—21no—
144C22×C61—21no—
145C22×C61—44no—
146C22×C61—42no—
147C22×C61—82no—
148C22×C61—81no—
149C22×C61—81no—
150C22×C61C2×C22⋊C811yescharacteristic
151C22×C61C2×#9911yescharacteristic
152C22×C61C2×#10011yescharacteristic
153C2×D61—22no—
154C2×D61—42no—
155C2×D61—41no—
156C2×D61—41no—
157C2×D61—42no—
158C4×C61—21no—
159C4×C61—21no—
160C4×C61—42no—
161C4×C61—42no—
162Dic61—42no—
163Dic61—42no—
164Dic61—42no—
165Dic61#32312yes—
166Dic61#32312yes—
167#2297—611no—
168#1458—612no—
169#1319—612no—
170#1318—611no—
171#1318D6111yescharacteristic, complemented
172C2×C61—21no—
173C2×C61—21no—
174C2×C61—82no—
175C2×C61C2×#32311yescharacteristic
176C2×C61#212611yescharacteristic
177C2×C61#213211yescharacteristic
178D61—42no—
179C22×#41—611no—
180C2×#117—611no—
181C22×C42—611no—
182#431—1222no—
183#391—611no—
184#391—611no—
185#390—611no—
186#390C2×D6111yescharacteristic
187#386—1222no—
188C4×C22⋊C4—1222no—
189#324—612no—
190#324C2×D6112yes—
191#323—1222no—
192C61#1143111yescharacteristic, complemented, Sylow
193C2×#41—614no—
194C2×#41—1222no—
195C2×#41—2442no—
196C2×#41—2441no—
197C2×#41—2441no—
198C22×Q8—611no—
199C22×D4—1221no—
200C22×D4C4×D6111yescharacteristic
201C23×C4—611no—
202C23×C4—1221no—
203C23×C4—1221no—
204#117—2441no—
205#117—2441no—
206C2×C4⋊C4—611no—
207C2×C4⋊C4C22×D6111yescharacteristic
208C2×C22⋊C4—22no—
209C2×C22⋊C4—612no—
210C2×C22⋊C4—1222no—
211C2×C42—612no—
212C2×C42—611no—
213C2×C42—1221no—
214C2×C42—2442no—
215C2×C42C4×D6111yescharacteristic
216C4⋊C8—2442no—
217C22⋊C8—22no—
218C22⋊C8—1222no—
219C4×C8—2442no—
220#95—1222no—
221#95—1222no—
222C24—21no—
223C24—611no—
224#41—2444no—
225#41—2444no—
226#41—2444no—
227#41—2444no—
228C2×Q8—612no—
229C2×Q8—1221no—
230C2×Q8—2441no—
231C2×Q8—2441no—
232C2×D4—21no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—1224no—
236C2×D4—2442no—
237C2×D4—2442no—
238C2×D4C8×D6112yes—
239C22×C4—21no—
240C22×C4—611no—
241C22×C4—612no—
242C22×C4—612no—
243C22×C4—612no—
244C22×C4—1222no—
245C22×C4—1221no—
246C22×C4—1221no—
247C22×C4—1224no—
248C22×C4—1221no—
249C22×C4—1221no—
250C22×C4—2442no—
251C22×C4—2442no—
252C22×C4—2442no—
253C22×C4—2441no—
254C22×C4—2441no—
255C22×C4—2442no—
256C22×C4C2×C4×D6111yescharacteristic
257C22×C4D4×D6112yes—
258C2×C8—42no—
259C2×C8—2442no—
260C4⋊C4—41no—
261C4⋊C4—2441no—
262C22⋊C4—44no—
263C22⋊C4—2442no—
264C22⋊C4—2444no—
265C42—41no—
266C42—1222no—
267C42—1222no—
268C42—2441no—
269C42—2442no—
270C23—22no—
271C23—42no—
272C23—42no—
273C23—41no—
274C23—41no—
275C23—612no—
276C23—1221no—
277C23—1221no—
278C23—2441no—
279C23—2441no—
280C23C22⋊C4×D6111yescharacteristic
281Q8—2442no—
282Q8—2442no—
283D4—42no—
284D4—42no—
285D4—2448no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—612no—
294C2×C4—611no—
295C2×C4—611no—
296C2×C4—612no—
297C2×C4—1222no—
298C2×C4—1222no—
299C2×C4—1222no—
300C2×C4—1222no—
301C2×C4—1222no—
302C2×C4—1222no—
303C2×C4—1222no—
304C2×C4—2448no—
305C2×C4—2444no—
306C2×C4—2444no—
307C2×C4—2441no—
308C2×C4—2441no—
309C2×C4—2441no—
310C2×C4—2441no—
311C2×C4—2442no—
312C2×C4C2×C8×D6111yescharacteristic
313C2×C4M16×D6111yescharacteristic
314C8—82no—
315C8—4882no—
316C22—21no—
317C22—21no—
318C22—44no—
319C22—42no—
320C22—82no—
321C22—81no—
322C22—81no—
323C22—1222no—
324C22—2442no—
325C22—2442no—
326C22—2441no—
327C22—2441no—
328C22C22⋊C8×D6111yescharacteristic
329C22#99×D6111yescharacteristic
330C22#100×D6111yescharacteristic
331C4—21no—
332C4—21no—
333C4—42no—
334C4—42no—
335C4—612no—
336C4—612no—
337C4—2442no—
338C4—2442no—
339C4—2442no—
340C2—21no—
341C2—21no—
342C2—82no—
343C2—2442no—
344C2#323×D6111yescharacteristic
345C2#2144748111yescharacteristic
346C2#2144750111yescharacteristic
subgroupG/H
centerC22#99×D61
commutatorC2×C4×C61C22×C8
FrattiniC2×C4⋊C4C22×D61
FittingC61×#1318C2
socleC22×C61C2×#99
prime radicalC22×#11758224C4
cosocleC2×C4⋊C4×C61C23
hypercenter#1318D61
hypocenterC61#11431
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1318

Complements

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

NH ≅ G/Ntype
2. #21444132C2semidirect
3. #21443752C2semidirect
4. #21443751C2semidirect
5. C61×#1318C2semidirect
23. C2×#11758224C8semidirect
171. #1318D61semidirect
192. C61#11431semidirect

Maximal subgroups

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

subgroupindex
#214483062
#214441322
#214437522
#214437512
C61×#13182
#1143161