Finite Group Databasebeta

← #29590252

Subgroups — #29590252

Order 15616.

456 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 87 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#467×D61C211yesmaximal, characteristic
2#21447428C211yesmaximal, characteristic
3#21445534C211yesmaximal, characteristic
4#21443832C211yesmaximal, characteristic, complemented
5#21443831C211yesmaximal, characteristic, complemented
6#21443830C211yesmaximal, characteristic, complemented
7C61×#1348C211yesmaximal, characteristic, complemented
8C2×Q16×D61C411yescharacteristic, complemented
9C22×C8×D61C2211yescharacteristic
10C22⋊Q8×D61—21no—
11C2×#14519735—21no—
12C2×#14519716—21no—
13#16840004C411yescharacteristic, complemented
14#16840003C2211yescharacteristic
15#16840002—21no—
16#16840001—21no—
17#16840000C411yescharacteristic, complemented
18#107×D61C411yescharacteristic, complemented
19#103×D61—21no—
20C22⋊#11758185C2211yescharacteristic
21#16839541—21no—
22#16839525C2211yescharacteristic
23#16839524—21no—
24C61×#467C2211yescharacteristic
25C61×C22⋊C16C2211yescharacteristic
26C61×#340C2211yescharacteristic
27C2×Q8×D61—21no—
28C22×C4×D61D411yescharacteristic
29C2×Dic244C2×C411yescharacteristic
30C2×#11758202—21no—
31Q16×D61—42no—
32C22×#92418D411yescharacteristic
33C2×C8×D61—42no—
34C2×C8×D61C2×C411yescharacteristic
35C2×C8×D61C2×C411yescharacteristic
36C2×C8×D61D411yescharacteristic
37C2×C8×D61D411yescharacteristic
38#14519757—21no—
39#14519756—21no—
40#14519755—41no—
41#14519754—41no—
42#14519753—21no—
43#14519735—24no—
44#14519735—42no—
45C2×C4⋊Dic61—21no—
46C4⋊C4×D61—21no—
47C4⋊C4×D61—41no—
48C2×#11758198—41no—
49C22⋊C4×D61—41no—
50C2×C4×Dic61—41no—
51#14519716—42no—
52C2×#11758185—21no—
53#14519711C2×C411yescharacteristic
54#14519710—21no—
55#14519702—21no—
56#14519701—21no—
57#14519700—21no—
58C2×Q16×C61C2×C411yescharacteristic
59C22×C8×C61C2311yescharacteristic
60C22⋊Q8×C61—21no—
61C2×C4⋊C4×C61—21no—
62C2×C16×C61—21no—
63#107×C61C2×C411yescharacteristic
64#103×C61—21no—
65C23×D61D811yescharacteristic
66C2×Q8×C61—21no—
67C22×C4×C61—41no—
68C22×C4×C61C2×D411yescharacteristic
69C22×Dic61—41no—
70C22×Dic61D811yescharacteristic
71Q8×D61—42no—
72Q8×D61—42no—
73C2×C4×D61—22no—
74C2×C4×D61—22no—
75C2×C4×D61—41no—
76C2×C4×D61—41no—
77C2×C4×D61C22⋊C411yescharacteristic
78C2×C4×D61C22⋊C411yescharacteristic
79C2×C4×D61SD1611yescharacteristic
80C2×C4×D61SD1611yescharacteristic
81C2×Dic122—21no—
82C2×Dic122—41no—
83Q16×C61—41no—
84C2×C8×C61—41no—
85C2×C8×C61C22×C411yescharacteristic
86C2×C8×C61C2×D411yescharacteristic
87C4⋊C4×C61—22no—
88C4⋊C4×C61—21no—
89C4⋊C4×C61—41no—
90C4⋊C4×C61—41no—
91C22⋊C4×C61—41no—
92C22⋊Dic61—41no—
93#11758202—42no—
94#11758198—44no—
95#11758198—41no—
96#11758198—41no—
97C4⋊Dic61—22no—
98C4⋊Dic61—21no—
99C4⋊Dic61—41no—
100C4⋊Dic61—41no—
101#11758196—41no—
102#11758196—41no—
103#11758196—41no—
104C4×Dic61—44no—
105C2×#92418—41no—
106C2×#92418C22⋊C411yescharacteristic
107C2×#92418C22⋊C411yescharacteristic
108Dic244—41no—
109C8×D61—22no—
110C8×D61—22no—
111C8×D61—41no—
112C8×D61—41no—
113C16×C61—41no—
114#11758185—41no—
115C23×C61C2×D811yescharacteristic
116C22×D61—21no—
117C22×D61—21no—
118C22×D61—22no—
119C22×D61—41no—
120C22×D61#10211yescharacteristic
121C22×D61#10211yescharacteristic
122Q8×C61—41no—
123Q8×C61—41no—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61—42no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—81no—
130C2×C4×C61—81no—
131C2×C4×C61C2×C22⋊C411yescharacteristic
132C2×C4×C61C2×SD1611yescharacteristic
133C2×Dic61—22no—
134C2×Dic61—42no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61—81no—
138C2×Dic61—81no—
139C2×Dic61#10211yescharacteristic
140C2×Dic61#10211yescharacteristic
141C4×D61—22no—
142C4×D61—21no—
143C4×D61—21no—
144C4×D61—22no—
145C4×D61—82no—
146C4×D61—82no—
147C4×D61#9912yes—
148C4×D61#10412yes—
149Dic122—41no—
150Dic122—41no—
151Dic122—82no—
152C8×C61—21no—
153C8×C61—21no—
154#92418—21no—
155#92418—21no—
156#11706—611nomaximal, Sylow
157C22×C61—21no—
158C22×C61—21no—
159C22×C61C2×#10211yescharacteristic
160C22×C61#51011yescharacteristic
161C22×C61#51011yescharacteristic
162C2×D61—21no—
163C2×D61—21no—
164C2×D61—21no—
165C2×D61—21no—
166C2×D61—21no—
167C2×D61—42no—
168C2×D61—42no—
169C2×D61#32712yes—
170C4×C61—21no—
171C4×C61—82no—
172C4×C61—81no—
173C4×C61—81no—
174C4×C61C2×#9911yescharacteristic
175C4×C61C2×#10411yescharacteristic
176Dic61—21no—
177Dic61—82no—
178Dic61—81no—
179Dic61—81no—
180Dic61#32712yes—
181C2×#467—611no—
182#2112—611no—
183#1735—611no—
184#1348—611no—
185#1348—611no—
186#1348—611no—
187#1348D6111yescharacteristic, complemented
188C2×C61—21no—
189C2×C61—21no—
190C2×C61C2×#32711yescharacteristic
191C2×C61#214111yescharacteristic
192C2×C61#214511yescharacteristic
193D61—22no—
194D61—41no—
195C22×Q16—611no—
196C23×C8—611no—
197C2×C22⋊Q8—1221no—
198C2×#117—1221no—
199C2×M32—1221no—
200#467—611no—
201#467—611no—
202#467—611no—
203#467—1221no—
204#467—1221no—
205#467C2×D6111yescharacteristic
206C2×#107—611no—
207C2×#103—1221no—
208C22⋊C16—611no—
209C22⋊C16—1221no—
210C22⋊C16C2×D6111yescharacteristic
211#340—611no—
212#340—1221no—
213#340C2×D6111yescharacteristic
214C61#1170611yescharacteristic, complemented, Sylow
215C22×Q8—1221no—
216C23×C4—611no—
217C2×Q16—611no—
218C2×Q16—1221no—
219C2×Q16—2442no—
220C2×Q16C4×D6111yescharacteristic
221C22×C8—611no—
222C22×C8—611no—
223C22×C8—611no—
224C22×C8—611no—
225C22×C8—611no—
226C22×C8—2442no—
227C22×C8C22×D6111yescharacteristic
228C22⋊Q8—21no—
229C22⋊Q8—1221no—
230C22⋊Q8—1221no—
231C22⋊Q8—1221no—
232C22⋊Q8—2441no—
233C22⋊Q8—2441no—
234#117—1224no—
235#117—2442no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—1221no—
238C2×C4⋊C4—1221no—
239C2×C4⋊C4—2441no—
240C2×C22⋊C4—2441no—
241C2×C22⋊C4—2441no—
242C2×C42—2441no—
243M32—2442no—
244C2×C16—21no—
245C2×C16—1221no—
246#107—611no—
247#107—1221no—
248#107C4×D6111yescharacteristic
249#103—21no—
250#103—1221no—
251#103—1221no—
252#103—1221no—
253C24—611no—
254C2×Q8—21no—
255C2×Q8—1221no—
256C2×Q8—2441no—
257C2×Q8—2442no—
258C2×Q8—2442no—
259C22×C4—41no—
260C22×C4—611no—
261C22×C4—611no—
262C22×C4—611no—
263C22×C4—611no—
264C22×C4—611no—
265C22×C4—1222no—
266C22×C4—1222no—
267C22×C4—2441no—
268C22×C4—2441no—
269C22×C4—2441no—
270C22×C4D4×D6111yescharacteristic
271Q16—41no—
272Q16—2441no—
273Q16—2442no—
274C2×C8—41no—
275C2×C8—611no—
276C2×C8—611no—
277C2×C8—1222no—
278C2×C8—1222no—
279C2×C8—2441no—
280C2×C8—2441no—
281C2×C8—2441no—
282C2×C8C2×C4×D6111yescharacteristic
283C2×C8D4×D6111yescharacteristic
284C4⋊C4—22no—
285C4⋊C4—21no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—1222no—
289C4⋊C4—1221no—
290C4⋊C4—2441no—
291C4⋊C4—2441no—
292C4⋊C4—2441no—
293C4⋊C4—2441no—
294C4⋊C4—2441no—
295C22⋊C4—41no—
296C22⋊C4—2441no—
297C22⋊C4—2441no—
298C22⋊C4—2444no—
299C22⋊C4—2441no—
300C42—2444no—
301C16—41no—
302C16—2441no—
303C23—611no—
304C23—611no—
305C23—1222no—
306C23—1221no—
307C23—1221no—
308C23—2441no—
309C23D8×D6111yescharacteristic
310Q8—41no—
311Q8—41no—
312Q8—4882no—
313Q8—2441no—
314Q8—2441no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—611no—
323C2×C4—612no—
324C2×C4—612no—
325C2×C4—611no—
326C2×C4—1222no—
327C2×C4—1222no—
328C2×C4—1222no—
329C2×C4—1221no—
330C2×C4—1221no—
331C2×C4—4882no—
332C2×C4—4882no—
333C2×C4—4881no—
334C2×C4—4881no—
335C2×C4—2442no—
336C2×C4—2441no—
337C2×C4—2441no—
338C2×C4C22⋊C4×D6111yescharacteristic
339C2×C4SD16×D6111yescharacteristic
340C8—21no—
341C8—21no—
342C8—1221no—
343C8—1221no—
344C22—21no—
345C22—21no—
346C22—612no—
347C22—1221no—
348C22—1221no—
349C22—1221no—
350C22—1221no—
351C22—1221no—
352C22—2442no—
353C22—2442no—
354C22#102×D6111yescharacteristic
355C22#1684020111yescharacteristic
356C22#1684020411yescharacteristic
357C4—21no—
358C4—82no—
359C4—81no—
360C4—81no—
361C4—612no—
362C4—1221no—
363C4—4882no—
364C4—4881no—
365C4—4881no—
366C4#99×D6111yescharacteristic
367C4#104×D6111yescharacteristic
368C2—21no—
369C2—21no—
370C2—1222no—
371C2—2441no—
372C2#327×D6111yescharacteristic
373C2#2144753711yescharacteristic
374C2#2144755311yescharacteristic
subgroupG/H
centerC22#102×D61
commutatorC2×C8×C61C22×C4
FrattiniC22×C8C22×D61
FittingC61×#1348C2
socleC22×C61C2×#102
prime radicalC23×D61D8
cosocleC22×C8×C61C23
hypercenter#1348D61
hypocenterC61#11706
LCSG ▷ C2×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1348

Complements

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

NH ≅ G/Ntype
4. #21443832C2semidirect
5. #21443831C2semidirect
6. #21443830C2semidirect
7. C61×#1348C2semidirect
8. C2×Q16×D61C4semidirect
13. #16840004C4semidirect
17. #16840000C4semidirect
18. #107×D61C4semidirect
187. #1348D61semidirect
214. C61#11706semidirect

Maximal subgroups

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

subgroupindex
#467×D612
#214474282
#214455342
#214438322
#214438312
#214438302
C61×#13482
#1170661