Finite Group Databasebeta

← #29580874

Subgroups — #29580874

Order 15616.

560 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 97 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#21455823C211yesmaximal, characteristic
2#21449102C211yesmaximal, characteristic
3#21446801C211yesmaximal, characteristic, complemented
4#21444248C211yesmaximal, characteristic, complemented
5#21444245C211yesmaximal, characteristic
6#21443647C211yesmaximal, characteristic, complemented
7C61×#3104C211yesmaximal, characteristic, complemented
8C2×C4×D244C2211yescharacteristic
9#16840090—21no—
10#16840049—21no—
11#16839964—21no—
12#16839912—21no—
13C4×#11758201—22no—
14#16839815C2211yescharacteristic
15#16839786C2211yescharacteristic
16#16839701—21no—
17#16839688—21no—
18#16839679—21no—
19#16839673—21no—
20C4×#11758198—22no—
21#16839608C2211yescharacteristic
22#16839521—22no—
23Q8⋊#92418—22no—
24#16839491—22no—
25C61×#533C2211yescharacteristic
26C2×C4×Q8×C61C2211yescharacteristic
27C61×#484—21no—
28C61×#477—21no—
29C61×#462—21no—
30C61×#451—21no—
31C4×Q16×C61—22no—
32C61×#431C2211yescharacteristic
33C22×D244D411yescharacteristic
34C22×C4×D61—21no—
35C2×#11758201—22no—
36C2×#11758194—21no—
37C4×D244—21no—
38C4×D244—21no—
39C4×D244—21no—
40C4×D244—21no—
41C2×C4⋊Dic61D411yescharacteristic
42C2×#11758198—22no—
43C2×#11758198—22no—
44C2×#11758198—21no—
45C2×C4×Dic61D412yes—
46#14519708—22no—
47C4⋊#92418—21no—
48C4⋊#92418—21no—
49C4⋊#92418—21no—
50#14519701—22no—
51#14519697—22no—
52C22⋊#92418—21no—
53C4×#92418D412yes—
54#14519684—22no—
55#14519684—22no—
56#14519684—21no—
57#14519684—21no—
58C22×Q8×C61D411yescharacteristic
59C2×Q16×C61—22no—
60C4⋊Q8×C61D411yescharacteristic
61C4⋊Q8×C61D411yescharacteristic
62#125×C61—21no—
63C22⋊Q8×C61—21no—
64C22⋊Q8×C61—21no—
65C4×Q8×C61—21no—
66C4×Q8×C61—21no—
67C4×Q8×C61—21no—
68C4×Q8×C61—21no—
69C4×Q8×C61—21no—
70C4×Q8×C61—21no—
71#117×C61—21no—
72C2×C4⋊C4×C61—21no—
73C2×C4⋊C4×C61D411yescharacteristic
74C2×C42×C61—21no—
75C2×C42×C61C2311yescharacteristic
76#107×C61—22no—
77C4⋊C8×C61—21no—
78#103×C61—22no—
79#103×C61—22no—
80C22⋊C8×C61—21no—
81C4×C8×C61D412yes—
82C23×D61—21no—
83C2×Q8×C61—21no—
84C2×Q8×C61—21no—
85C2×Q8×C61—21no—
86C2×Q8×C61—21no—
87C2×Q8×C61—42no—
88C22×C4×C61—21no—
89C22×C4×C61—21no—
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×Dic61—21no—
94C22×Dic61—22no—
95C2×D244—21no—
96C2×D244—21no—
97C2×D244—42no—
98C2×C4×D61—41no—
99C2×C4×D61—41no—
100C2×C4×D61—42no—
101C2×C4×D61—42no—
102Q16×C61—44no—
103C2×C8×C61—22no—
104C4⋊C4×C61—21no—
105C4⋊C4×C61—21no—
106C4⋊C4×C61—21no—
107C4⋊C4×C61—21no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61—41no—
111C4⋊C4×C61—41no—
112C4⋊C4×C61—41no—
113C4⋊C4×C61—41no—
114C22⋊C4×C61—41no—
115C42×C61—41no—
116C42×C61—41no—
117C42×C61—41no—
118C42×C61C2×D411yescharacteristic
119C42×C61C2×D411yescharacteristic
120C42×C61C2×D411yescharacteristic
121C42×C61#4111yescharacteristic
122#11758201—44no—
123#11758198—44no—
124#11758198—44no—
125#11758198—41no—
126#11758198—41no—
127C4⋊Dic61—21no—
128C4⋊Dic61—21no—
129C4×Dic61—22no—
130C4×Dic61—22no—
131#11758194—42no—
132C2×#92418—22no—
133C2×#92418—41no—
134C23×C61#12011yescharacteristic
135C22×D61—22no—
136C22×D61—42no—
137C22×D61—41no—
138C22×D61—41no—
139C22×D61—42no—
140Q8×C61—22no—
141Q8×C61—22no—
142Q8×C61—41no—
143Q8×C61—41no—
144Q8×C61—42no—
145Q8×C61—42no—
146C2×C4×C61—22no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—42no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61#12011yescharacteristic
156C2×C4×C61#12011yescharacteristic
157C2×C4×C61#12111yescharacteristic
158C2×C4×C61#12111yescharacteristic
159C2×C4×C61#12111yescharacteristic
160C2×C4×C61#12711yescharacteristic
161C2×C4×C61#13512yes—
162C2×C4×C61#13712yes—
163C2×Dic61—42no—
164C2×Dic61—42no—
165C2×Dic61—41no—
166C2×Dic61—42no—
167C2×Dic61—42no—
168D244—22no—
169D244—22no—
170D244—41no—
171D244—41no—
172C4×D61—82no—
173C4×D61—81no—
174C4×D61—81no—
175C8×C61—42no—
176#92418—42no—
177#92418—81no—
178#10016—611nomaximal, Sylow
179C22×C61—21no—
180C22×C61—21no—
181C22×C61#39211yescharacteristic
182C22×C61#45212yes—
183C2×D61—44no—
184C2×D61—42no—
185C2×D61—82no—
186C2×D61—81no—
187C2×D61—81no—
188C4×C61—21no—
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—21no—
192C4×C61—42no—
193C4×C61—81no—
194C4×C61—81no—
195C4×C61—81no—
196C4×C61#46412yes—
197C4×C61#49511yescharacteristic
198C4×C61#49711yescharacteristic
199Dic61—82no—
200Dic61—81no—
201#3104D6111yescharacteristic, complemented
202#3101—611no—
203#2398—611no—
204#1995—611no—
205#1487—611no—
206#1486—611no—
207#1274—611no—
208C2×C61—22no—
209C2×C61#201011yescharacteristic
210C2×C61#201011yescharacteristic
211C2×C61#201511yescharacteristic
212D61—82no—
213#533C2×D6111yescharacteristic
214C2×C4×Q8C2×D6111yescharacteristic
215C2×C4×D4—611no—
216#484—21no—
217#483—1221no—
218#477—21no—
219#476—1221no—
220#462—21no—
221#461—1221no—
222#451—21no—
223#450—1221no—
224C4×Q16—22no—
225C4×SD16—1222no—
226#431—611no—
227#431C2×D6111yescharacteristic
228#423—611no—
229#400—1221no—
230#397—1221no—
231#394—1221no—
232#393—1221no—
233C4×C22⋊C4—1222no—
234#376—611no—
235#339—1222no—
236Q8⋊C8—1222no—
237D4⋊C8—1222no—
238C61#1001611yescharacteristic, complemented, Sylow
239C22×Q8C22⋊D6111yescharacteristic
240C22×D4—611no—
241C23×C4—1221no—
242C2×Q16—22no—
243C2×SD16—1222no—
244C2×M16—1221no—
245C4⋊Q8C22⋊D6111yescharacteristic
246C4⋊Q8C22⋊D6111yescharacteristic
247#125—21no—
248C22⋊Q8—21no—
249C22⋊Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8—21no—
254C4×Q8—21no—
255C4×Q8—21no—
256C4×D4—1221no—
257C4×D4—1221no—
258C4×D4—1221no—
259C4×D4—1221no—
260#117—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—611no—
263C2×C4⋊C4C22⋊D6111yescharacteristic
264C2×C22⋊C4—1222no—
265C2×C22⋊C4—1222no—
266C2×C22⋊C4—1221no—
267C2×C42—21no—
268C2×C42—612no—
269C2×C42C22×D6111yescharacteristic
270#107—22no—
271#106—1222no—
272C4⋊C8—21no—
273C4⋊C8—1221no—
274C4⋊C8—1221no—
275C4⋊C8—1221no—
276#103—22no—
277#103—22no—
278#103—1222no—
279#102—1222no—
280C22⋊C8—21no—
281C22⋊C8—1221no—
282C4×C8—612no—
283C4×C8C22⋊D6112yes—
284#95—1222no—
285#95—1222no—
286#95—1221no—
287#95—1221no—
288C24—1221no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—42no—
294C2×D4—1221no—
295C2×D4—1221no—
296C2×D4—2442no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—1222no—
300C22×C4—1221no—
301C22×C4—2442no—
302C22×C4—2441no—
303C22×C4—2441no—
304C22×C4—2442no—
305C22×C4D4×D6111yescharacteristic
306C22×C4D4×D6111yescharacteristic
307C22×C4C2×C22⋊D6111yescharacteristic
308Q16—44no—
309SD16—2444no—
310M16—2442no—
311C2×C8—22no—
312C2×C8—1222no—
313C2×C8—2441no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—1221no—
325C4⋊C4—1221no—
326C22⋊C4—41no—
327C22⋊C4—2444no—
328C22⋊C4—2444no—
329C22⋊C4—2441no—
330C22⋊C4—2441no—
331C42—41no—
332C42—41no—
333C42—41no—
334C42—1222no—
335C42—1222no—
336C42D4×D6111yescharacteristic
337C42#1175822411yescharacteristic
338C42C2×C22⋊D6111yescharacteristic
339C42C2×C22⋊D6111yescharacteristic
340C23—1222no—
341C23—2442no—
342C23—2441no—
343C23—2442no—
344C23—2441no—
345C23#1451974511yescharacteristic
346Q8—22no—
347Q8—22no—
348Q8—41no—
349Q8—41no—
350Q8—42no—
351Q8—42no—
352D4—1222no—
353D4—1222no—
354D4—2441no—
355D4—2441no—
356C2×C4—22no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—4882no—
366C2×C4—4881no—
367C2×C4—4881no—
368C2×C4—2442no—
369C2×C4—2442no—
370C2×C4—2442no—
371C2×C4—2442no—
372C2×C4—2441no—
373C2×C4#1451974511yescharacteristic
374C2×C4#1451974611yescharacteristic
375C2×C4#1451974811yescharacteristic
376C2×C4#1451974911yescharacteristic
377C2×C4#1451974911yescharacteristic
378C2×C4#1451977311yescharacteristic
379C2×C4#1451980012yes—
380C2×C4#1451980912yes—
381C8—42no—
382C8—4881no—
383C8—2442no—
384C22—21no—
385C22—21no—
386C22—4882no—
387C22—4881no—
388C22—4881no—
389C22—2444no—
390C22—2442no—
391C22#1683967011yescharacteristic
392C22#1683992312yes—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—42no—
398C4—81no—
399C4—81no—
400C4—81no—
401C4—4882no—
402C4—4881no—
403C4#1683998612yes—
404C4#1684014711yescharacteristic
405C4#1684015511yescharacteristic
406C2—22no—
407C2—4882no—
408C2#2144688311yescharacteristic
409C2#2144688411yescharacteristic
410C2#2144691011yescharacteristic
subgroupG/H
centerC2×C4#14519773
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#3104C2
socleC22×C61#392
prime radicalC22×D244D4
hypercenter#3104D61
hypocenterC61#10016
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3104

Complements

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

NH ≅ G/Ntype
3. #21446801C2semidirect
4. #21444248C2semidirect
6. #21443647C2semidirect
7. C61×#3104C2semidirect
201. #3104D61semidirect
238. C61#10016semidirect

Maximal subgroups

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

subgroupindex
#214558232
#214491022
#214468012
#214442482
#214442452
#214436472
C61×#31042
#1001661