Finite Group Databasebeta

← #29577889

Subgroups — #29577889

Order 15616.

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

normal subgroups only

This group has 544 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#21454236C211yesmaximal, characteristic
2#21451390C211yesmaximal, characteristic
3#21444266C211yesmaximal, characteristic, complemented
4#21444151C211yesmaximal, characteristic
5#21443685C211yesmaximal, characteristic, complemented
6#21443684C211yesmaximal, characteristic, complemented
7C61×#1491C211yesmaximal, characteristic, complemented
8C2×M16×D61—21no—
9C2×#14519734C2211yescharacteristic
10#16839819C2211yescharacteristic
11#16839719—22no—
12#16839716—22no—
13C4×#11758194C2211yescharacteristic
14M16×Dic61—21no—
15#16839693C411yescharacteristic
16#16839688C411yescharacteristic
17#16839681C411yescharacteristic
18#16839673C411yescharacteristic
19#16839653—21no—
20#16839646—21no—
21#16839643—21no—
22#16839633—21no—
23#16839630C2211yescharacteristic
24Q8⋊#92418—22no—
25#16839487—21no—
26#16839485—21no—
27C2×C4×Q8×C61C2211yescharacteristic
28C61×#432C2211yescharacteristic
29C4×M16×C61C2211yescharacteristic
30C61×Q8⋊C8—22no—
31C22×C4×D61D411yescharacteristic
32C2×#11758194—21no—
33C2×#11758189—22no—
34M16×D61—24no—
35M16×D61—24no—
36C2×C8×D61—22no—
37#14519734—21no—
38#14519734—21no—
39#14519734—21no—
40#14519734—21no—
41C2×#11758196—21no—
42C2×#11758196—21no—
43C2×#11758196C2×C411yescharacteristic
44C2×#11758196D411yescharacteristic
45C2×#11758198—21no—
46C2×#11758198—21no—
47C2×#11758198C2×C411yescharacteristic
48C2×#11758198D411yescharacteristic
49C2×C4×Dic61D411yescharacteristic
50C4⋊#92418—21no—
51C22⋊#92418—21no—
52#14519688—21no—
53#14519688—21no—
54#14519687—22no—
55C4×#92418D412yes—
56C8×Dic61—22no—
57#14519684—21no—
58#14519684—21no—
59#14519684—21no—
60#14519684—21no—
61#14519684—21no—
62#14519684—21no—
63#14519684C2×C411yescharacteristic
64#14519684C2×C411yescharacteristic
65C22×Q8×C61C2×C411yescharacteristic
66C2×M16×C61—21no—
67C4×Q8×C61—21no—
68C4×Q8×C61—21no—
69C4×Q8×C61—21no—
70C4×Q8×C61—21no—
71C2×C4⋊C4×C61—21no—
72C2×C4⋊C4×C61C2×C411yescharacteristic
73C2×C42×C61—21no—
74C2×C42×C61C2311yescharacteristic
75C4⋊C8×C61—21no—
76C22⋊C8×C61—21no—
77#97×C61—21no—
78#97×C61—21no—
79C4×C8×C61D412yes—
80C23×D61C22⋊C411yescharacteristic
81C2×Q8×C61—21no—
82C2×Q8×C61—21no—
83C2×Q8×C61—42no—
84C22×C4×C61—21no—
85C22×C4×C61—21no—
86C22×C4×C61C22×C411yescharacteristic
87C22×C4×C61C2×D411yescharacteristic
88C22×C4×C61C2×D411yescharacteristic
89C22×Dic61—21no—
90C22×Dic61—21no—
91C22×Dic61C22⋊C411yescharacteristic
92C22×Dic61C22⋊C411yescharacteristic
93C22×Dic61C22⋊C411yescharacteristic
94C2×C4×D61—22no—
95C2×C4×D61—21no—
96C2×C4×D61—21no—
97C2×C4×D61—22no—
98M16×C61—22no—
99M16×C61—22no—
100C2×C8×C61—22no—
101C2×C8×C61—41no—
102C4⋊C4×C61—21no—
103C4⋊C4×C61—21no—
104C4⋊C4×C61—41no—
105C4⋊C4×C61—41no—
106C42×C61—41no—
107C42×C61—41no—
108C42×C61C2×D411yescharacteristic
109C42×C61C2×D411yescharacteristic
110C42×C61#4111yescharacteristic
111C42×C61#4111yescharacteristic
112#11758198—21no—
113#11758198—21no—
114#11758198—21no—
115#11758198—21no—
116#11758198—42no—
117#11758198—42no—
118#11758196—21no—
119#11758196—21no—
120#11758196—21no—
121#11758196—21no—
122#11758196—42no—
123#11758196—42no—
124C4×Dic61—21no—
125C4×Dic61—21no—
126#11758194—22no—
127#11758194—22no—
128C2×#92418—22no—
129C2×#92418—41no—
130#11758189—28no—
131C8×D61—28no—
132C23×C61C2×C22⋊C411yescharacteristic
133C22×D61—21no—
134C22×D61—21no—
135C22×D61—41no—
136C22×D61—41no—
137C22×D61#10412yes—
138Q8×C61—22no—
139Q8×C61—22no—
140Q8×C61—41no—
141Q8×C61—41no—
142C2×C4×C61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—22no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61C4×D411yescharacteristic
153C2×C4×C61C4×D411yescharacteristic
154C2×C4×C61#12011yescharacteristic
155C2×C4×C61#12111yescharacteristic
156C2×C4×C61#12111yescharacteristic
157C2×C4×C61#12311yescharacteristic
158C2×Dic61—21no—
159C2×Dic61—21no—
160C2×Dic61—21no—
161C2×Dic61—21no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61#10412yes—
171C4×D61—22no—
172C4×D61—22no—
173C4×D61—24no—
174C8×C61—24no—
175C8×C61—81no—
176#92418—24no—
177#92418—81no—
178#9491—611nomaximal, Sylow
179C22×C61—21no—
180C22×C61—21no—
181C22×C61#38611yescharacteristic
182C22×C61C2×#10411yescharacteristic
183C22×C61C2×#10411yescharacteristic
184C2×D61—22no—
185C2×D61—41no—
186C2×D61—41no—
187C2×D61—42no—
188C2×D61—81no—
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—21no—
192C4×C61—21no—
193C4×C61—21no—
194C4×C61—21no—
195C4×C61—42no—
196C4×C61—81no—
197Dic61—22no—
198Dic61—41no—
199Dic61—41no—
200Dic61—81no—
201Dic61—81no—
202#2873—611no—
203#2599—611no—
204#1491—611no—
205#1491D6111yescharacteristic, complemented
206#1464—611no—
207#1288—611no—
208#1288—611no—
209C2×C61—21no—
210C2×C61—21no—
211C2×C61#189111yescharacteristic
212C2×C61#189611yescharacteristic
213C2×C61#190911yescharacteristic
214D61—42no—
215C22×M16—1221no—
216C2×C4×Q8C2×D6111yescharacteristic
217C2×#117—611no—
218#432—611no—
219#432C2×D6111yescharacteristic
220#405—1222no—
221#405—1222no—
222C4×M16—611no—
223C4×M16—1221no—
224C4×M16C2×D6111yescharacteristic
225#398—611no—
226#397—611no—
227#395—611no—
228#393—611no—
229#388—1221no—
230#387—1221no—
231#387—1221no—
232#385—1221no—
233#383—611no—
234Q8⋊C8—22no—
235Q8⋊C8—1222no—
236#324—1221no—
237#323—1221no—
238C61#949111yescharacteristic, complemented, Sylow
239C22×Q8C4×D6111yescharacteristic
240C23×C4—611no—
241C2×M16—21no—
242C2×M16—1222no—
243C2×M16—1224no—
244C2×M16—1224no—
245C2×M16—1221no—
246C22×C8—1222no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8—21no—
250C4×Q8—21no—
251#117—1221no—
252#117—1221no—
253#117—1221no—
254#117—1221no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—611no—
257C2×C4⋊C4—611no—
258C2×C4⋊C4—1221no—
259C2×C4⋊C4—1221no—
260C2×C4⋊C4C4×D6111yescharacteristic
261C2×C22⋊C4—611no—
262C2×C22⋊C4—611no—
263C2×C22⋊C4—1221no—
264C2×C22⋊C4—1221no—
265C2×C42—21no—
266C2×C42—611no—
267C2×C42C22×D6111yescharacteristic
268C4⋊C8—21no—
269C4⋊C8—1221no—
270C22⋊C8—21no—
271C22⋊C8—1221no—
272#97—21no—
273#97—21no—
274#97—1222no—
275#97—1221no—
276#97—1221no—
277C4×C8—612no—
278C4×C8—1222no—
279C4×C8C22⋊D6112yes—
280#95—611no—
281#95—611no—
282#95—1221no—
283#95—1221no—
284#95—1221no—
285#95—1221no—
286#95—1221no—
287#95—1221no—
288C24—611no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—42no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—611no—
295C22×C4—611no—
296C22×C4—611no—
297C22×C4—1222no—
298C22×C4—1222no—
299C22×C4—1221no—
300C22×C4—1221no—
301C22×C4—1221no—
302C22×C4—1221no—
303C22×C4C2×C4×D6111yescharacteristic
304C22×C4D4×D6111yescharacteristic
305C22×C4D4×D6111yescharacteristic
306M16—22no—
307M16—22no—
308M16—1228no—
309M16—1222no—
310M16—1222no—
311C2×C8—22no—
312C2×C8—41no—
313C2×C8—1228no—
314C2×C8—1222no—
315C2×C8—2441no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—1221no—
321C4⋊C4—1221no—
322C4⋊C4—1221no—
323C4⋊C4—1221no—
324C4⋊C4—2442no—
325C4⋊C4—2442no—
326C22⋊C4—1221no—
327C22⋊C4—1221no—
328C22⋊C4—1221no—
329C22⋊C4—1221no—
330C22⋊C4—2442no—
331C22⋊C4—2442no—
332C42—41no—
333C42—41no—
334C42—1221no—
335C42—1221no—
336C42#1175822211yescharacteristic
337C42D4×D6111yescharacteristic
338C42#1175822411yescharacteristic
339C42C2×C22⋊D6111yescharacteristic
340C23—612no—
341C23—1221no—
342C23—1221no—
343C23—2441no—
344C23—2441no—
345C23C22⋊C4×D6111yescharacteristic
346Q8—22no—
347Q8—22no—
348Q8—41no—
349Q8—41no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—612no—
361C2×C4—1222no—
362C2×C4—1222no—
363C2×C4—1224no—
364C2×C4—1221no—
365C2×C4—1221no—
366C2×C4—1221no—
367C2×C4—1221no—
368C2×C4—2441no—
369C2×C4—2441no—
370C2×C4—2441no—
371C2×C4—2441no—
372C2×C4—2441no—
373C2×C4—2441no—
374C2×C4—2441no—
375C2×C4—2441no—
376C2×C4#1451973711yescharacteristic
377C2×C4C4×C22⋊D6111yescharacteristic
378C2×C4#1451974511yescharacteristic
379C2×C4#1451974811yescharacteristic
380C2×C4#1451975011yescharacteristic
381C2×C4#1451975911yescharacteristic
382C8—24no—
383C8—81no—
384C8—1224no—
385C8—4881no—
386C22—21no—
387C22—21no—
388C22—1222no—
389C22—4881no—
390C22—2442no—
391C22—2441no—
392C22—2441no—
393C22#1683963811yescharacteristic
394C22#104×D6111yescharacteristic
395C22#104×D6111yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—42no—
403C4—81no—
404C4—1222no—
405C4—4881no—
406C4—4881no—
407C4—2441no—
408C4—2441no—
409C2—21no—
410C2—21no—
411C2—2442no—
412C2#2144623411yescharacteristic
413C2#2144625911yescharacteristic
414C2#2144633811yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1491C2
socleC22×C61#386
prime radicalC23×D61C22⋊C4
cosocleC2×C42×C61C23
hypercenter#1491D61
hypocenterC61#9491
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1491

Complements

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

NH ≅ G/Ntype
3. #21444266C2semidirect
5. #21443685C2semidirect
6. #21443684C2semidirect
7. C61×#1491C2semidirect
205. #1491D61semidirect
238. C61#9491semidirect

Maximal subgroups

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

subgroupindex
#214542362
#214513902
#214442662
#214441512
#214436852
#214436842
C61×#14912
#949161