Finite Group Databasebeta

← #29576325

Subgroups — #29576325

Order 15616.

452 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C2×#16840103C211yesmaximal, characteristic
2#21454477C211yesmaximal, characteristic
3#21444338C211yesmaximal, characteristic
4#21444184C211yesmaximal, characteristic
5#21443695C211yesmaximal, characteristic
6#21443667C211yesmaximal, characteristic
7C61×#1826C211yesmaximal, characteristic
8C2×#14519756C2211yescharacteristic
9#16840103—21no—
10#16840103—21no—
11#16840103—21no—
12#16840103—21no—
13#16839823—22no—
14#16839819—21no—
15#16839815—21no—
16C2×#14519710C411yescharacteristic, complemented
17C2×#14519708C411yescharacteristic, complemented
18#16839789—21no—
19C2×#14519700C411yescharacteristic, complemented
20C2×#14519700C411yescharacteristic, complemented
21#16839726C2211yescharacteristic
22C2×C22⋊#92418C2211yescharacteristic
23C2×C22⋊#92418C2211yescharacteristic
24#16839718—21no—
25#16839498—22no—
26#16839487—21no—
27#16839487—21no—
28C2×#117×C61C2211yescharacteristic
29C61×#407C2211yescharacteristic
30C4×C22⋊C4×C61C2211yescharacteristic
31C61×#339—21no—
32C61×#339—21no—
33C22×Dic122C2×C411yescharacteristic
34C2×#11758194—21no—
35C22×#92418—21no—
36C22×#92418C2×C411yescharacteristic
37C22×#92418C2×C411yescharacteristic
38#14519756—21no—
39#14519756—21no—
40#14519756—21no—
41#14519756—21no—
42C2×C4⋊Dic61C2×C411yescharacteristic
43C2×#11758196—21no—
44C2×C22⋊Dic61—21no—
45#14519710—21no—
46#14519710—21no—
47#14519708—21no—
48#14519708—21no—
49C4⋊#92418—22no—
50C4⋊#92418—22no—
51#14519700—21no—
52#14519700—21no—
53#14519700—21no—
54#14519700—21no—
55C22⋊#92418—21no—
56C22⋊#92418—21no—
57C22⋊#92418—21no—
58C22⋊#92418—21no—
59C22⋊#92418D411yescharacteristic
60C22⋊#92418D411yescharacteristic
61C22⋊#92418D412yes—
62#14519688—22no—
63C4×#92418—22no—
64C23×C4×C61C2311yescharacteristic
65C2×M16×C61—21no—
66#117×C61—21no—
67#117×C61—21no—
68#117×C61D411yescharacteristic
69#117×C61D411yescharacteristic
70#117×C61D411yescharacteristic
71#117×C61D411yescharacteristic
72C2×C4⋊C4×C61C2×C411yescharacteristic
73C2×C4⋊C4×C61C2×C411yescharacteristic
74C2×C22⋊C4×C61—21no—
75C2×C22⋊C4×C61—21no—
76C2×C42×C61—21no—
77C2×C42×C61—21no—
78C22⋊C8×C61—21no—
79C22⋊C8×C61—21no—
80#95×C61—21no—
81C24×C61#4111yescharacteristic
82C22×C4×C61—21no—
83C22×C4×C61—21no—
84C22×C4×C61—41no—
85C22×C4×C61—41no—
86C22×C4×C61—41no—
87C22×C4×C61C22×C411yescharacteristic
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61#4111yescharacteristic
93C22×Dic61—21no—
94C22×Dic61—21no—
95C2×Dic122—21no—
96C2×Dic122—21no—
97C2×Dic122—41no—
98C2×Dic122—41no—
99M16×C61—41no—
100M16×C61—41no—
101C2×C8×C61—41no—
102C4⋊C4×C61—21no—
103C4⋊C4×C61—21no—
104C4⋊C4×C61C22⋊C411yescharacteristic
105C4⋊C4×C61C22⋊C411yescharacteristic
106C4⋊C4×C61C22⋊C411yescharacteristic
107C4⋊C4×C61C22⋊C411yescharacteristic
108C22⋊C4×C61—21no—
109C22⋊C4×C61—21no—
110C22⋊C4×C61—21no—
111C22⋊C4×C61—21no—
112C22⋊C4×C61—22no—
113C22⋊C4×C61—41no—
114C42×C61—21no—
115C42×C61—21no—
116C42×C61—21no—
117C42×C61—21no—
118C42×C61—41no—
119C42×C61—41no—
120C22⋊Dic61—41no—
121C22⋊Dic61—41no—
122C4⋊Dic61—21no—
123C4⋊Dic61—21no—
124#11758196—41no—
125#11758196—41no—
126#11758194—41no—
127#11758194—41no—
128C2×#92418—21no—
129C2×#92418—21no—
130C2×#92418—21no—
131C2×#92418—22no—
132C2×#92418—21no—
133C2×#92418—41no—
134C2×#92418—41no—
135C2×#92418—41no—
136C2×#92418—41no—
137C23×C61—21no—
138C23×C61—21no—
139C23×C61—41no—
140C23×C61—41no—
141C23×C61C4×D411yescharacteristic
142C23×C61#12111yescharacteristic
143C23×C61#12311yescharacteristic
144C2×C4×C61—21no—
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—42no—
160C2×C4×C61—81no—
161C2×C4×C61—81no—
162C2×C4×C61—81no—
163C2×C4×C61C2×C22⋊C411yescharacteristic
164C2×C4×C61C4×D411yescharacteristic
165C2×C4×C61#12011yescharacteristic
166C2×C4×C61#12111yescharacteristic
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61—41no—
171C2×Dic61—81no—
172Dic122—41no—
173Dic122—41no—
174Dic122—81no—
175C8×C61—81no—
176#92418—41no—
177#92418—41no—
178#92418—42no—
179#92418—41no—
180#92418—41no—
181#9240—611nomaximal, Sylow
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—81no—
191C22×C61—81no—
192C22×C61#38611yescharacteristic
193C22×C61#44311yescharacteristic
194C22×C61#44411yescharacteristic
195C4×C61—21no—
196C4×C61—21no—
197C4×C61—21no—
198C4×C61—41no—
199C4×C61—41no—
200C4×C61—41no—
201C4×C61—41no—
202C4×C61—82no—
203C4×C61#41011yescharacteristic
204C4×C61#41011yescharacteristic
205Dic61—81no—
206Dic61—81no—
207C2×#485—611no—
208#2924—611no—
209#1826D6111yescharacteristic
210#1508—611no—
211#1472—611no—
212#1293—611no—
213#1282—611no—
214C2×C61—21no—
215C2×C61—21no—
216C2×C61—41no—
217C2×C61—41no—
218C2×C61#187611yescharacteristic
219C2×C61#189911yescharacteristic
220C2×C61#190211yescharacteristic
221C2×C22⋊Q8—611no—
222C2×#117C2×D6111yescharacteristic
223#485—1221no—
224#485—1221no—
225#485—1221no—
226#485—1221no—
227#433—1222no—
228#432—1221no—
229#431—1221no—
230C2×#107—611no—
231C2×#106—611no—
232#424—1221no—
233C2×#103—611no—
234C2×#103—611no—
235#407—611no—
236#407C2×D6111yescharacteristic
237C2×C22⋊C8—611no—
238C2×C22⋊C8—611no—
239#405—1221no—
240C4×C22⋊C4C2×D6111yescharacteristic
241#339—21no—
242#339—21no—
243#328—1222no—
244#324—1221no—
245#324—1221no—
246C61#924011yescharacteristic, complemented, Sylow
247C22×Q8—611no—
248C23×C4C22×D6111yescharacteristic
249C2×M16—21no—
250C2×M16—1221no—
251C22×C8—611no—
252C22×C8—611no—
253C22×C8—1221no—
254C22⋊Q8—1221no—
255C22⋊Q8—1221no—
256C22⋊Q8—1221no—
257C22⋊Q8—1221no—
258#117—21no—
259#117—21no—
260#117D24411yescharacteristic
261#117D24411yescharacteristic
262#117C22⋊D6111yescharacteristic
263#117C22⋊D6111yescharacteristic
264C2×C4⋊C4—611no—
265C2×C4⋊C4—1221no—
266C2×C4⋊C4C4×D6111yescharacteristic
267C2×C4⋊C4C4×D6111yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—1221no—
271C2×C42—21no—
272C2×C42—21no—
273#107—1221no—
274#107—1221no—
275#106—1221no—
276#106—1221no—
277C4⋊C8—1222no—
278C4⋊C8—1222no—
279#103—1221no—
280#103—1221no—
281#103—1221no—
282#103—1221no—
283C22⋊C8—21no—
284C22⋊C8—21no—
285C22⋊C8—611no—
286C22⋊C8—611no—
287C22⋊C8—612no—
288C22⋊C8—1221no—
289C22⋊C8—1221no—
290C22⋊C8—1221no—
291C22⋊C8—1221no—
292#97—1222no—
293C4×C8—1222no—
294#95—21no—
295C24#1175822411yescharacteristic
296C2×Q8—1221no—
297C2×Q8—1221no—
298C2×Q8—2441no—
299C2×Q8—2441no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—1221no—
306C22×C4—1221no—
307C22×C4C2×C4×D6111yescharacteristic
308C22×C4C2×D24411yescharacteristic
309C22×C4D4×D6111yescharacteristic
310C22×C4D4×D6111yescharacteristic
311C22×C4#1175822411yescharacteristic
312C22×C4C2×C22⋊D6111yescharacteristic
313M16—41no—
314M16—41no—
315M16—2441no—
316M16—2441no—
317C2×C8—41no—
318C2×C8—1221no—
319C2×C8—1221no—
320C2×C8—1221no—
321C2×C8—1222no—
322C2×C8—1221no—
323C2×C8—2441no—
324C2×C8—2441no—
325C2×C8—2441no—
326C2×C8—2441no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—1221no—
330C4⋊C4—1221no—
331C4⋊C4—2441no—
332C4⋊C4—2441no—
333C4⋊C4#1175819811yescharacteristic
334C4⋊C4#1175819811yescharacteristic
335C4⋊C4#1175819811yescharacteristic
336C4⋊C4#1175819811yescharacteristic
337C22⋊C4—21no—
338C22⋊C4—21no—
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—22no—
342C22⋊C4—41no—
343C22⋊C4—2441no—
344C22⋊C4—2441no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—41no—
350C42—41no—
351C23—21no—
352C23—21no—
353C23—41no—
354C23—41no—
355C23#1451973711yescharacteristic
356C23#1451974911yescharacteristic
357C23#1451976111yescharacteristic
358Q8—4881no—
359Q8—2441no—
360Q8—2441no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—4881no—
381C2×C4—2441no—
382C2×C4—2441no—
383C2×C4—2441no—
384C2×C4—2441no—
385C2×C4C2×#1175819811yescharacteristic
386C2×C4#1451973711yescharacteristic
387C2×C4#1451974711yescharacteristic
388C2×C4#1451974911yescharacteristic
389C8—81no—
390C8—2441no—
391C8—2441no—
392C8—2442no—
393C8—2441no—
394C8—2441no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—81no—
404C22—81no—
405C22#1683963911yescharacteristic
406C22#1683987611yescharacteristic
407C22#1683988111yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—82no—
416C4—4881no—
417C4—4881no—
418C4#1683973511yescharacteristic
419C4#1683973511yescharacteristic
420C2—21no—
421C2—21no—
422C2—41no—
423C2—41no—
424C2#2144614411yescharacteristic
425C2#2144627511yescharacteristic
426C2#2144629011yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1826C2
socleC22×C61#386
prime radicalC24×C61#41
cosocleC23×C4×C61C23
hypercenter#1826D61
hypocenterC61#9240
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1826

Complements

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

NH ≅ G/Ntype
16. C2×#14519710C4semidirect
17. C2×#14519708C4semidirect
19. C2×#14519700C4semidirect
20. C2×#14519700C4semidirect
246. C61#9240semidirect

Maximal subgroups

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

subgroupindex
C2×#168401032
#214544772
#214443382
#214441842
#214436952
#214436672
C61×#18262
#924061