Finite Group Databasebeta

← #29580914

Subgroups — #29580914

Order 15616.

576 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 81 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#21456208C211yesmaximal, characteristic, complemented
2#21451671C211yesmaximal, characteristic
3#21444442C211yesmaximal, characteristic, complemented
4#21444370C211yesmaximal, characteristic, complemented
5#21444262C211yesmaximal, characteristic, complemented
6#21443685C211yesmaximal, characteristic, complemented
7C61×#3110C211yesmaximal, characteristic, complemented
8C2×#14519765C2211yescharacteristic
9#16840124—21no—
10#16840124—21no—
11#16840119—22no—
12#16839819C2211yescharacteristic
13#16839770—21no—
14#16839766—21no—
15C4×#11758194C2211yescharacteristic
16#16839688—21no—
17#16839674—21no—
18#16839671—21no—
19#16839670—21no—
20#16839641—22no—
21#16839630C2211yescharacteristic
22#16839501—22no—
23#16839499—22no—
24D4⋊#92418—22no—
25C61×#533C2211yescharacteristic
26C2×C4×D4×C61C2211yescharacteristic
27C61×#484—21no—
28C61×#474—21no—
29C61×#461—21no—
30C61×#450—21no—
31C61×#440—22no—
32C61×#432C2211yescharacteristic
33C22×Dic122D411yescharacteristic
34C22×D244D411yescharacteristic
35C22×C22⋊D61—22no—
36C2×#11758194—21no—
37#14519765—21no—
38#14519765—21no—
39#14519765—21no—
40#14519765—21no—
41C2×C22⋊Dic61—22no—
42C2×#11758198—21no—
43C2×#11758198—21no—
44C2×#11758198—22no—
45C4⋊#92418—21no—
46#14519700—22no—
47#14519700—22no—
48#14519697—22no—
49#14519697—22no—
50C22⋊#92418—21no—
51#14519688—21no—
52#14519688—21no—
53C4×#92418D412yescomplemented
54#14519684—22no—
55#14519684—21no—
56#14519684—21no—
57#14519684D412yes—
58C22×D4×C61D411yescharacteristic
59C23×C4×C61—21no—
60C2×SD16×C61—22no—
61C4⋊Q8×C61D411yescharacteristic, complemented
62C4⋊Q8×C61D411yescharacteristic, complemented
63#125×C61—21no—
64C22⋊Q8×C61—21no—
65C22⋊Q8×C61—21no—
66C4×Q8×C61—21no—
67C4×Q8×C61—21no—
68C4×D4×C61—21no—
69C4×D4×C61—21no—
70C4×D4×C61—21no—
71C4×D4×C61—21no—
72#117×C61—21no—
73C2×C4⋊C4×C61D411yescharacteristic
74C2×C22⋊C4×C61—21no—
75C2×C42×C61C2311yescharacteristic
76#107×C61—22no—
77C4⋊C8×C61—21no—
78#103×C61—22no—
79#102×C61—22no—
80C22⋊C8×C61—21no—
81#97×C61D412yescomplemented
82C24×C61—21no—
83C23×D61—21no—
84C2×Q8×C61—21no—
85C2×Q8×C61—21no—
86C2×D4×C61—21no—
87C2×D4×C61—21no—
88C2×D4×C61—42no—
89C22×C4×C61—21no—
90C22×C4×C61—41no—
91C22×C4×C61—41no—
92C22×C4×C61—82no—
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C2×C22⋊D61—44no—
97C2×C22⋊D61—82no—
98C2×C22⋊D61—82no—
99C22×Dic61—22no—
100C22×Dic61—21no—
101C2×D244—21no—
102C2×D244—21no—
103C2×D244—42no—
104C2×Dic122—21no—
105C2×Dic122—21no—
106C2×Dic122—42no—
107SD16×C61—44no—
108C2×C8×C61—22no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61—21no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61—21no—
113C4⋊C4×C61—21no—
114C4⋊C4×C61—21no—
115C4⋊C4×C61—41no—
116C4⋊C4×C61—41no—
117C22⋊C4×C61—41no—
118C22⋊C4×C61—41no—
119C22⋊C4×C61—41no—
120C42×C61—41no—
121C42×C61C2×D411yescharacteristic
122C42×C61C2×D411yescharacteristic
123C42×C61C2×D411yescharacteristic
124C42×C61#4111yescharacteristic
125C22⋊Dic61—82no—
126#11758198—41no—
127#11758198—41no—
128#11758198—41no—
129#11758198—41no—
130#11758198—82no—
131#11758194—42no—
132C2×#92418—22no—
133C2×#92418—41no—
134C23×C61—22no—
135C23×C61—42no—
136C23×C61—41no—
137C23×C61—41no—
138C23×C61—42no—
139C23×C61#12011yescharacteristic
140C22×D61—41no—
141C22×D61—42no—
142C22×D61—41no—
143C22×D61—41no—
144C22×D61—42no—
145Q8×C61—42no—
146Q8×C61—81no—
147D4×C61—22no—
148D4×C61—22no—
149D4×C61—41no—
150D4×C61—41no—
151C2×C4×C61—22no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—42no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—82no—
159C2×C4×C61—82no—
160C2×C4×C61#12011yescharacteristic
161C2×C4×C61#12011yescharacteristic
162C2×C4×C61#12111yescharacteristic
163C2×C4×C61#12111yescharacteristic
164C2×C4×C61#12111yescharacteristic
165C2×C4×C61#12711yescharacteristic
166C22⋊D61—88no—
167C2×Dic61—42no—
168C2×Dic61—42no—
169C2×Dic61—41no—
170C2×Dic61—42no—
171C2×Dic61—42no—
172D244—42no—
173D244—81no—
174Dic122—42no—
175Dic122—81no—
176C8×C61—42no—
177#92418—42no—
178#92418—81no—
179#10022—611nomaximal, Sylow
180C22×C61—21no—
181C22×C61—21no—
182C22×C61—44no—
183C22×C61—42no—
184C22×C61—82no—
185C22×C61—81no—
186C22×C61—81no—
187C22×C61#39211yescharacteristic
188C22×C61#45512yes—
189C2×D61—82no—
190C2×D61—82no—
191C2×D61—81no—
192C2×D61—81no—
193C2×D61—81no—
194C4×C61—21no—
195C4×C61—21no—
196C4×C61—21no—
197C4×C61—21no—
198C4×C61—21no—
199C4×C61—21no—
200C4×C61—81no—
201C4×C61—81no—
202C4×C61—81no—
203Dic61—82no—
204Dic61—81no—
205#3140—611no—
206#3110D6111yescharacteristic, complemented
207#2631—611no—
208#1533—611no—
209#1515—611no—
210#1490—611no—
211#1288—611no—
212C2×C61—22no—
213C2×C61—82no—
214C2×C61#200411yescharacteristic
215C2×C61#200811yescharacteristic
216C2×C61#201011yescharacteristic
217D61—161no—
218#533C2×D6111yescharacteristic
219C2×#124—611no—
220C2×C4×D4C2×D6111yescharacteristic
221#489—1221no—
222#489—1221no—
223#488—1222no—
224#484—21no—
225#474—21no—
226#461—21no—
227#450—21no—
228#440—22no—
229#432—611no—
230#432C2×D6111yescharacteristic
231#419—1221no—
232#418—1221no—
233C4×M16—611no—
234#397—1221no—
235#393—1221no—
236#392—1221no—
237#392—1221no—
238#386—1222no—
239#383—611no—
240#329—1222no—
241#328—1222no—
242D4⋊C8—1222no—
243C61#1002211yescharacteristic, complemented, Sylow
244C22×Q8—611no—
245C22×D4—611no—
246C22×D4—1222no—
247C22×D4C22⋊D6111yescharacteristic
248C23×C4—21no—
249C2×SD16—22no—
250C2×M16—1221no—
251C4⋊Q8C22⋊D6111yescharacteristic, complemented
252C4⋊Q8C22⋊D6111yescharacteristic, complemented
253#125—21no—
254#124—1221no—
255#124—1221no—
256#124—1221no—
257#124—1221no—
258C22⋊Q8—21no—
259C22⋊Q8—21no—
260C4×Q8—21no—
261C4×Q8—21no—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266#117—21no—
267C2×C4⋊C4C22⋊D6111yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—1222no—
270C2×C22⋊C4—1222no—
271C2×C22⋊C4—1221no—
272C2×C22⋊C4—1221no—
273C2×C42C22×D6111yescharacteristic
274#107—22no—
275C4⋊C8—21no—
276C4⋊C8—1221no—
277#103—22no—
278#103—1222no—
279#103—1222no—
280#102—22no—
281#102—1222no—
282#102—1222no—
283C22⋊C8—21no—
284C22⋊C8—1221no—
285#97—1221no—
286#97—1221no—
287#97C22⋊D6112yescomplemented
288C4×C8—612no—
289#95—612no—
290#95—1222no—
291#95—1221no—
292#95—1221no—
293C24—21no—
294C24—1221no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—1221no—
298C2×Q8—1221no—
299C2×Q8—2442no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—42no—
303C2×D4—1221no—
304C2×D4—1221no—
305C2×D4—4882no—
306C2×D4—4882no—
307C2×D4—2442no—
308C2×D4—2444no—
309C22×C4—21no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—82no—
313C22×C4—1222no—
314C22×C4—1221no—
315C22×C4D4×D6111yescharacteristic
316C22×C4D4×D6111yescharacteristic
317C22×C4C2×C22⋊D6111yescharacteristic
318SD16—44no—
319M16—2442no—
320C2×C8—22no—
321C2×C8—1222no—
322C2×C8—2441no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—4882no—
335C22⋊C4—4882no—
336C22⋊C4—2441no—
337C22⋊C4—2441no—
338C22⋊C4—2441no—
339C22⋊C4—2441no—
340C42—41no—
341C42D4×D6111yescharacteristic
342C42#1175822411yescharacteristic
343C42C2×C22⋊D6111yescharacteristic
344C42C2×C22⋊D6111yescharacteristic
345C23—22no—
346C23—42no—
347C23—41no—
348C23—42no—
349C23—41no—
350C23—2441no—
351C23—2441no—
352C23—2442no—
353C23—2442no—
354C23—2441no—
355C23#1451974511yescharacteristic
356Q8—42no—
357Q8—81no—
358Q8—4881no—
359Q8—2442no—
360D4—22no—
361D4—22no—
362D4—41no—
363D4—41no—
364D4—4881no—
365D4—4888no—
366D4—2442no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—2442no—
377C2×C4—2442no—
378C2×C4—2442no—
379C2×C4—2442no—
380C2×C4—2441no—
381C2×C4#1451974511yescharacteristic
382C2×C4#1451974611yescharacteristic
383C2×C4#1451974811yescharacteristic
384C2×C4#1451974911yescharacteristic
385C2×C4#1451974911yescharacteristic
386C2×C4#1451977311yescharacteristic
387C8—42no—
388C8—4881no—
389C8—2442no—
390C22—21no—
391C22—21no—
392C22—44no—
393C22—42no—
394C22—82no—
395C22—81no—
396C22—81no—
397C22—4882no—
398C22—4882no—
399C22—4881no—
400C22—4881no—
401C22—4881no—
402C22#1683967011yescharacteristic
403C22#1683993812yes—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—81no—
411C4—81no—
412C4—81no—
413C4—4882no—
414C4—4881no—
415C2—22no—
416C2—82no—
417C2—9761no—
418C2#2144684511yescharacteristic
419C2#2144687011yescharacteristic
420C2#2144688411yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#3110C2
socleC22×C61#392
prime radical#21451671C2
hypercenter#3110D61
hypocenterC61#10022
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3110

Complements

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

NH ≅ G/Ntype
1. #21456208C2semidirect
3. #21444442C2semidirect
4. #21444370C2semidirect
5. #21444262C2semidirect
6. #21443685C2semidirect
7. C61×#3110C2semidirect
53. C4×#92418D4semidirect
61. C4⋊Q8×C61D4semidirect
62. C4⋊Q8×C61D4semidirect
81. #97×C61D4semidirect
206. #3110D61semidirect
243. C61#10022semidirect
251. C4⋊Q8C22⋊D61semidirect
252. C4⋊Q8C22⋊D61semidirect
287. #97C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214562082
#214516712
#214444422
#214443702
#214442622
#214436852
C61×#31102
#1002261