Finite Group Databasebeta

← #29577072

Subgroups — #29577072

Order 15616.

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

normal subgroups only

This group has 444 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#21455763C211yesmaximal, characteristic
2#21454479C211yesmaximal, characteristic
3#21445915C211yesmaximal, characteristic
4#21444391C211yesmaximal, characteristic
5#21443726C211yesmaximal, characteristic
6#21443707C211yesmaximal, characteristic
7C61×#1472C211yesmaximal, characteristic
8C2×#14519736C2211yescharacteristic
9#16840100—21no—
10#16840100—21no—
11#16840095—21no—
12#16840090—21no—
13#16839820—22no—
14#16839816—21no—
15#16839812—21no—
16#16839802C411yescharacteristic, complemented
17#16839802C411yescharacteristic, complemented
18#16839790—21no—
19#16839771C411yescharacteristic, complemented
20#16839771C411yescharacteristic, complemented
21#16839726C2211yescharacteristic
22C2×C22⋊#92418C2211yescharacteristic
23#16839720—21no—
24#16839622C2211yescharacteristic
25#16839525—21no—
26#16839525—21no—
27#16839499—21no—
28#16839499—21no—
29C2×C22⋊Q8×C61C2211yescharacteristic
30C61×#407C2211yescharacteristic
31C2×C22⋊C8×C61C2211yescharacteristic
32C61×#324—22no—
33C2×#11758194C2×C411yescharacteristic
34C2×#11758194C2×C411yescharacteristic
35C22×#92418—21no—
36#14519736—21no—
37#14519736—21no—
38#14519736C2×C411yescharacteristic
39#14519736C2×C411yescharacteristic
40#14519736D411yescharacteristic
41#14519736D411yescharacteristic
42C2×C4⋊Dic61—21no—
43C2×C4⋊Dic61D411yescharacteristic
44C2×C4⋊Dic61D411yescharacteristic
45C2×C22⋊Dic61—21no—
46C2×C22⋊Dic61—21no—
47C2×C4×Dic61—21no—
48#14519710—21no—
49#14519710—21no—
50#14519708—21no—
51#14519708—21no—
52#14519706—22no—
53#14519706—22no—
54#14519701—21no—
55#14519701—21no—
56#14519701—21no—
57#14519701—21no—
58C22⋊#92418—21no—
59C22⋊#92418—21no—
60C22⋊#92418—21no—
61C22⋊#92418D411yescharacteristic
62C22⋊#92418D411yescharacteristic
63#14519687—22no—
64C8×Dic61—22no—
65#14519684—21no—
66C22×Q8×C61C2×C411yescharacteristic
67C23×C4×C61C2311yescharacteristic
68C2×M16×C61—21no—
69C22×C8×C61—21no—
70C22⋊Q8×C61—21no—
71C22⋊Q8×C61—21no—
72C22⋊Q8×C61—21no—
73C22⋊Q8×C61—21no—
74C2×C4⋊C4×C61—21no—
75C2×C4⋊C4×C61C2×C411yescharacteristic
76C2×C22⋊C4×C61—21no—
77C22⋊C8×C61—21no—
78C22⋊C8×C61—21no—
79C22⋊C8×C61—21no—
80C22⋊C8×C61D412yes—
81C24×C61#4111yescharacteristic
82C2×Q8×C61—21no—
83C2×Q8×C61—21no—
84C2×Q8×C61—41no—
85C2×Q8×C61—41no—
86C22×C4×C61—21no—
87C22×C4×C61—21no—
88C22×C4×C61—41no—
89C22×C4×C61—41no—
90C22×C4×C61C22×C411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61#4111yescharacteristic
96C22×Dic61—21no—
97C22×Dic61—21no—
98C22×Dic61—41no—
99M16×C61—41no—
100M16×C61—41no—
101C2×C8×C61—22no—
102C2×C8×C61—41no—
103C2×C8×C61—41no—
104C2×C8×C61—41no—
105C4⋊C4×C61—21no—
106C4⋊C4×C61—21no—
107C4⋊C4×C61—41no—
108C4⋊C4×C61—41no—
109C22⋊C4×C61—41no—
110C22⋊C4×C61—41no—
111C22⋊Dic61—21no—
112C22⋊Dic61—21no—
113C22⋊Dic61—21no—
114C22⋊Dic61—21no—
115C22⋊Dic61—81no—
116C4⋊Dic61—21no—
117C4⋊Dic61—21no—
118C4⋊Dic61—41no—
119C4⋊Dic61—41no—
120C4⋊Dic61C22⋊C411yescharacteristic
121C4⋊Dic61C22⋊C411yescharacteristic
122C4⋊Dic61C22⋊C411yescharacteristic
123C4⋊Dic61C22⋊C411yescharacteristic
124C4×Dic61—21no—
125C4×Dic61—21no—
126C4×Dic61—21no—
127C4×Dic61—21no—
128#11758194—41no—
129#11758194—41no—
130C2×#92418—21no—
131C2×#92418—21no—
132C2×#92418—21no—
133C2×#92418—21no—
134C2×#92418—41no—
135C2×#92418—41no—
136C23×C61—21no—
137C23×C61—21no—
138C23×C61—41no—
139C23×C61—41no—
140C23×C61C4×D411yescharacteristic
141C23×C61#12111yescharacteristic
142C23×C61#12311yescharacteristic
143Q8×C61—41no—
144Q8×C61—41no—
145Q8×C61—81no—
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—41no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—42no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—81no—
159C2×C4×C61C2×C22⋊C411yescharacteristic
160C2×C4×C61C4×D411yescharacteristic
161C2×C4×C61#12011yescharacteristic
162C2×C4×C61#12111yescharacteristic
163C2×Dic61—21no—
164C2×Dic61—21no—
165C2×Dic61—21no—
166C2×Dic61—21no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—81no—
170C2×Dic61—81no—
171C2×Dic61—81no—
172C2×Dic61—81no—
173C8×C61—42no—
174C8×C61—81no—
175#92418—41no—
176#92418—41no—
177#92418—41no—
178#92418—41no—
179#9346—611nomaximal, Sylow
180C22×C61—21no—
181C22×C61—21no—
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—81no—
189C22×C61—81no—
190C22×C61#38611yescharacteristic
191C22×C61#47111yescharacteristic
192C22×C61#47311yescharacteristic
193C4×C61—21no—
194C4×C61—21no—
195C4×C61—21no—
196C4×C61—81no—
197C4×C61—81no—
198C4×C61#41011yescharacteristic
199C4×C61#41011yescharacteristic
200Dic61—41no—
201Dic61—41no—
202Dic61—41no—
203Dic61—41no—
204Dic61—161no—
205#3095—611no—
206#2924—611no—
207#1825—611no—
208#1520—611no—
209#1472D6111yescharacteristic
210#1305—611no—
211#1298—611no—
212C2×C61—21no—
213C2×C61—21no—
214C2×C61—41no—
215C2×C61—41no—
216C2×C61#188111yescharacteristic
217C2×C61#189911yescharacteristic
218C2×C61#190811yescharacteristic
219C2×C22⋊Q8C2×D6111yescharacteristic
220C2×#117—611no—
221#485—1221no—
222#485—1221no—
223#484—1221no—
224#483—1221no—
225#433—1222no—
226#432—1221no—
227#431—1221no—
228#428—611no—
229#428—611no—
230#424—1221no—
231#419—611no—
232#419—611no—
233#407—611no—
234#407C2×D6111yescharacteristic
235C2×C22⋊C8—611no—
236C2×C22⋊C8C2×D6111yescharacteristic
237#405—1221no—
238#380—611no—
239#340—1221no—
240#340—1221no—
241#328—1221no—
242#328—1221no—
243#324—22no—
244C61#934611yescharacteristic, complemented, Sylow
245C22×Q8C4×D6111yescharacteristic
246C23×C4C22×D6111yescharacteristic
247C2×M16—21no—
248C2×M16—611no—
249C2×M16—611no—
250C22×C8—21no—
251C22×C8—1221no—
252C22⋊Q8—21no—
253C22⋊Q8—21no—
254C22⋊Q8—21no—
255C22⋊Q8—21no—
256#117—611no—
257#117—611no—
258#117—611no—
259#117—611no—
260#117—1221no—
261#117—1221no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—611no—
264C2×C4⋊C4—611no—
265C2×C4⋊C4—1221no—
266C2×C4⋊C4C4×D6111yescharacteristic
267C2×C22⋊C4—21no—
268C2×C22⋊C4—1221no—
269C2×C22⋊C4—1221no—
270C2×C42—1221no—
271#107—1221no—
272#107—1221no—
273#106—1221no—
274#106—1221no—
275C4⋊C8—1222no—
276C4⋊C8—1222no—
277#103—1221no—
278#103—1221no—
279#103—1221no—
280#103—1221no—
281C22⋊C8—21no—
282C22⋊C8—21no—
283C22⋊C8—21no—
284C22⋊C8—611no—
285C22⋊C8—611no—
286C22⋊C8—1221no—
287C22⋊C8—1221no—
288C22⋊C8—1221no—
289C22⋊C8C22⋊D6112yes—
290#97—1222no—
291C4×C8—1222no—
292#95—1221no—
293C24#1175822211yescharacteristic
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—41no—
297C2×Q8—41no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—1221no—
303C22×C4—1221no—
304C22×C4—2441no—
305C22×C4C2×C4×D6111yescharacteristic
306C22×C4D4×D6111yescharacteristic
307C22×C4D4×D6111yescharacteristic
308C22×C4D4×D6111yescharacteristic
309C22×C4#1175822411yescharacteristic
310C22×C4C2×C22⋊D6111yescharacteristic
311M16—41no—
312M16—41no—
313M16—2441no—
314M16—2441no—
315C2×C8—22no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—1221no—
320C2×C8—1221no—
321C2×C8—1221no—
322C2×C8—1221no—
323C2×C8—2441no—
324C2×C8—2441no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—611no—
330C4⋊C4—611no—
331C4⋊C4—611no—
332C4⋊C4—611no—
333C4⋊C4—1221no—
334C4⋊C4—1221no—
335C4⋊C4—2441no—
336C4⋊C4—2441no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—1221no—
340C22⋊C4—1221no—
341C22⋊C4—1221no—
342C22⋊C4—1221no—
343C22⋊C4—4881no—
344C42—1221no—
345C42—1221no—
346C42—1221no—
347C42—1221no—
348C23—21no—
349C23—21no—
350C23—41no—
351C23—41no—
352C23C4×C22⋊D6111yescharacteristic
353C23#1451975011yescharacteristic
354C23#1451975911yescharacteristic
355Q8—41no—
356Q8—41no—
357Q8—81no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—81no—
371C2×C4—1221no—
372C2×C4—1221no—
373C2×C4—1221no—
374C2×C4—1221no—
375C2×C4—4881no—
376C2×C4—4881no—
377C2×C4—4881no—
378C2×C4—4881no—
379C2×C4—2441no—
380C2×C4—2441no—
381C2×C4C22⋊C4×D6111yescharacteristic
382C2×C4#1451973711yescharacteristic
383C2×C4#1451974511yescharacteristic
384C2×C4#1451974811yescharacteristic
385C8—42no—
386C8—81no—
387C8—2441no—
388C8—2441no—
389C8—2441no—
390C8—2441no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—81no—
400C22—81no—
401C22#1683963811yescharacteristic
402C22#1684002811yescharacteristic
403C22#1684003611yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—81no—
408C4—81no—
409C4—9761no—
410C4—2441no—
411C4—2441no—
412C4—2441no—
413C4—2441no—
414C4#1683973411yescharacteristic
415C4#1683973411yescharacteristic
416C2—21no—
417C2—21no—
418C2—41no—
419C2—41no—
420C2#2144618411yescharacteristic
421C2#2144627411yescharacteristic
422C2#2144633111yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1472C2
socleC22×C61#386
prime radicalC24×C61#41
cosocleC23×C4×C61C23
hypercenter#1472D61
hypocenterC61#9346
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1472

Complements

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

NH ≅ G/Ntype
16. #16839802C4semidirect
17. #16839802C4semidirect
19. #16839771C4semidirect
20. #16839771C4semidirect
244. C61#9346semidirect

Maximal subgroups

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

subgroupindex
#214557632
#214544792
#214459152
#214443912
#214437262
#214437072
C61×#14722
#934661