Finite Group Databasebeta

← #29577067

Subgroups — #29577067

Order 15616.

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

normal subgroups only

This group has 444 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
1#21455767C211yesmaximal, characteristic
2#21454477C211yesmaximal, characteristic
3#21444390C211yesmaximal, characteristic
4#21444182C211yesmaximal, characteristic
5#21443725C211yesmaximal, characteristic
6#21443707C211yesmaximal, characteristic
7C61×#1825C211yesmaximal, characteristic
8C2×#14519756C2211yescharacteristic
9#16840103—21no—
10#16840103—21no—
11#16840097—21no—
12#16840092—21no—
13#16839823—22no—
14#16839819—21no—
15#16839815—21no—
16#16839805C411yescharacteristic, complemented
17#16839805C411yescharacteristic, complemented
18#16839789—21no—
19#16839770C411yescharacteristic, complemented
20#16839770C411yescharacteristic, complemented
21#16839726C2211yescharacteristic
22#16839726C2211yescharacteristic
23C2×C22⋊#92418C2211yescharacteristic
24#16839718—21no—
25#16839498—21no—
26#16839498—21no—
27#16839487—22no—
28C2×#117×C61C2211yescharacteristic
29C2×C22⋊C8×C61C2211yescharacteristic
30C61×#380C2211yescharacteristic
31C61×#340—21no—
32C61×#340—21no—
33C22×Dic122C2×C411yescharacteristic
34C2×#11758194—21no—
35C2×#11758194C2×C411yescharacteristic
36C2×#11758194C2×C411yescharacteristic
37C22×#92418—21no—
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
65C22×C8×C61—21no—
66#117×C61—21no—
67#117×C61—21no—
68#117×C61C2×C411yescharacteristic
69#117×C61C2×C411yescharacteristic
70#117×C61D411yescharacteristic
71#117×C61D411yescharacteristic
72C2×C4⋊C4×C61—21no—
73C2×C4⋊C4×C61D411yescharacteristic
74C2×C4⋊C4×C61D411yescharacteristic
75C2×C22⋊C4×C61—21no—
76C2×C22⋊C4×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—
99C2×C8×C61—41no—
100C2×C8×C61—41no—
101C2×C8×C61—41no—
102C4⋊C4×C61—21no—
103C4⋊C4×C61—21no—
104C4⋊C4×C61—41no—
105C4⋊C4×C61—41no—
106C4⋊C4×C61C22⋊C411yescharacteristic
107C4⋊C4×C61C22⋊C411yescharacteristic
108C4⋊C4×C61C22⋊C411yescharacteristic
109C4⋊C4×C61C22⋊C411yescharacteristic
110C22⋊C4×C61—21no—
111C22⋊C4×C61—21no—
112C22⋊C4×C61—21no—
113C22⋊C4×C61—21no—
114C22⋊C4×C61—81no—
115C42×C61—21no—
116C42×C61—21no—
117C42×C61—21no—
118C42×C61—21no—
119C22⋊Dic61—41no—
120C22⋊Dic61—41no—
121C4⋊Dic61—21no—
122C4⋊Dic61—21no—
123#11758196—41no—
124#11758196—41no—
125#11758194—41no—
126#11758194—41no—
127#11758194—41no—
128#11758194—41no—
129C2×#92418—21no—
130C2×#92418—21no—
131C2×#92418—21no—
132C2×#92418—22no—
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
143C2×C4×C61—21no—
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—41no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—81no—
159C2×C4×C61—81no—
160C2×C4×C61—81no—
161C2×C4×C61—81no—
162C2×C4×C61C2×C22⋊C411yescharacteristic
163C2×C4×C61C4×D411yescharacteristic
164C2×C4×C61#12011yescharacteristic
165C2×C4×C61#12111yescharacteristic
166C2×Dic61—42no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—81no—
170Dic122—41no—
171Dic122—41no—
172Dic122—81no—
173C8×C61—81no—
174#92418—41no—
175#92418—41no—
176#92418—42no—
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—41no—
197C4×C61—41no—
198C4×C61—41no—
199C4×C61—41no—
200C4×C61—161no—
201C4×C61#41011yescharacteristic
202C4×C61#41011yescharacteristic
203Dic61—81no—
204Dic61—81no—
205#3095—611no—
206#2924—611no—
207#1825D6111yescharacteristic
208#1520—611no—
209#1472—611no—
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⋊Q8—611no—
220C2×#117C2×D6111yescharacteristic
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#407—611no—
235C2×C22⋊C8—611no—
236C2×C22⋊C8C2×D6111yescharacteristic
237#405—1221no—
238#380C2×D6111yescharacteristic
239#340—21no—
240#340—21no—
241#328—1221no—
242#328—1221no—
243#324—1222no—
244C61#934611yescharacteristic, complemented, Sylow
245C22×Q8—611no—
246C23×C4C22×D6111yescharacteristic
247C2×M16—611no—
248C2×M16—611no—
249C2×M16—1221no—
250C22×C8—21no—
251C22×C8—1221no—
252C22⋊Q8—1221no—
253C22⋊Q8—1221no—
254C22⋊Q8—1221no—
255C22⋊Q8—1221no—
256#117—21no—
257#117—21no—
258#117C4×D6111yescharacteristic
259#117C4×D6111yescharacteristic
260#117D24411yescharacteristic
261#117D24411yescharacteristic
262C2×C4⋊C4—21no—
263C2×C4⋊C4—611no—
264C2×C4⋊C4—1221no—
265C2×C4⋊C4C22⋊D6111yescharacteristic
266C2×C4⋊C4C22⋊D6111yescharacteristic
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—1221no—
270C2×C42—21no—
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—612no—
284C22⋊C8—611no—
285C22⋊C8—611no—
286C22⋊C8—1221no—
287C22⋊C8—1221no—
288C22⋊C8—1221no—
289C22⋊C8—1221no—
290#97—1222no—
291C4×C8—1222no—
292#95—21no—
293C24#1175822411yescharacteristic
294C2×Q8—1221no—
295C2×Q8—1221no—
296C2×Q8—2441no—
297C2×Q8—2441no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—1221no—
304C22×C4—1221no—
305C22×C4C2×C4×D6111yescharacteristic
306C22×C4C2×D24411yescharacteristic
307C22×C4D4×D6111yescharacteristic
308C22×C4D4×D6111yescharacteristic
309C22×C4#1175822411yescharacteristic
310C22×C4C2×C22⋊D6111yescharacteristic
311M16—2441no—
312M16—2441no—
313M16—2441no—
314M16—2441no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—1221no—
319C2×C8—1221no—
320C2×C8—1221no—
321C2×C8—1222no—
322C2×C8—1221no—
323C2×C8—2441no—
324C2×C8—2441no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—41no—
328C4⋊C4—41no—
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—81no—
342C22⋊C4—2441no—
343C22⋊C4—2441no—
344C42—21no—
345C42—21no—
346C42—21no—
347C42—21no—
348C23—21no—
349C23—21no—
350C23—41no—
351C23—41no—
352C23#1451973711yescharacteristic
353C23#1451974911yescharacteristic
354C23#1451976111yescharacteristic
355Q8—4881no—
356Q8—2441no—
357Q8—2441no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—4881no—
378C2×C4—2442no—
379C2×C4—2441no—
380C2×C4—2441no—
381C2×C4C2×#1175819811yescharacteristic
382C2×C4#1451973711yescharacteristic
383C2×C4#1451974711yescharacteristic
384C2×C4#1451974911yescharacteristic
385C8—81no—
386C8—2441no—
387C8—2441no—
388C8—2442no—
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#1683963911yescharacteristic
402C22#1684002311yescharacteristic
403C22#1684003511yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—41no—
411C4—161no—
412C4—4881no—
413C4—4881no—
414C4#1683973511yescharacteristic
415C4#1683973511yescharacteristic
416C2—21no—
417C2—21no—
418C2—41no—
419C2—41no—
420C2#2144617911yescharacteristic
421C2#2144627511yescharacteristic
422C2#2144633211yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1825C2
socleC22×C61#386
prime radicalC24×C61#41
cosocleC23×C4×C61C23
hypercenter#1825D61
hypocenterC61#9346
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1825

Complements

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

NH ≅ G/Ntype
16. #16839805C4semidirect
17. #16839805C4semidirect
19. #16839770C4semidirect
20. #16839770C4semidirect
244. C61#9346semidirect

Maximal subgroups

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

subgroupindex
#214557672
#214544772
#214443902
#214441822
#214437252
#214437072
C61×#18252
#934661