Finite Group Databasebeta

← #29577654

Subgroups — #29577654

Order 15616.

474 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 89 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#21454454C211yesmaximal, characteristic
2#21450273C211yesmaximal, characteristic
3#21445894C211yesmaximal, characteristic
4#21444352C211yesmaximal, characteristic
5#21443673C211yesmaximal, characteristic
6#21443671C211yesmaximal, characteristic
7C61×#1511C211yesmaximal, characteristic
8C2×#14519736C2211yescharacteristic
9#16839820—22no—
10#16839820—22no—
11#16839785—21no—
12#16839726C2211yescharacteristic
13M16×Dic61—21no—
14#16839690—21no—
15#16839690—21no—
16#16839669C411yescharacteristic, complemented
17#16839669C411yescharacteristic, complemented
18#16839643—21no—
19#16839643—21no—
20#16839631C411yescharacteristic, complemented
21#16839627C411yescharacteristic, complemented
22#16839625C2211yescharacteristic
23C4×C22⋊Dic61C2211yescharacteristic
24#16839528—21no—
25#16839528—21no—
26#16839499—22no—
27C2×C22⋊Q8×C61C2211yescharacteristic
28C61×#407C2211yescharacteristic
29C61×#381C2211yescharacteristic
30C61×#328—22no—
31C2×#11758194—21no—
32#14519736—21no—
33#14519736—21no—
34#14519736—21no—
35#14519736—21no—
36C2×C4⋊Dic61C2×C411yescharacteristic
37C2×C4⋊Dic61C2×C411yescharacteristic
38C2×#11758196—21no—
39C2×#11758196—21no—
40C2×C22⋊Dic61—21no—
41C2×C22⋊Dic61—21no—
42C2×C22⋊Dic61D411yescharacteristic
43C2×C22⋊Dic61D411yescharacteristic
44C2×C4×Dic61C2×C411yescharacteristic
45C2×C4×Dic61C2×C411yescharacteristic
46C2×C4×Dic61D411yescharacteristic
47C2×C4×Dic61D411yescharacteristic
48#14519706—24no—
49C22⋊#92418—21no—
50C22⋊#92418D412yes—
51#14519687—22no—
52C8×Dic61—22no—
53#14519684—21no—
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57#14519684—21no—
58#14519684—21no—
59#14519684—21no—
60#14519684—21no—
61#14519684—21no—
62C22×Q8×C61C2×C411yescharacteristic
63C23×C4×C61C2311yescharacteristic
64C2×M16×C61—21no—
65C22⋊Q8×C61—21no—
66C22⋊Q8×C61—21no—
67C22⋊Q8×C61—21no—
68C22⋊Q8×C61—21no—
69C2×C4⋊C4×C61—21no—
70C2×C4⋊C4×C61C2×C411yescharacteristic
71C2×C22⋊C4×C61—21no—
72C2×C22⋊C4×C61—21no—
73C22⋊C8×C61—21no—
74C22⋊C8×C61D412yes—
75#95×C61—21no—
76#95×C61—21no—
77C24×C61C2×D411yescharacteristic
78C2×Q8×C61—21no—
79C2×Q8×C61—21no—
80C2×Q8×C61—41no—
81C2×Q8×C61—41no—
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×C61#4111yescharacteristic
92C22×C4×C61#4111yescharacteristic
93C22×Dic61—21no—
94C22×Dic61—21no—
95C22×Dic61—41no—
96C22×Dic61C22⋊C411yescharacteristic
97C22×Dic61C22⋊C411yescharacteristic
98C22×Dic61C22⋊C411yescharacteristic
99C22×Dic61C22⋊C411yescharacteristic
100M16×C61—41no—
101M16×C61—41no—
102C2×C8×C61—22no—
103C4⋊C4×C61—21no—
104C4⋊C4×C61—21no—
105C4⋊C4×C61—41no—
106C4⋊C4×C61—41no—
107C22⋊C4×C61—41no—
108C22⋊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—42no—
116C22⋊Dic61—41no—
117C22⋊Dic61—41no—
118C4⋊Dic61—21no—
119C4⋊Dic61—21no—
120C4⋊Dic61—21no—
121C4⋊Dic61—21no—
122#11758196—42no—
123#11758196—42no—
124C4×Dic61—21no—
125C4×Dic61—21no—
126C4×Dic61—21no—
127C4×Dic61—21no—
128C4×Dic61—41no—
129C4×Dic61—41no—
130#11758194—41no—
131#11758194—41no—
132C2×#92418—22no—
133C23×C61—21no—
134C23×C61—21no—
135C23×C61—41no—
136C23×C61—41no—
137C23×C61C2×C22⋊C411yescharacteristic
138C23×C61#12111yescharacteristic
139C23×C61#12111yescharacteristic
140Q8×C61—81no—
141Q8×C61—81no—
142C2×C4×C61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—21no—
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×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—81no—
157C2×C4×C61—81no—
158C2×C4×C61—81no—
159C2×C4×C61C4×D411yescharacteristic
160C2×C4×C61C4×D411yescharacteristic
161C2×C4×C61#12011yescharacteristic
162C2×C4×C61#12311yescharacteristic
163C2×Dic61—22no—
164C2×Dic61—21no—
165C2×Dic61—21no—
166C2×Dic61—21no—
167C2×Dic61—21no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61—41no—
171C2×Dic61—41no—
172C2×Dic61—41no—
173C2×Dic61—41no—
174C2×Dic61—41no—
175C2×Dic61—41no—
176C2×Dic61—41no—
177C2×Dic61—81no—
178C2×Dic61—81no—
179C2×Dic61—81no—
180C8×C61—42no—
181#92418—42no—
182#9444—611nomaximal, Sylow
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—81no—
192C22×C61—81no—
193C22×C61#38611yescharacteristic
194C22×C61#41011yescharacteristic
195C22×C61#41011yescharacteristic
196C4×C61—21no—
197C4×C61—21no—
198C4×C61—21no—
199C4×C61—21no—
200C4×C61—81no—
201C4×C61—81no—
202C4×C61—161no—
203Dic61—42no—
204Dic61—41no—
205Dic61—41no—
206Dic61—41no—
207Dic61—41no—
208Dic61—81no—
209Dic61—161no—
210#2921—611no—
211#2499—611no—
212#1820—611no—
213#1511—611no—
214#1511D6111yescharacteristic
215#1284—611no—
216#1284—611no—
217C2×C61—21no—
218C2×C61—21no—
219C2×C61—41no—
220C2×C61—41no—
221C2×C61#188411yescharacteristic
222C2×C61#189911yescharacteristic
223C2×C61#190911yescharacteristic
224C2×C22⋊Q8C2×D6111yescharacteristic
225C2×#117—611no—
226#433—1222no—
227#433—1222no—
228#423—1221no—
229#407—611no—
230#407C2×D6111yescharacteristic
231C4×M16—1221no—
232#397—1221no—
233#397—1221no—
234#391—611no—
235#391—611no—
236#387—1221no—
237#387—1221no—
238#384—611no—
239#382—611no—
240#381—611no—
241#381C2×D6111yescharacteristic
242C4×C22⋊C4—611no—
243#342—1221no—
244#342—1221no—
245#328—22no—
246#328—1222no—
247C61#944411yescharacteristic, complemented, Sylow
248C22×Q8C4×D6111yescharacteristic
249C23×C4C22×D6111yescharacteristic
250C2×M16—21no—
251C2×M16—1221no—
252C22⋊Q8—21no—
253C22⋊Q8—21no—
254C22⋊Q8—21no—
255C22⋊Q8—21no—
256#117—1221no—
257#117—1221no—
258#117—1221no—
259#117—1221no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—611no—
262C2×C4⋊C4—611no—
263C2×C4⋊C4—1221no—
264C2×C4⋊C4—1221no—
265C2×C4⋊C4C4×D6111yescharacteristic
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—611no—
269C2×C22⋊C4—611no—
270C2×C22⋊C4—1221no—
271C2×C22⋊C4—1221no—
272C2×C42—611no—
273C2×C42—611no—
274C2×C42—611no—
275C2×C42—611no—
276C4⋊C8—1224no—
277C22⋊C8—21no—
278C22⋊C8—612no—
279C22⋊C8—1221no—
280C22⋊C8C22⋊D6112yes—
281#97—1222no—
282C4×C8—1222no—
283#95—21no—
284#95—21no—
285#95—1221no—
286#95—1221no—
287#95—1221no—
288#95—1221no—
289#95—1221no—
290#95—1221no—
291#95—1221no—
292#95—1221no—
293#95—1221no—
294C24D4×D6111yescharacteristic
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—41no—
298C2×Q8—41no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—611no—
305C22×C4—611no—
306C22×C4—611no—
307C22×C4—611no—
308C22×C4—1221no—
309C22×C4—1221no—
310C22×C4—2441no—
311C22×C4C2×C4×D6111yescharacteristic
312C22×C4#1175822211yescharacteristic
313C22×C4D4×D6111yescharacteristic
314C22×C4D4×D6111yescharacteristic
315C22×C4#1175822411yescharacteristic
316C22×C4C2×C22⋊D6111yescharacteristic
317M16—41no—
318M16—41no—
319M16—2441no—
320M16—2441no—
321C2×C8—22no—
322C2×C8—1222no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—1221no—
328C4⋊C4—1221no—
329C4⋊C4—1221no—
330C4⋊C4—1221no—
331C4⋊C4—2442no—
332C4⋊C4—2442no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—1221no—
338C22⋊C4—1221no—
339C22⋊C4—1221no—
340C22⋊C4—1221no—
341C22⋊C4—2442no—
342C22⋊C4—2441no—
343C22⋊C4—2441no—
344C42—1221no—
345C42—1221no—
346C42—1221no—
347C42—1221no—
348C42—2441no—
349C42—2441no—
350C23—21no—
351C23—21no—
352C23—41no—
353C23—41no—
354C23C22⋊C4×D6111yescharacteristic
355C23#1451974811yescharacteristic
356C23#1451975011yescharacteristic
357Q8—81no—
358Q8—81no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
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—1222no—
377C2×C4—1221no—
378C2×C4—1221no—
379C2×C4—1221no—
380C2×C4—1221no—
381C2×C4—4881no—
382C2×C4—4881no—
383C2×C4—4881no—
384C2×C4—2441no—
385C2×C4—2441no—
386C2×C4—2441no—
387C2×C4—2441no—
388C2×C4—2441no—
389C2×C4—2441no—
390C2×C4—2441no—
391C2×C4—2441no—
392C2×C4—2441no—
393C2×C4#1451973711yescharacteristic
394C2×C4C4×C22⋊D6111yescharacteristic
395C2×C4#1451974511yescharacteristic
396C2×C4#1451975911yescharacteristic
397C8—42no—
398C8—2442no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—81no—
408C22—81no—
409C22#1683963811yescharacteristic
410C22#1683973411yescharacteristic
411C22#1683973411yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—81no—
417C4—81no—
418C4—161no—
419C4—9761no—
420C4—4881no—
421C4—2442no—
422C4—2441no—
423C4—2441no—
424C4—2441no—
425C4—2441no—
426C2—21no—
427C2—21no—
428C2—41no—
429C2—41no—
430C2#2144619911yescharacteristic
431C2#2144627411yescharacteristic
432C2#2144633811yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1511C2
socleC22×C61#386
prime radicalC24×C61C2×D4
cosocleC23×C4×C61C23
hypercenter#1511D61
hypocenterC61#9444
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1511

Complements

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

NH ≅ G/Ntype
16. #16839669C4semidirect
17. #16839669C4semidirect
20. #16839631C4semidirect
21. #16839627C4semidirect
247. C61#9444semidirect

Maximal subgroups

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

subgroupindex
#214544542
#214502732
#214458942
#214443522
#214436732
#214436712
C61×#15112
#944461