Finite Group Databasebeta

← #30599776

Subgroups — #30599776

Order 15616.

476 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 106 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#21462352C211yesmaximal, characteristic
2#21461946C211yesmaximal, characteristic
3#21461931C211yesmaximal, characteristic
4#21461930C211yesmaximal, characteristic
5#21460653C211yesmaximal, characteristic
6#21460653C211yesmaximal, characteristic
7#333×D61C211yesmaximal, characteristic
8C2×#14519846—21no—
9C2×#14519845—21no—
10C2×#14519840—22no—
11C4×D61⋊C8C2211yescharacteristic
12(C2×#6923)⋊Q8—22no—
13#16840713—21no—
14#16840713—21no—
15#16840708C2211yescharacteristic
16C4⋊C4×#6923—21no—
17C4⋊C4×#6923—21no—
18C4×C4⋊#6923C2211yescharacteristic
19C4×C4⋊#6923C2211yescharacteristic
20#16840636—21no—
21#16840636—21no—
22C4⋊Q8×D61C2211yescharacteristic
23C4⋊C8×D61C2211yescharacteristic
24C4⋊C8×D61C2211yescharacteristic
25#16839511C411yescharacteristic, complemented
26#16839510C411yescharacteristic, complemented
27#16839510C411yescharacteristic, complemented
28C61×#333C411yescharacteristic, complemented
29C2×D61⋊C8—21no—
30C2×C4⋊#6923—21no—
31C2×C4⋊#6923—21no—
32C2×C4⋊#6923—21no—
33C2×C4⋊#6923—22no—
34C2×C4⋊#6923D411yescharacteristic
35C2×C4⋊#6923D411yescharacteristic
36C2×C4⋊#6923D411yescharacteristic
37C2×C4⋊#6923D411yescharacteristic
38C2×C4×#6923—21no—
39C2×C4×#6923—21no—
40C2×C4×#6923—22no—
41#14519846—42no—
42#14519845—42no—
43#14519840—44no—
44C4×#92420—21no—
45C4×#92420—21no—
46(C2×#6923)⋊C4—22no—
47(C2×#6923)⋊C4—22no—
48(C2×#6923)⋊C4—21no—
49(C2×#6923)⋊C4—21no—
50C2×Q8×D61D412yes—
51C2×C8×D61—21no—
52C2×C8×D61—21no—
53#14519777C2×C411yescharacteristic
54#14519776—21no—
55#14519776—21no—
56#14519775Q812yes—
57C4⋊C4×D61—21no—
58C4⋊C4×D61—21no—
59C42×D61C2311yescharacteristic
60C4⋊#92418C2×C411yescharacteristic
61C4⋊#92418C2×C411yescharacteristic
62#14519706—21no—
63#14519706—21no—
64C4⋊Q8×C61C2×C411yescharacteristic
65C4⋊C8×C61C2×C411yescharacteristic
66C4⋊C8×C61C2×C411yescharacteristic
67C22×#6923—21no—
68C22×#6923—21no—
69C22×#6923—21no—
70C22×#6923—21no—
71C22×#6923—41no—
72C2×Q8×C61C4⋊C412yes—
73Q8×D61—42no—
74Q8×D61—42no—
75C2×C4×D61—22no—
76C2×C4×D61C2×D411yescharacteristic
77C2×C4×D61C2×D411yescharacteristic
78C2×C4×D61C2×D411yescharacteristic
79C2×Dic122—22no—
80C2×Dic122C4⋊C412yes—
81C2×#92420—41no—
82C4⋊#6923—21no—
83C4⋊#6923—21no—
84C4⋊#6923—21no—
85C4⋊#6923—21no—
86C4⋊#6923—21no—
87C4⋊#6923—21no—
88C4⋊#6923—42no—
89C4⋊#6923—41no—
90C4⋊#6923—41no—
91C4⋊#6923—41no—
92C4⋊#6923—41no—
93C4⋊#6923—82no—
94C4⋊#6923SD1612yes—
95C4⋊#6923Q1612yes—
96C4×#6923—41no—
97C4×#6923—41no—
98C4×#6923—41no—
99C4×#6923—41no—
100C4×#6923—82no—
101D61⋊C8—22no—
102D61⋊C8—41no—
103C2×C8×C61—21no—
104C2×C8×C61—21no—
105C4⋊C4×C61—21no—
106C4⋊C4×C61—21no—
107C42×C61C22×C411yescharacteristic
108C4⋊Dic61—21no—
109C4⋊Dic61—21no—
110#11758196—22no—
111#11758196—22no—
112C4×Dic61C2×Q811yescharacteristic
113C4×Dic61#4111yescharacteristic
114C4×Dic61#4111yescharacteristic
115C2×#92418—21no—
116C2×#92418—21no—
117C8×D61—41no—
118C8×D61—41no—
119C8×D61—41no—
120C8×D61—41no—
121C22×D61#12011yescharacteristic
122C2×#6923—22no—
123C2×#6923—22no—
124C2×#6923—41no—
125C2×#6923—41no—
126C2×#6923—41no—
127C2×#6923—41no—
128C2×#6923—41no—
129C2×#6923—41no—
130C2×#6923—41no—
131C2×#6923—41no—
132C2×#6923—41no—
133C2×#6923—41no—
134C2×#6923—81no—
135C2×#6923—81no—
136C2×#6923—81no—
137Q8×C61—42no—
138C2×C4×C61—22no—
139C2×C4×C61C2×C4⋊C411yescharacteristic
140C2×C4×C61C4×D411yescharacteristic
141C2×C4×C61C4×D411yescharacteristic
142C2×Dic61—22no—
143C2×Dic61C22⋊Q811yescharacteristic
144C2×Dic61C22⋊Q811yescharacteristic
145C2×Dic61#12311yescharacteristic
146C4×D61—21no—
147C4×D61—21no—
148C4×D61—42no—
149C4×D61—42no—
150C4×D61C2×SD1611yescharacteristic
151C4×D61C2×Q1611yescharacteristic
152C4×D61#13511yescharacteristic
153C4×D61#13511yescharacteristic
154C4×D61AGL(1,ℤ/8)11yescharacteristic
155C4×D61#13711yescharacteristic
156C4×D61#13711yescharacteristic
157C4×D61#13711yescharacteristic
158Dic122—42no—
159Dic122—82no—
160#92420—42no—
161C8×C61—41no—
162C8×C61—41no—
163#92418—41no—
164#92418—41no—
165#8927—611nomaximal, Sylow
166C22×C61#38511yescharacteristic
167C2×D61#44811yescharacteristic
168C2×D61#44911yescharacteristic
169C2×D61#45111yescharacteristic
170C2×D61#45211yescharacteristic
171C2×D61#45311yescharacteristic
172C2×D61#45511yescharacteristic
173#6923—42no—
174#6923—42no—
175#6923—81no—
176#6923—81no—
177#6923—161no—
178C4×C61—21no—
179C4×C61—42no—
180C4×C61C4×SD1611yescharacteristic
181C4×C61C4×Q1611yescharacteristic
182C4×C61#44011yescharacteristic
183C4×C61#44111yescharacteristic
184Dic61—21no—
185Dic61—42no—
186Dic61#47511yescharacteristic
187Dic61#47711yescharacteristic
188Dic61#47911yescharacteristic
189Dic61#47911yescharacteristic
190#2369—611no—
191#1956—611no—
192#1948—611no—
193#1947—611no—
194C2×#333—611no—
195#1279—611no—
196#1279—611no—
197C2×C61#185611yescharacteristic
198C2×C61#186311yescharacteristic
199C2×C61#186511yescharacteristic
200D61#162211yescharacteristic
201D61#163211yescharacteristic
202D61#164511yescharacteristic
203D61#164611yescharacteristic
204C2×C4⋊Q8—611no—
205C2×#107—1221no—
206C2×#106—1221no—
207C2×C4⋊C8—611no—
208C2×C4⋊C8—611no—
209C2×#103—1222no—
210C2×C4×C8—611no—
211#391—1222no—
212#389—1221no—
213#389—1221no—
214#384—611no—
215C4×C4⋊C4—611no—
216C4×C4⋊C4—611no—
217C4×C4⋊C4—1221no—
218C4×C4⋊C4—1221no—
219#340—1221no—
220#340—1221no—
221#333—611no—
222#333—611no—
223#333—611no—
224#333#692311yescharacteristic, complemented
225C61#892711yescharacteristic, complemented, Sylow
226C22×Q8—612no—
227C22×C8—1221no—
228C22×C8—1221no—
229C22×C8—1221no—
230C4⋊Q8—612no—
231C4⋊Q8—611no—
232C4⋊Q8—1221no—
233C4⋊Q8—1221no—
234C4⋊Q8C2×#692311yescharacteristic
235C2×C4⋊C4—611no—
236C2×C4⋊C4—611no—
237C2×C4⋊C4—611no—
238C2×C4⋊C4—611no—
239C2×C4⋊C4—1221no—
240C2×C4⋊C4—1221no—
241C2×C4⋊C4—1222no—
242C2×C4⋊C4—1221no—
243C2×C4⋊C4—1221no—
244C2×C4⋊C4—1221no—
245C2×C42—611no—
246C2×C42—1222no—
247C2×C42—1221no—
248C2×C42—1221no—
249#107—2442no—
250#106—2442no—
251C4⋊C8—611no—
252C4⋊C8—611no—
253C4⋊C8—1221no—
254C4⋊C8—1221no—
255C4⋊C8C2×#692311yescharacteristic
256C4⋊C8C2×#692311yescharacteristic
257#103—2444no—
258C4×C8—1221no—
259C4×C8—1221no—
260#95—1221no—
261#95—1221no—
262#95—1222no—
263#95—1222no—
264C2×Q8—612no—
265C2×Q8—1222no—
266C2×Q8—2442no—
267C2×Q8—2442no—
268C2×Q8C4⋊#692312yes—
269C22×C4—611no—
270C22×C4—611no—
271C22×C4—611no—
272C22×C4—1222no—
273C22×C4—1221no—
274C22×C4—1221no—
275C22×C4—1221no—
276C22×C4—1221no—
277C22×C4—2441no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—1222no—
281C2×C8—1221no—
282C2×C8—1221no—
283C2×C8—2441no—
284C2×C8—2441no—
285C2×C8—2441no—
286C2×C8—2441no—
287C2×C8—2441no—
288C2×C8—2441no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—612no—
292C4⋊C4—612no—
293C4⋊C4—1222no—
294C4⋊C4—1222no—
295C4⋊C4—1221no—
296C4⋊C4—1221no—
297C4⋊C4—1221no—
298C4⋊C4—1221no—
299C4⋊C4—1221no—
300C4⋊C4—1221no—
301C4⋊C4—1221no—
302C4⋊C4—1221no—
303C4⋊C4—4882no—
304C4⋊C4—2441no—
305C4⋊C4—2441no—
306C4⋊C4—2441no—
307C4⋊C4—2441no—
308C4⋊C4—2442no—
309C42—611no—
310C42—611no—
311C42—611no—
312C42—4882no—
313C42—2441no—
314C42—2441no—
315C42—2441no—
316C42—2441no—
317C42C22×#692311yescharacteristic
318C23—611no—
319Q8—42no—
320Q8—4882no—
321Q8—2442no—
322C2×C4—22no—
323C2×C4—611no—
324C2×C4—611no—
325C2×C4—611no—
326C2×C4—611no—
327C2×C4—611no—
328C2×C4—611no—
329C2×C4—611no—
330C2×C4—611no—
331C2×C4—611no—
332C2×C4—611no—
333C2×C4—611no—
334C2×C4—1221no—
335C2×C4—1221no—
336C2×C4—1222no—
337C2×C4—1222no—
338C2×C4—1222no—
339C2×C4—4881no—
340C2×C4—4881no—
341C2×C4—4881no—
342C2×C4—2442no—
343C2×C4—2442no—
344C2×C4—2441no—
345C2×C4—2441no—
346C2×C4—2441no—
347C2×C4—2441no—
348C2×C4—2441no—
349C2×C4—2441no—
350C2×C4—2441no—
351C2×C4—2441no—
352C2×C4—2441no—
353C2×C4—2441no—
354C2×C4C2×C4⋊#692311yescharacteristic
355C2×C4D4×#692311yescharacteristic
356C2×C4D4×#692311yescharacteristic
357C8—41no—
358C8—41no—
359C8—2442no—
360C8—2441no—
361C8—2441no—
362C22—611no—
363C22—611no—
364C22—611no—
365C22—611no—
366C22—611no—
367C22—611no—
368C22#1684070911yescharacteristic
369C4—21no—
370C4—42no—
371C4—611no—
372C4—611no—
373C4—611no—
374C4—611no—
375C4—1221no—
376C4—9761no—
377C4—4881no—
378C4—4881no—
379C4—2442no—
380C4—2442no—
381C4—2442no—
382C4SD16×#692311yescharacteristic
383C4Q16×#692311yescharacteristic
384C4#1684080111yescharacteristic
385C4#1684080211yescharacteristic
386C2—611no—
387C2—611no—
388C2—611no—
389C2—611no—
390C2#2146177411yescharacteristic
391C2#2146178311yescharacteristic
392C2#2146178511yescharacteristic
subgroupG/H
centerC22#16840709
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#333C4
socleC22×C61#385
prime radicalC22×D61#120
cosocleC42×D61C23
hypercenter#333#6923
hypocenterC61#8927
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
25. #16839511C4semidirect
26. #16839510C4semidirect
27. #16839510C4semidirect
28. C61×#333C4semidirect
224. #333#6923semidirect
225. C61#8927semidirect

Maximal subgroups

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

subgroupindex
#214623522
#214619462
#214619312
#214619302
#214606532
#333×D612
#892761