Finite Group Databasebeta

← #30607608

Subgroups — #30607608

Order 15616.

474 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 86 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
1C2×#16840802C211yesmaximal, characteristic, complemented
2C2×#16840681C211yesmaximal, characteristic, complemented
3C2×#16840678C211yesmaximal, characteristic
4#21461974C211yesmaximal, characteristic
5#21460884C211yesmaximal, characteristic, complemented
6#21460859C211yesmaximal, characteristic, complemented
7C2×#16840201C211yesmaximal, characteristic
8C2×Q8×#6923—21no—
9C2×#14519852C2211yescharacteristic
10#16840802—22no—
11#16840802—22no—
12C2×#14519847C2211yescharacteristic
13C2×#14519845C2211yescharacteristic
14C2×#14519841—21no—
15C2×#14519840—21no—
16#16840736—21no—
17C2×#14519831C2211yescharacteristic
18#16840681—22no—
19#16840678—22no—
20C2×#14519796C2211yescharacteristic
21C2×Q16×D61C2211yescharacteristic
22#16840201—22no—
23#16840201—22no—
24C2×Dic488C411yescharacteristic, complemented
25C2×#14519723C411yescharacteristic, complemented
26C2×#14519721C411yescharacteristic, complemented
27C2×#14519716C2211yescharacteristic
28C2×SD32×C61C411yescharacteristic, complemented
29C2×#11758213—21no—
30C2×D61⋊C8D411yescharacteristic
31Q8×#6923—24no—
32Q8×#6923—42no—
33C2×C4⋊#6923—21no—
34C2×C4⋊#6923—41no—
35C2×C4×#6923—41no—
36C2×C4×#6923D411yescharacteristic
37#14519852—21no—
38#14519852—21no—
39C2×#11758187—21no—
40#14519847—21no—
41#14519847—21no—
42#14519845—21no—
43#14519845D412yes—
44#14519841—42no—
45#14519840—24no—
46C22⋊#92420—42no—
47#14519831—21no—
48#14519831—21no—
49C2×#11758224—21no—
50C2×Q8×D61—21no—
51#14519796—21no—
52#14519796—21no—
53#14519796—21no—
54#14519796—21no—
55C2×Dic244C2×C411yescharacteristic
56C2×Dic244C2×C411yescharacteristic
57C2×#11758202—21no—
58Q16×D61—21no—
59Q16×D61—21no—
60Q16×D61D412yes—
61Q16×D61D412yes—
62C2×#11758200—21no—
63C2×C8×D61C2311yescharacteristic
64Dic488—22no—
65#14519723—22no—
66#14519721—22no—
67#14519716—21no—
68#14519716—21no—
69C2×#11758185C2×C411yescharacteristic
70C2×Q16×C61C2×C411yescharacteristic
71C2×D8×C61C2×C411yescharacteristic
72SD32×C61—22no—
73C2×C16×C61C2×C411yescharacteristic
74C22×#6923—21no—
75C22×#6923—41no—
76C2×Q8×C61—21no—
77C2×D4×C61—21no—
78C2×C22⋊D61—41no—
79C22×Dic61—41no—
80Q8×D61—22no—
81Q8×D61—22no—
82Q8×D61—41no—
83Q8×D61—41no—
84#11758224—41no—
85#11758224—41no—
86#11758224—41no—
87#11758224—41no—
88C2×C4×D61—41no—
89C2×C4×D61C2×D411yescharacteristic
90C2×Dic122—21no—
91C2×Dic122—21no—
92C2×Dic122—41no—
93C2×#92420—21no—
94C2×#92420—41no—
95C4⋊#6923—22no—
96C4⋊#6923—44no—
97C4⋊#6923—41no—
98C4×#6923—21no—
99C4×#6923—44no—
100C4×#6923D812yes—
101#11758213—41no—
102#11758213—41no—
103D61⋊C8—21no—
104D61⋊C8—21no—
105Q16×C61—21no—
106Q16×C61C22⋊C412yes—
107D8×C61—21no—
108D8×C61—21no—
109C2×C8×C61C22×C411yescharacteristic
110#11758202—22no—
111#11758202—22no—
112#11758200—41no—
113#11758200—41no—
114C2×#92418#4111yescharacteristic
115Dic244—21no—
116Dic244—21no—
117Dic244—21no—
118Dic244C22⋊C412yes—
119C8×D61C2×D411yescharacteristic
120C8×D61C2×D411yescharacteristic
121C8×D61C2×D411yescharacteristic
122C8×D61#4111yescharacteristic
123#11758187—41no—
124C16×C61—21no—
125#11758185—21no—
126C23×C61—41no—
127C22×D61AGL(1,ℤ/8)11yescharacteristic
128C2×#6923—22no—
129C2×#6923—41no—
130C2×#6923—41no—
131C2×#6923—42no—
132C2×#6923—81no—
133C2×#6923—81no—
134Q8×C61—22no—
135Q8×C61—41no—
136D4×C61—41no—
137D4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61C4×D411yescharacteristic
140C22⋊D61—81no—
141C22⋊D61—81no—
142C2×Dic61—41no—
143C2×Dic61—41no—
144C2×Dic61—81no—
145C2×Dic61—81no—
146C2×Dic61AGL(1,ℤ/8)11yescharacteristic
147C4×D61—42no—
148C4×D61—42no—
149C4×D61#12011yescharacteristic
150C4×D61#12111yescharacteristic
151C4×D61C2×D811yescharacteristic
152C4×D61#13511yescharacteristic
153Dic122—22no—
154Dic122—41no—
155Dic122—42no—
156Dic122—42no—
157Dic122—41no—
158Dic122—41no—
159#92420—41no—
160#92420—81no—
161C8×C61C2×C22⋊C411yescharacteristic
162C8×C61C4×D411yescharacteristic
163#92418#12111yescharacteristic
164#92418#12311yescharacteristic
165#13038—611nomaximal, Sylow
166C22×C61—81no—
167C22×C61—81no—
168C22×C61—81no—
169C22×C61#44211yescharacteristic
170C2×D61—21no—
171C2×D61—21no—
172C2×D61#44711yescharacteristic
173C2×D61#46311yescharacteristic
174#6923—42no—
175#6923—82no—
176C4×C61—42no—
177C4×C61#38611yescharacteristic
178C4×C61C4×D811yescharacteristic
179Dic61—42no—
180Dic61—81no—
181Dic61#44911yescharacteristic
182Dic61#45911yescharacteristic
183C2×#510—611no—
184C2×#441—611no—
185C2×#362—611no—
186C2×#360—611no—
187#1975—611no—
188#1383—611no—
189#1375—611no—
190C2×C61—161no—
191C2×C61#187911yescharacteristic
192C2×C61#218011yescharacteristic
193C2×C61#218011yescharacteristic
194D61—21no—
195D61—21no—
196C2×#135—611no—
197C22×Q16—611no—
198C2×C4×Q8—1221no—
199#510—1222no—
200#510—1222no—
201C2×Q32—611no—
202C2×Q32—611no—
203C2×SD32—611no—
204C2×SD32#692311yescharacteristic, complemented
205C2×M32—611no—
206C2×M32—611no—
207#441—1222no—
208#441—1222no—
209C2×#108—611no—
210C2×#106—611no—
211C2×#104—1221no—
212C2×#103—1221no—
213#408—1221no—
214C2×#97—611no—
215#362—1222no—
216#360—1222no—
217C61#1303811yescharacteristic, complemented, Sylow
218C2×#41—1221no—
219C22×Q8—1221no—
220#135—1221no—
221#135—1221no—
222#135—1221no—
223#135—1221no—
224C2×Q16—612no—
225C2×Q16—612no—
226C2×Q16—611no—
227C2×Q16—611no—
228C2×Q16—1221no—
229C2×Q16—1221no—
230C2×Q16—1221no—
231C2×Q16C2×#692311yescharacteristic
232C2×SD16—1221no—
233C2×D8C2×#692311yescharacteristic
234C2×M16—1221no—
235C22×C8—611no—
236C22×C8—611no—
237C4×Q8—1224no—
238C4×Q8—2442no—
239C2×C4⋊C4—1221no—
240C2×C4⋊C4—2441no—
241C2×C42—611no—
242C2×C42—2441no—
243Q32—1222no—
244Q32—1222no—
245SD32—22no—
246SD32—1222no—
247M32—1221no—
248M32—1221no—
249M32—1221no—
250M32—1221no—
251C2×C16—611no—
252C2×C16—1221no—
253C2×C16C2×#692311yescharacteristic
254#108—1221no—
255#108—1221no—
256#106—612no—
257#106—1221no—
258#104—2442no—
259#103—1224no—
260C22⋊C8—2442no—
261#97—1221no—
262#97—1221no—
263#41—2441no—
264#41—2441no—
265#41—2441no—
266#41—2441no—
267C2×Q8—21no—
268C2×Q8—1222no—
269C2×Q8—1222no—
270C2×Q8—1221no—
271C2×Q8—1221no—
272C2×Q8—2441no—
273C2×Q8—2441no—
274C2×Q8—2441no—
275C2×D4—21no—
276C2×D4—2441no—
277C22×C4—611no—
278C22×C4—1221no—
279C22×C4—2441no—
280C22×C4—2441no—
281C22×C4—2441no—
282Q16—21no—
283Q16—612no—
284Q16—1222no—
285Q16—1222no—
286Q16—1221no—
287Q16—1221no—
288Q16—1221no—
289Q16C22⋊#692312yes—
290SD16—2441no—
291SD16—2441no—
292D8—21no—
293D8—21no—
294M16—2441no—
295M16—2441no—
296C2×C8—611no—
297C2×C8—611no—
298C2×C8—611no—
299C2×C8—611no—
300C2×C8—611no—
301C2×C8—1221no—
302C2×C8—1221no—
303C2×C8—1221no—
304C2×C8—2441no—
305C2×C8C22×#692311yescharacteristic
306C4⋊C4—1222no—
307C4⋊C4—2441no—
308C4⋊C4—2444no—
309C42—612no—
310C42—1221no—
311C42—2444no—
312C16—21no—
313C16—1221no—
314C16—2441no—
315C23—41no—
316C23—611no—
317Q8—22no—
318Q8—41no—
319Q8—1222no—
320Q8—2442no—
321Q8—2442no—
322Q8—2441no—
323Q8—2441no—
324Q8—2441no—
325D4—41no—
326D4—41no—
327D4—4881no—
328D4—4881no—
329C2×C4—41no—
330C2×C4—611no—
331C2×C4—611no—
332C2×C4—611no—
333C2×C4—611no—
334C2×C4—611no—
335C2×C4—1222no—
336C2×C4—4881no—
337C2×C4—4881no—
338C2×C4—4881no—
339C2×C4—4881no—
340C2×C4—2442no—
341C2×C4—2442no—
342C2×C4—2442no—
343C2×C4—2441no—
344C2×C4—2441no—
345C2×C4—2441no—
346C2×C4—2441no—
347C2×C4D4×#692311yescharacteristic
348C8—611no—
349C8—611no—
350C8—4881no—
351C8—2441no—
352C8C2×C22⋊#692311yescharacteristic
353C8D4×#692311yescharacteristic
354C22—81no—
355C22—81no—
356C22—81no—
357C22—611no—
358C22—611no—
359C22—1221no—
360C22—1221no—
361C22#1684080311yescharacteristic
362C4—42no—
363C4—611no—
364C4—611no—
365C4—4882no—
366C4—4881no—
367C4—2442no—
368C4—2442no—
369C4#1684071011yescharacteristic
370C4D8×#692311yescharacteristic
371C2—161no—
372C2—1221no—
373C2—1221no—
374C2#2146181411yescharacteristic
375C2#2146222411yescharacteristic
376C2#2146222511yescharacteristic
subgroupG/H
centerC22#16840803
commutatorC2×C8×C61C22×C4
FrattiniC2×C8C22×#6923
FittingC2×SD32×C61C4
socleC22×C61#442
prime radicalC2×#14519796C22
cosocleC2×C8×D61C23
hypercenterC2×SD32#6923
hypocenterC61#13038
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×SD32

Complements

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

NH ≅ G/Ntype
1. C2×#16840802C2semidirect
2. C2×#16840681C2semidirect
5. #21460884C2semidirect
6. #21460859C2semidirect
24. C2×Dic488C4semidirect
25. C2×#14519723C4semidirect
26. C2×#14519721C4semidirect
28. C2×SD32×C61C4semidirect
204. C2×SD32#6923semidirect
217. C61#13038semidirect

Maximal subgroups

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

subgroupindex
C2×#168408022
C2×#168406812
C2×#168406782
#214619742
#214608842
#214608592
C2×#168402012
#1303861