Finite Group Databasebeta

← #30598999

Subgroups — #30598999

Order 15616.

524 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 98 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#21461696C211yesmaximal, characteristic, complemented
2#21461695C211yesmaximal, characteristic, complemented
3#21461679C212yesmaximal
4#21460682C212yesmaximal, complemented
5C2×#16839831C211yesmaximal, characteristic
6C2×#14519846—21no—
7C2×#14519845—21no—
8C2×#14519835—22no—
9#16840704C2212yes—
10#16840704C2212yes—
11#16840640—22no—
12#16840636—21no—
13#16840636—21no—
14C22×C8×D61C2211yescharacteristic
15C2×C4×C22⋊D61C2211yescharacteristic
16#16839831—21no—
17#16839831—21no—
18#16839831—21no—
19#16839831—21no—
20C2×#14519706C411yescharacteristic, complemented
21C2×C22⋊#92418C411yescharacteristic, complemented
22C2×#14519689C2211yescharacteristic
23C2×#14519687C411yescharacteristic, complemented
24C22×M16×C61C411yescharacteristic, complemented
25C2×C4⋊#6923—22no—
26C2×C4⋊#6923—22no—
27C2×C22⋊#6923—22no—
28C2×C22⋊#6923—22no—
29#14519846—42no—
30#14519845—42no—
31#14519835—44no—
32(C2×#6923)⋊C4—22no—
33(C2×#6923)⋊C4—22no—
34C22×C22⋊D61D411yescharacteristic
35C22×C4×D61C2311yescharacteristic
36C22×#92418C2×C411yescharacteristic
37C22×#92418C2×C411yescharacteristic
38C2×C8×D61—42no—
39C2×C8×D61D411yescharacteristic
40C2×C8×D61D411yescharacteristic
41C2×C8×D61D411yescharacteristic
42C2×C8×D61D411yescharacteristic
43C4×C22⋊D61—21no—
44C4×C22⋊D61—21no—
45C4×C22⋊D61—21no—
46C4×C22⋊D61—21no—
47C2×#11758196Q811yescharacteristic
48C2×C22⋊Dic61Q811yescharacteristic
49C2×#11758198D411yescharacteristic
50C2×C4×Dic61C2×C411yescharacteristic
51#14519706—21no—
52#14519706—21no—
53C22⋊#92418—21no—
54C22⋊#92418—21no—
55#14519689—21no—
56#14519689—21no—
57#14519687—21no—
58#14519687—21no—
59C23×C4×C61C2×C411yescharacteristic
60C2×M16×C61—21no—
61C2×M16×C61—21no—
62C2×M16×C61—42no—
63C22×C8×C61C2×C411yescharacteristic
64C22×C8×C61C2×C411yescharacteristic
65C24×C61C4⋊C411yescharacteristic
66C23×D61C2×D411yescharacteristic
67C22×#6923—42no—
68C22×#6923—42no—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61—41no—
72C22×C4×C61—41no—
73C22×C4×C61C4⋊C411yescharacteristic
74C22×C4×C61C22×C411yescharacteristic
75C2×C22⋊D61—21no—
76C2×C22⋊D61—21no—
77C2×C22⋊D61—22no—
78C2×C22⋊D61—22no—
79C22×Dic61C4⋊C411yescharacteristic
80C22×Dic61C4⋊C411yescharacteristic
81C22×Dic61C2×Q811yescharacteristic
82C2×C4×D61—41no—
83C2×C4×D61—41no—
84C2×C4×D61C2×D411yescharacteristic
85C2×C4×D61C2×D411yescharacteristic
86C2×C4×D61#4111yescharacteristic
87C2×C4×D61#4111yescharacteristic
88C22⋊#6923—42no—
89C22⋊#6923—42no—
90C22⋊#6923—42no—
91C22⋊#6923—42no—
92C4⋊#6923—42no—
93C4⋊#6923—42no—
94C4⋊#6923—42no—
95C4⋊#6923—42no—
96M16×C61—42no—
97M16×C61—42no—
98C2×C8×C61—22no—
99C2×C8×C61—21no—
100C2×C8×C61—41no—
101C2×C8×C61C22⋊C411yescharacteristic
102C2×C8×C61C22⋊C411yescharacteristic
103C22⋊Dic61—21no—
104C22⋊Dic61—21no—
105#11758198—21no—
106#11758198—21no—
107#11758196—21no—
108#11758196—21no—
109C4×Dic61—21no—
110C4×Dic61—21no—
111C2×#92418—21no—
112C2×#92418—41no—
113C2×#92418—41no—
114C2×#92418C22⋊C411yescharacteristic
115C2×#92418C22⋊C411yescharacteristic
116C8×D61—22no—
117C8×D61—22no—
118C8×D61—41no—
119C8×D61—41no—
120C23×C61—21no—
121C23×C61—22no—
122C23×C61—41no—
123C23×C61—41no—
124C23×C61C2×C4⋊C411yescharacteristic
125C22×D61—41no—
126C22×D61—41no—
127C22×D61#12111yescharacteristic
128C22×D61#12111yescharacteristic
129C22×D61AGL(1,ℤ/8)12yes—
130C22×D61#13712yes—
131C2×#6923—82no—
132C2×#6923—82no—
133C2×#6923—82no—
134C2×#6923—82no—
135C2×#6923—82no—
136C2×#6923—82no—
137C2×C4×C61—21no—
138C2×C4×C61—21no—
139C2×C4×C61—21no—
140C2×C4×C61—42no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61C2×C22⋊C411yescharacteristic
145C2×C4×C61#11711yescharacteristic
146C22⋊D61—42no—
147C22⋊D61—41no—
148C22⋊D61—41no—
149C2×Dic61—21no—
150C2×Dic61—21no—
151C2×Dic61—22no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61C22⋊Q811yescharacteristic
155C2×Dic61C22⋊Q811yescharacteristic
156C4×D61—21no—
157C4×D61—21no—
158C4×D61—21no—
159C4×D61—21no—
160C4×D61—41no—
161C4×D61—41no—
162C8×C61—21no—
163C8×C61—41no—
164C8×C61#9912yescomplemented
165#92418—21no—
166#92418—21no—
167#92418—41no—
168#8610—611nomaximal, Sylow
169C22×C61—21no—
170C22×C61—21no—
171C22×C61—41no—
172C22×C61—41no—
173C22×C61—41no—
174C22×C61—82no—
175C22×C61#38011yescharacteristic
176C22×C61#42812yes—
177C2×D61—21no—
178C2×D61—21no—
179C2×D61—21no—
180C2×D61—21no—
181C2×D61—42no—
182C2×D61—42no—
183C2×D61#46812yes—
184C2×D61#46911yescharacteristic
185C2×D61#47011yescharacteristic
186#6923—162no—
187#6923—162no—
188C4×C61—21no—
189C4×C61—41no—
190C4×C61C2×#9911yescharacteristic
191C4×C61#41011yescharacteristic
192Dic61—21no—
193Dic61—21no—
194Dic61—41no—
195Dic61—41no—
196C2×#435—611no—
197#1808—611no—
198#1807—611no—
199#1799—612no—
200#1289—612no—
201C2×C61—22no—
202C2×C61—81no—
203C2×C61#179911yescharacteristic
204C2×C61#180911yescharacteristic
205C2×C61#181211yescharacteristic
206D61—21no—
207D61—22no—
208D61—21no—
209C22×M16#692311yescharacteristic, complemented
210C23×C8—611no—
211C2×C4×D4—611no—
212#435—1221no—
213#435—1221no—
214#435—1221no—
215#435—1221no—
216C2×#107—1221no—
217C2×#106—1221no—
218C2×C4⋊C8—611no—
219C2×#99—1222no—
220C2×C22⋊C8—611no—
221C2×C22⋊C8—611no—
222C2×#97—611no—
223#380—612no—
224#380—612no—
225#342—1222no—
226#340—1221no—
227#340—1221no—
228C61#861011yescharacteristic, complemented, Sylow
229C22×D4—611no—
230C23×C4—611no—
231C23×C4C2×#692311yescharacteristic
232C2×M16—21no—
233C2×M16—21no—
234C2×M16—42no—
235C22×C8—611no—
236C22×C8—611no—
237C22×C8—611no—
238C22×C8—611no—
239C22×C8—611no—
240C22×C8—611no—
241C22×C8—2442no—
242C22×C8C2×#692311yescharacteristic
243C22×C8C2×#692311yescharacteristic
244C4×D4—1221no—
245C4×D4—1221no—
246C4×D4—1221no—
247C4×D4—1221no—
248C2×C4⋊C4—611no—
249C2×C4⋊C4—1222no—
250C2×C4⋊C4—1222no—
251C2×C22⋊C4—611no—
252C2×C22⋊C4—611no—
253C2×C22⋊C4—1222no—
254C2×C22⋊C4—1222no—
255C2×C42—611no—
256#107—2442no—
257#106—2442no—
258C4⋊C8—1221no—
259C4⋊C8—1221no—
260#99—2444no—
261C22⋊C8—1221no—
262C22⋊C8—1221no—
263C22⋊C8—1221no—
264C22⋊C8—1221no—
265#97—1221no—
266#97—1221no—
267#95—1222no—
268#95—1222no—
269C24—611no—
270C24C4⋊#692311yescharacteristic
271C2×D4—1222no—
272C2×D4—1221no—
273C2×D4—1221no—
274C2×D4—1222no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—611no—
280C22×C4—611no—
281C22×C4—611no—
282C22×C4—611no—
283C22×C4—611no—
284C22×C4—611no—
285C22×C4—611no—
286C22×C4—2441no—
287C22×C4—2441no—
288C22×C4—2442no—
289C22×C4—2442no—
290C22×C4C4⋊#692311yescharacteristic
291C22×C4C22×#692311yescharacteristic
292M16—42no—
293M16—42no—
294C2×C8—22no—
295C2×C8—21no—
296C2×C8—41no—
297C2×C8—611no—
298C2×C8—611no—
299C2×C8—1222no—
300C2×C8—1222no—
301C2×C8—1221no—
302C2×C8—2441no—
303C2×C8—2441no—
304C2×C8—2441no—
305C2×C8—2441no—
306C2×C8C22⋊#692311yescharacteristic
307C2×C8C22⋊#692311yescharacteristic
308C4⋊C4—1221no—
309C4⋊C4—1221no—
310C4⋊C4—2442no—
311C4⋊C4—2442no—
312C4⋊C4—2442no—
313C4⋊C4—2442no—
314C22⋊C4—1221no—
315C22⋊C4—1221no—
316C22⋊C4—1221no—
317C22⋊C4—1221no—
318C22⋊C4—2442no—
319C22⋊C4—2442no—
320C22⋊C4—2442no—
321C22⋊C4—2442no—
322C42—1221no—
323C42—1221no—
324C23—21no—
325C23—22no—
326C23—41no—
327C23—41no—
328C23—611no—
329C23—611no—
330C23—612no—
331C23—612no—
332C23—2441no—
333C23—2441no—
334C23C2×C4⋊#692311yescharacteristic
335D4—2441no—
336D4—2442no—
337D4—2441no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—611no—
346C2×C4—611no—
347C2×C4—1221no—
348C2×C4—1221no—
349C2×C4—1221no—
350C2×C4—1221no—
351C2×C4—1221no—
352C2×C4—1221no—
353C2×C4—1222no—
354C2×C4—4882no—
355C2×C4—4882no—
356C2×C4—4882no—
357C2×C4—4882no—
358C2×C4—4882no—
359C2×C4—4882no—
360C2×C4—2441no—
361C2×C4—2441no—
362C2×C4—2441no—
363C2×C4—2441no—
364C2×C4C2×C22⋊#692311yescharacteristic
365C2×C4#1451985611yescharacteristic
366C8—21no—
367C8—41no—
368C8—1221no—
369C8—1221no—
370C8—2441no—
371C8#1451983512yescomplemented
372C22—21no—
373C22—21no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—82no—
378C22—611no—
379C22—612no—
380C22—611no—
381C22—1221no—
382C22—1221no—
383C22—1221no—
384C22—1221no—
385C22—2442no—
386C22—2442no—
387C22#1684070411yescharacteristic
388C22#1684077312yes—
389C4—21no—
390C4—41no—
391C4—1221no—
392C4—1221no—
393C4—9762no—
394C4—9762no—
395C4—2441no—
396C4—2441no—
397C4C2×#1451983511yescharacteristic
398C4#1684074111yescharacteristic
399C2—22no—
400C2—81no—
401C2—1221no—
402C2—1222no—
403C2—1221no—
404C2#2146167911yescharacteristic
405C2#2146169711yescharacteristic
406C2#2146170211yescharacteristic
subgroupG/H
centerC22#16840704
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC22×M16×C61C4
socleC22×C61#380
prime radicalC22×C22⋊D61D4
cosocleC22×C4×D61C23
hypercenterC22×M16#6923
hypocenterC61#8610
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
1. #21461696C2semidirect
2. #21461695C2semidirect
4. #21460682C2semidirect
20. C2×#14519706C4semidirect
21. C2×C22⋊#92418C4semidirect
23. C2×#14519687C4semidirect
24. C22×M16×C61C4semidirect
164. C8×C61#99semidirect
209. C22×M16#6923semidirect
228. C61#8610semidirect
371. C8#14519835semidirect

Maximal subgroups

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

subgroupindex
#214616962
#214616952
#214616792
#214606822
C2×#168398312
#861061