Finite Group Databasebeta

← #30600313

Subgroups — #30600313

Order 15616.

632 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 114 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×#16840781C211yesmaximal, characteristic
2C2×#16840616C212yesmaximal
3#21460995C211yesmaximal, characteristic
4#21460692C212yesmaximal
5#21454160C211yesmaximal, characteristic
6#16840781—24no—
7C2×C4⋊#92420—21no—
8C2×#14519834C2212yes—
9C2×C4×#92420—21no—
10C2×(C2×#6923)⋊C4—21no—
11C2×(C2×#6923)⋊C4—21no—
12C2×(C2×#6923)⋊C4C2212yes—
13#16840616—24no—
14C2×#14519753C2211yescharacteristic
15C2×C42×D61C2211yescharacteristic
16#16839669C411yescharacteristic, complemented
17#16839661C411yescharacteristic, complemented
18#16839660C411yescharacteristic, complemented
19#16839645—21no—
20#16839644—21no—
21#16839636—21no—
22#16839624—21no—
23#16839619C2211yescharacteristic
24C61×#391C411yescharacteristic, complemented
25C23×#6923—22no—
26C22×#92420—22no—
27C4⋊#92420—24no—
28#14519834—22no—
29#14519834D414yes—
30C4×#92420—24no—
31(C2×#6923)⋊C4—24no—
32(C2×#6923)⋊C4—22no—
33(C2×#6923)⋊C4—22no—
34(C2×#6923)⋊C4—42no—
35C22×C4×D61D411yescharacteristic
36C22×C4×D61D411yescharacteristic
37C22×C4×D61C2311yescharacteristic
38#14519753—22no—
39#14519753—22no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61C2×C411yescharacteristic
42C2×C4⋊Dic61C2×C411yescharacteristic
43C2×#11758196—21no—
44C2×#11758196—21no—
45C2×#11758198—21no—
46C2×#11758198—21no—
47C2×C4×Dic61C2×C411yescharacteristic
48C2×C4×Dic61D411yescharacteristic
49C2×C4×Dic61D411yescharacteristic
50C42×D61—22no—
51C42×D61—22no—
52#14519684—21no—
53#14519684—21no—
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57C22×Q8×C61C2×C411yescharacteristic
58C2×C4⋊C4×C61C2×C411yescharacteristic
59C2×C42×C61C2×C411yescharacteristic
60#95×C61—21no—
61#95×C61—21no—
62C23×D61C2×D411yescharacteristic
63C22×#6923—24no—
64C22×#6923—42no—
65C22×#6923—42no—
66C22×#6923—42no—
67C22×#6923—42no—
68C22×#6923—42no—
69C22×#6923—42no—
70C2×Q8×C61—22no—
71C2×Q8×C61—41no—
72C2×Q8×C61—41no—
73C22×C4×C61—21no—
74C22×C4×C61—21no—
75C22×C4×C61C22⋊C411yescharacteristic
76C22×C4×C61C22⋊C411yescharacteristic
77C22×C4×C61C22×C411yescharacteristic
78C22×Dic61—21no—
79C22×Dic61—21no—
80C22×Dic61C22⋊C411yescharacteristic
81C22×Dic61C22⋊C411yescharacteristic
82C22×Dic61C2×D411yescharacteristic
83C2×C4×D61—22no—
84C2×C4×D61—22no—
85C2×C4×D61—22no—
86C2×C4×D61—22no—
87C2×C4×D61—41no—
88C2×C4×D61—41no—
89C2×C4×D61—41no—
90C2×C4×D61SD1612yes—
91C2×C4×D61Q1612yes—
92C2×C4×D61C2×D412yes—
93C2×C4×D61#4112yes—
94C2×#92420—24no—
95C2×#92420—42no—
96C2×#92420—42no—
97C4⋊C4×C61—41no—
98C42×C61—22no—
99#11758198—42no—
100#11758198—42no—
101C4⋊Dic61—22no—
102C4⋊Dic61—42no—
103C4⋊Dic61—41no—
104#11758196—42no—
105#11758196—41no—
106#11758196—41no—
107C4×Dic61—22no—
108C4×Dic61—21no—
109C4×Dic61—21no—
110C4×Dic61—21no—
111C4×Dic61—21no—
112C23×C61C2×C22⋊C411yescharacteristic
113C22×D61—21no—
114C22×D61—21no—
115C22×D61—21no—
116C22×D61—21no—
117C22×D61#12112yes—
118C22×D61C2×SD1611yescharacteristic
119C22×D61C2×Q1611yescharacteristic
120C22×D61#13512yes—
121C2×#6923—44no—
122C2×#6923—44no—
123C2×#6923—44no—
124C2×#6923—82no—
125C2×#6923—82no—
126C2×#6923—82no—
127C2×#6923—82no—
128Q8×C61—82no—
129C2×C4×C61—21no—
130C2×C4×C61—21no—
131C2×C4×C61—21no—
132C2×C4×C61—21no—
133C2×C4×C61—21no—
134C2×C4×C61—21no—
135C2×C4×C61—41no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61#10312yes—
142C2×C4×C61#10412yes—
143C2×C4×C61C4×D412yes—
144C2×Dic61—21no—
145C2×Dic61—21no—
146C2×Dic61—21no—
147C2×Dic61—21no—
148C2×Dic61—21no—
149C2×Dic61—21no—
150C2×Dic61—21no—
151C2×Dic61—41no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61#10312yes—
158C2×Dic61#12011yescharacteristic
159C2×Dic61#12311yescharacteristic
160C4×D61—24no—
161C4×D61—24no—
162C4×D61—42no—
163C4×D61—42no—
164C4×D61—44no—
165#92420—44no—
166#9112—611nomaximal, Sylow
167C22×C61#38611yescharacteristic
168C22×C61#41011yescharacteristic
169C22×C61#41011yescharacteristic
170C22×C61C2×#10311yescharacteristic
171C22×C61#41611yescharacteristic
172C22×C61C2×#10411yescharacteristic
173C22×C61#42111yescharacteristic
174C2×D61—22no—
175C2×D61—22no—
176C2×D61—22no—
177C2×D61—21no—
178C2×D61—21no—
179C2×D61—21no—
180C2×D61—21no—
181C2×D61—22no—
182C2×D61#46512yes—
183C2×D61#46712yes—
184#6923—84no—
185C4×C61—22no—
186C4×C61—22no—
187C4×C61—22no—
188C4×C61—81no—
189C4×C61—81no—
190Dic61—22no—
191Dic61—21no—
192Dic61—21no—
193Dic61—41no—
194Dic61—81no—
195Dic61—81no—
196C2×#431—611no—
197#2855—611no—
198C2×#328—612no—
199#1434—611no—
200#1294—612no—
201C2×C61#150011yescharacteristic
202C2×C61#150311yescharacteristic
203C2×C61#151811yescharacteristic
204C2×C61#152111yescharacteristic
205C2×C61#187111yescharacteristic
206C2×C61#189211yescharacteristic
207C2×C61#189911yescharacteristic
208D61—22no—
209D61—22no—
210C2×C22⋊Q8—611no—
211C22×C42—611no—
212#431—1224no—
213C2×C4⋊C8—1221no—
214C2×C22⋊C8—612no—
215C2×C4×C8—1221no—
216#391—611no—
217#391#692311yescharacteristic, complemented
218#389—611no—
219#389—611no—
220#387—1221no—
221#387—1221no—
222#385—1221no—
223#381—1221no—
224#380—611no—
225C2×#95—612no—
226C2×#95—1221no—
227C2×#95—1221no—
228#328—1224no—
229C61#911211yescharacteristic, complemented, Sylow
230C22×Q8C2×#692311yescharacteristic
231C23×C4—611no—
232C23×C4—611no—
233C23×C4—611no—
234C23×C4—1222no—
235C22×C8—1222no—
236C22⋊Q8—1222no—
237C22⋊Q8—1222no—
238C2×C4⋊C4—611no—
239C2×C4⋊C4—611no—
240C2×C4⋊C4—1221no—
241C2×C4⋊C4—1221no—
242C2×C4⋊C4—1221no—
243C2×C4⋊C4C2×#692311yescharacteristic
244C2×C22⋊C4—1221no—
245C2×C22⋊C4—1221no—
246C2×C42—611no—
247C2×C42—611no—
248C2×C42—611no—
249C2×C42—1222no—
250C2×C42—1222no—
251C2×C42C2×#692311yescharacteristic
252C4⋊C8—1224no—
253C22⋊C8—614no—
254C22⋊C8—1222no—
255C4×C8—1224no—
256#95—21no—
257#95—21no—
258#95—1221no—
259#95—1221no—
260#95—1221no—
261#95—1221no—
262#95—1222no—
263#95—1222no—
264#95—1224no—
265#95—1221no—
266#95—2442no—
267C24—611no—
268C2×Q8—22no—
269C2×Q8—41no—
270C2×Q8—41no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—611no—
274C22×C4—611no—
275C22×C4—612no—
276C22×C4—612no—
277C22×C4—612no—
278C22×C4—612no—
279C22×C4—611no—
280C22×C4—1222no—
281C22×C4—1222no—
282C22×C4—1222no—
283C22×C4—1222no—
284C22×C4—1224no—
285C22×C4—1221no—
286C22×C4—1221no—
287C22×C4—2441no—
288C22×C4—2441no—
289C22×C4—2441no—
290C22×C4—2442no—
291C22×C4—2442no—
292C22×C4—2442no—
293C22×C4—2442no—
294C22×C4—2442no—
295C22×C4—2442no—
296C22×C4C22⋊#692311yescharacteristic
297C22×C4C22⋊#692311yescharacteristic
298C22×C4C22×#692311yescharacteristic
299C2×C8—1224no—
300C2×C8—2442no—
301C2×C8—2442no—
302C4⋊C4—41no—
303C4⋊C4—1222no—
304C4⋊C4—2441no—
305C4⋊C4—2441no—
306C4⋊C4—2441no—
307C4⋊C4—2442no—
308C4⋊C4—2442no—
309C22⋊C4—2442no—
310C22⋊C4—2442no—
311C42—22no—
312C42—1221no—
313C42—1221no—
314C42—1221no—
315C42—1221no—
316C42—1222no—
317C23—612no—
318C23—611no—
319C23—611no—
320C23—612no—
321C23—1221no—
322C23—1221no—
323C23—1221no—
324C23—1221no—
325C23C2×C22⋊#692311yescharacteristic
326Q8—82no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—611no—
340C2×C4—611no—
341C2×C4—612no—
342C2×C4—1224no—
343C2×C4—1224no—
344C2×C4—1221no—
345C2×C4—1221no—
346C2×C4—1221no—
347C2×C4—1221no—
348C2×C4—1221no—
349C2×C4—1221no—
350C2×C4—1221no—
351C2×C4—4882no—
352C2×C4—4882no—
353C2×C4—4882no—
354C2×C4—4882no—
355C2×C4—2442no—
356C2×C4—2442no—
357C2×C4—2444no—
358C2×C4—2444no—
359C2×C4—2444no—
360C2×C4—2444no—
361C2×C4—2441no—
362C2×C4—2441no—
363C2×C4—2441no—
364C2×C4—2441no—
365C2×C4—2441no—
366C2×C4—2441no—
367C2×C4#1451984012yes—
368C2×C4#1451984212yes—
369C2×C4D4×#692312yes—
370C8—2444no—
371C22—612no—
372C22—612no—
373C22—1221no—
374C22—1221no—
375C22—1222no—
376C22—1222no—
377C22—1222no—
378C22—1222no—
379C22—1221no—
380C22—1221no—
381C22#1684071011yescharacteristic
382C22#1684074111yescharacteristic
383C22#1684074211yescharacteristic
384C22C2×#1451984011yescharacteristic
385C22#1684075211yescharacteristic
386C22C2×#1451984211yescharacteristic
387C22#1684075911yescharacteristic
388C4—22no—
389C4—22no—
390C4—22no—
391C4—81no—
392C4—81no—
393C4—1222no—
394C4—1221no—
395C4—1221no—
396C4—4884no—
397C4—4881no—
398C4—4881no—
399C4—2441no—
400C2—1222no—
401C2—1222no—
402C2#2146119211yescharacteristic
403C2#2146120311yescharacteristic
404C2#2146125211yescharacteristic
405C2#2146126011yescharacteristic
406C2#2146180011yescharacteristic
407C2#2146183211yescharacteristic
408C2#2146184211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#391C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenter#391#6923
hypocenterC61#9112
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
16. #16839669C4semidirect
17. #16839661C4semidirect
18. #16839660C4semidirect
24. C61×#391C4semidirect
217. #391#6923semidirect
229. C61#9112semidirect

Maximal subgroups

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

subgroupindex
C2×#168407812
C2×#168406162
#214609952
#214606922
#214541602
#911261