Finite Group Databasebeta

← #30600376

Subgroups — #30600376

Order 15616.

588 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 122 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×#16840779C211yesmaximal, characteristic
2#21461675C211yesmaximal, characteristic
3C2×#16840622C212yesmaximal, complemented
4#21460697C212yesmaximal, complemented
5#21454167C211yesmaximal, characteristic
6#16840779—24no—
7C2×#14519844C2212yes—
8C2×#14519844C2212yes—
9C2×C22⋊#92420—21no—
10C2×C22⋊#92420—21no—
11C2×C4×#92420—21no—
12#16840628—21no—
13#16840622—24no—
14C2×#14519775C2211yescharacteristic
15C2×#14519756C411yescharacteristic
16C22×C4×Dic61C2211yescharacteristic
17#16839669C411yescharacteristic
18#16839667C411yescharacteristic
19#16839657—21no—
20#16839657—21no—
21#16839631C2211yescharacteristic
22#16839620—21no—
23#16839620—21no—
24C2×C22⋊Q8×C61C411yescharacteristic
25C22×#92420—22no—
26C22×#92420—22no—
27#14519844—22no—
28#14519844—22no—
29#14519844—22no—
30#14519844D414yes—
31C22⋊#92420—24no—
32C22⋊#92420—42no—
33C4×#92420—24no—
34C22×Dic122C2×C411yescharacteristic
35C23×Dic61D411yescharacteristic
36C23×Dic61D411yescharacteristic
37#14519775—22no—
38#14519775—22no—
39#14519756—42no—
40C2×C4⋊Dic61C2×C411yescharacteristic
41C2×#11758196—21no—
42C2×#11758196—21no—
43C2×#11758196—21no—
44C2×#11758196—21no—
45C2×#11758196—21no—
46C2×C22⋊Dic61—21no—
47C2×C4×Dic61—22no—
48C2×C4×Dic61—22no—
49C2×C4×Dic61C2×C411yescharacteristic
50C2×C4×Dic61D411yescharacteristic
51C2×C4×Dic61D411yescharacteristic
52C2×C4×Dic61C2311yescharacteristic
53#14519684—21no—
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57C22×Q8×C61C2×C411yescharacteristic
58C23×C4×C61C2×C411yescharacteristic
59C22⋊Q8×C61—42no—
60C2×C4⋊C4×C61—21no—
61C2×C4⋊C4×C61C2×C411yescharacteristic
62C2×C22⋊C4×C61—21no—
63C24×C61C22⋊C411yescharacteristic
64C2×Q8×C61—22no—
65C2×Q8×C61—41no—
66C2×Q8×C61—41no—
67C22×C4×C61—22no—
68C22×C4×C61—22no—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61—41no—
72C22×C4×C61C22⋊C411yescharacteristic
73C22×C4×C61C22×C411yescharacteristic
74C22×Dic61—22no—
75C22×Dic61—22no—
76C22×Dic61—21no—
77C22×Dic61—21no—
78C22×Dic61—22no—
79C22×Dic61—22no—
80C22×Dic61—21no—
81C22×Dic61—21no—
82C22×Dic61C22⋊C411yescharacteristic
83C22×Dic61C22⋊C411yescharacteristic
84C22×Dic61SD1612yes—
85C22×Dic61Q1612yes—
86C22×Dic61C2×D411yescharacteristic
87C22×Dic61C2×D411yescharacteristic
88C2×Dic122—22no—
89C2×Dic122—41no—
90C2×Dic122—41no—
91C2×#92420—24no—
92C2×#92420—42no—
93C2×#92420—42no—
94C2×#92420—42no—
95C2×#92420—42no—
96C2×#92420—42no—
97C4⋊C4×C61—41no—
98C4⋊C4×C61—42no—
99C22⋊C4×C61—42no—
100C22⋊Dic61—42no—
101C4⋊Dic61—41no—
102#11758196—42no—
103#11758196—41no—
104#11758196—42no—
105#11758196—41no—
106#11758196—42no—
107#11758196—42no—
108C4×Dic61—22no—
109C4×Dic61—21no—
110C4×Dic61—22no—
111C4×Dic61—21no—
112C4×Dic61C2×D412yes—
113C4×Dic61#4112yes—
114C23×C61—21no—
115C23×C61—21no—
116C23×C61—21no—
117C23×C61—21no—
118C23×C61—21no—
119C23×C61—21no—
120C23×C61#10312yes—
121C23×C61C2×C22⋊C411yescharacteristic
122Q8×C61—82no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—44no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61#10012yes—
135C2×C4×C61C4×D412yes—
136C2×Dic61—21no—
137C2×Dic61—21no—
138C2×Dic61—24no—
139C2×Dic61—24no—
140C2×Dic61—21no—
141C2×Dic61—21no—
142C2×Dic61—21no—
143C2×Dic61—42no—
144C2×Dic61—42no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61—41no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61#10312yes—
152C2×Dic61#12011yescharacteristic
153C2×Dic61#12112yes—
154C2×Dic61#12311yescharacteristic
155C2×Dic61C2×SD1611yescharacteristic
156C2×Dic61C2×Q1611yescharacteristic
157C2×Dic61#13512yes—
158C2×Dic61#13512yes—
159C2×Dic61#13712yes—
160Dic122—82no—
161#92420—44no—
162#92420—82no—
163#9128—611nomaximal, Sylow
164C22×C61—22no—
165C22×C61—22no—
166C22×C61—22no—
167C22×C61—41no—
168C22×C61—41no—
169C22×C61—41no—
170C22×C61—41no—
171C22×C61#38611yescharacteristic
172C22×C61C2×#10011yescharacteristic
173C22×C61#41311yescharacteristic
174C22×C61C2×#10311yescharacteristic
175C22×C61#41611yescharacteristic
176C22×C61#41611yescharacteristic
177C22×C61#41911yescharacteristic
178C4×C61—22no—
179C4×C61—22no—
180C4×C61—81no—
181C4×C61—81no—
182Dic61—22no—
183Dic61—41no—
184Dic61—81no—
185Dic61—81no—
186Dic61#45212yes—
187Dic61#46512yes—
188Dic61#46712yes—
189Dic61#48512yes—
190C2×#431—611no—
191#2857—611no—
192#1797—611no—
193C2×#333—612no—
194#1296—612no—
195C2×C61—42no—
196C2×C61#154211yescharacteristic
197C2×C61#154311yescharacteristic
198C2×C61#154811yescharacteristic
199C2×C61#155411yescharacteristic
200C2×C61#187311yescharacteristic
201C2×C61#189211yescharacteristic
202C2×C61#190011yescharacteristic
203C2×C4⋊Q8—611no—
204C2×C22⋊Q8—611no—
205C2×C22⋊Q8#692311yescharacteristic
206C22×C42—611no—
207#431—1224no—
208C2×C4⋊C8—612no—
209C2×C4⋊C8—612no—
210C2×C22⋊C8—1221no—
211C2×C22⋊C8—1221no—
212C2×C4×C8—1221no—
213#391—611no—
214#391—611no—
215#389—1221no—
216#389—1221no—
217#384—611no—
218#380—1221no—
219#380—1221no—
220#336—1221no—
221#333—1224no—
222C61#912811yescharacteristic, complemented, Sylow
223C22×Q8—611no—
224C22×Q8C2×#692311yescharacteristic
225C23×C4—611no—
226C23×C4—611no—
227C23×C4C2×#692311yescharacteristic
228C22×C8—1222no—
229C22×C8—1222no—
230C4⋊Q8—1222no—
231C4⋊Q8—1222no—
232C22⋊Q8—42no—
233C22⋊Q8—2442no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—611no—
236C2×C4⋊C4—1221no—
237C2×C4⋊C4—1221no—
238C2×C4⋊C4—1221no—
239C2×C4⋊C4—1221no—
240C2×C4⋊C4—1221no—
241C2×C4⋊C4C2×#692311yescharacteristic
242C2×C22⋊C4—21no—
243C2×C22⋊C4—1221no—
244C2×C42—611no—
245C2×C42—611no—
246C2×C42—611no—
247C2×C42—611no—
248C2×C42—1222no—
249C2×C42—1222no—
250C4⋊C8—614no—
251C4⋊C8—1222no—
252C4⋊C8—1222no—
253C4⋊C8—1222no—
254C22⋊C8—1224no—
255C22⋊C8—2442no—
256C4×C8—1224no—
257#95—1221no—
258#95—1221no—
259#95—1221no—
260#95—1221no—
261C24C22⋊#692311yescharacteristic
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—1222no—
266C2×Q8—2441no—
267C2×Q8—2441no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—41no—
273C22×C4—611no—
274C22×C4—611no—
275C22×C4—612no—
276C22×C4—612no—
277C22×C4—611no—
278C22×C4—611no—
279C22×C4—1222no—
280C22×C4—1222no—
281C22×C4—1221no—
282C22×C4—1221no—
283C22×C4—1222no—
284C22×C4—1222no—
285C22×C4—1221no—
286C22×C4—1221no—
287C22×C4C22⋊#692311yescharacteristic
288C22×C4C22×#692311yescharacteristic
289C2×C8—1224no—
290C2×C8—2442no—
291C2×C8—2442no—
292C2×C8—2442no—
293C2×C8—2442no—
294C2×C8—2442no—
295C4⋊C4—41no—
296C4⋊C4—42no—
297C4⋊C4—2441no—
298C4⋊C4—2442no—
299C4⋊C4—2442no—
300C4⋊C4—2441no—
301C4⋊C4—2442no—
302C4⋊C4—2441no—
303C4⋊C4—2442no—
304C22⋊C4—42no—
305C22⋊C4—2442no—
306C42—612no—
307C42—612no—
308C42—1221no—
309C42—1222no—
310C42—1221no—
311C42—1222no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23#1451984012yes—
319C23C2×C22⋊#692311yescharacteristic
320Q8—82no—
321Q8—4882no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—44no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—611no—
334C2×C4—612no—
335C2×C4—611no—
336C2×C4—611no—
337C2×C4—611no—
338C2×C4—612no—
339C2×C4—612no—
340C2×C4—612no—
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—2442no—
350C2×C4—2442no—
351C2×C4—2441no—
352C2×C4—2441no—
353C2×C4—2441no—
354C2×C4—2441no—
355C2×C4—2441no—
356C2×C4—2441no—
357C2×C4#1451983712yes—
358C2×C4D4×#692312yes—
359C8—4882no—
360C8—2444no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22#1684071011yescharacteristic
369C22C2×#1451983711yescharacteristic
370C22#1684074911yescharacteristic
371C22C2×#1451984011yescharacteristic
372C22#1684075211yescharacteristic
373C22#1684075211yescharacteristic
374C22#1684075511yescharacteristic
375C4—22no—
376C4—22no—
377C4—81no—
378C4—81no—
379C4—612no—
380C4—612no—
381C4—612no—
382C4—612no—
383C4—1222no—
384C4—4881no—
385C4—4881no—
386C4—2441no—
387C2—42no—
388C2#2146132111yescharacteristic
389C2#2146132211yescharacteristic
390C2#2146133211yescharacteristic
391C2#2146134511yescharacteristic
392C2#2146180511yescharacteristic
393C2#2146183211yescharacteristic
394C2#2146184411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C22⋊Q8×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenterC2×C22⋊Q8#6923
hypocenterC61#9128
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×#16840622C2semidirect
4. #21460697C2semidirect
222. C61#9128semidirect

Maximal subgroups

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

subgroupindex
C2×#168407792
#214616752
C2×#168406222
#214606972
#214541672
#912861