Finite Group Databasebeta

← #30600399

Subgroups — #30600399

Order 15616.

568 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 110 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×#16840785C211yesmaximal, characteristic
2#21461676C211yesmaximal, characteristic
3C2×#16840621C212yesmaximal, complemented
4#21460697C212yesmaximal, complemented
5#21454170C211yesmaximal, characteristic
6#16840785—24no—
7C2×#14519844C2212yes—
8C2×#14519844C2212yes—
9C2×#14519834—21no—
10C2×#14519834—21no—
11C2×#14519832—21no—
12#16840628—21no—
13#16840621—24no—
14C2×#14519776C2211yescharacteristic
15C2×#14519755—21no—
16C2×C42×D61C2211yescharacteristic
17#16839666—21no—
18#16839660C411yescharacteristic
19#16839657—21no—
20#16839631C2211yescharacteristic
21#16839621—21no—
22#16839619C411yescharacteristic
23#16839619C411yescharacteristic
24C61×#389C411yescharacteristic
25C22×#92420—22no—
26C22×#92420—22no—
27#14519844—22no—
28#14519844—22no—
29#14519844—22no—
30#14519844D414yes—
31#14519834—24no—
32#14519834—42no—
33#14519832—24no—
34C22×Dic122—21no—
35C22×C4×D61C2×C411yescharacteristic
36C22×C4×D61D411yescharacteristic
37C22×C4×D61D411yescharacteristic
38#14519776—22no—
39#14519776—22no—
40#14519755—44no—
41C2×C4⋊Dic61C2×C411yescharacteristic
42C2×C4⋊Dic61C2×C411yescharacteristic
43C2×#11758196—21no—
44C2×#11758196—21no—
45C2×#11758196—21no—
46C2×#11758196—21no—
47C2×#11758198—21no—
48C2×#11758198—21no—
49C2×C4×Dic61D411yescharacteristic
50C2×C4×Dic61D411yescharacteristic
51C2×C4×Dic61C2311yescharacteristic
52C42×D61—22no—
53C42×D61—22no—
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57C2×C4⋊C4×C61—21no—
58C2×C4⋊C4×C61C2×C411yescharacteristic
59C2×C4⋊C4×C61C2×C411yescharacteristic
60C2×C42×C61C2×C411yescharacteristic
61#95×C61—21no—
62C23×D61C22⋊C411yescharacteristic
63C22×C4×C61—21no—
64C22×C4×C61—21no—
65C22×C4×C61C22⋊C411yescharacteristic
66C22×C4×C61C22⋊C411yescharacteristic
67C22×C4×C61C22×C411yescharacteristic
68C22×Dic61—21no—
69C22×Dic61—21no—
70C22×Dic61C22⋊C411yescharacteristic
71C22×Dic61C2×D411yescharacteristic
72C22×Dic61C2×D411yescharacteristic
73C2×C4×D61—22no—
74C2×C4×D61—22no—
75C2×C4×D61—21no—
76C2×C4×D61—22no—
77C2×C4×D61—21no—
78C2×C4×D61—22no—
79C2×C4×D61—21no—
80C2×C4×D61—22no—
81C2×C4×D61—22no—
82C2×C4×D61—21no—
83C2×C4×D61—41no—
84C2×Dic122—41no—
85C2×Dic122—41no—
86C2×Dic122—42no—
87C2×Dic122—42no—
88C2×#92420—24no—
89C2×#92420—42no—
90C2×#92420—42no—
91C2×#92420—42no—
92C2×#92420—42no—
93C2×#92420—42no—
94C4⋊C4×C61—22no—
95C4⋊C4×C61—42no—
96C4⋊C4×C61—41no—
97C42×C61—22no—
98#11758198—42no—
99#11758198—42no—
100C4⋊Dic61—22no—
101C4⋊Dic61—41no—
102#11758196—42no—
103#11758196—42no—
104#11758196—41no—
105#11758196—42no—
106#11758196—41no—
107C4×Dic61—21no—
108C4×Dic61—22no—
109C4×Dic61—21no—
110C4×Dic61C2×D412yes—
111C4×Dic61#4112yes—
112C23×C61C2×C22⋊C411yescharacteristic
113C22×D61—21no—
114C22×D61—21no—
115C22×D61—21no—
116C22×D61—21no—
117C22×D61—21no—
118C22×D61—21no—
119C22×D61#10112yes—
120C2×C4×C61—21no—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—21no—
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×C61C4×D412yes—
135C2×Dic61—21no—
136C2×Dic61—21no—
137C2×Dic61—21no—
138C2×Dic61—41no—
139C2×Dic61—41no—
140C2×Dic61—41no—
141C2×Dic61—41no—
142C2×Dic61—41no—
143C2×Dic61—41no—
144C2×Dic61#12011yescharacteristic
145C2×Dic61#12112yes—
146C2×Dic61#12311yescharacteristic
147C2×Dic61#13512yes—
148C2×Dic61AGL(1,ℤ/8)12yes—
149C2×Dic61AGL(1,ℤ/8)11yescharacteristic
150C2×Dic61#13712yes—
151C2×Dic61#13711yescharacteristic
152C4×D61—44no—
153C4×D61—42no—
154C4×D61—42no—
155C4×D61—42no—
156C4×D61—42no—
157Dic122—82no—
158Dic122—82no—
159#92420—44no—
160#92420—82no—
161#9140—611nomaximal, Sylow
162C22×C61#38611yescharacteristic
163C22×C61C2×#10111yescharacteristic
164C22×C61#41311yescharacteristic
165C22×C61#41611yescharacteristic
166C22×C61#41711yescharacteristic
167C22×C61#41811yescharacteristic
168C22×C61#41911yescharacteristic
169C2×D61—22no—
170C2×D61—22no—
171C2×D61—22no—
172C2×D61—41no—
173C2×D61—41no—
174C2×D61—41no—
175C2×D61—41no—
176C4×C61—22no—
177C4×C61—41no—
178C4×C61—41no—
179C4×C61—81no—
180C4×C61—81no—
181Dic61—41no—
182Dic61—81no—
183Dic61—81no—
184Dic61#44912yes—
185Dic61#46812yes—
186Dic61#47012yes—
187Dic61#48212yes—
188C2×#432—611no—
189#2857—611no—
190#1797—611no—
191C2×#332—612no—
192#1296—612no—
193C2×C61#155211yescharacteristic
194C2×C61#155411yescharacteristic
195C2×C61#155611yescharacteristic
196C2×C61#155711yescharacteristic
197C2×C61#187511yescharacteristic
198C2×C61#189411yescharacteristic
199C2×C61#190111yescharacteristic
200D61—42no—
201C2×C4⋊Q8—611no—
202C2×C22⋊Q8—1221no—
203C22×C42—611no—
204#432—1224no—
205C2×C4⋊C8—612no—
206C2×C4⋊C8—612no—
207C2×C22⋊C8—1221no—
208C2×C22⋊C8—1221no—
209C2×#97—1221no—
210#391—1221no—
211#389—611no—
212#389—1221no—
213#389#692311yescharacteristic
214#384—611no—
215#380—611no—
216#380—611no—
217#380—1221no—
218#336—1221no—
219#332—1224no—
220C61#914011yescharacteristic, complemented, Sylow
221C22×Q8—1221no—
222C23×C4—611no—
223C23×C4—611no—
224C23×C4—611no—
225C22×C8—1222no—
226C22×C8—1222no—
227C4⋊Q8—1222no—
228C4⋊Q8—1222no—
229C22⋊Q8—2444no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—611no—
232C2×C4⋊C4—611no—
233C2×C4⋊C4—1221no—
234C2×C4⋊C4—1221no—
235C2×C4⋊C4—1221no—
236C2×C4⋊C4—1221no—
237C2×C4⋊C4C2×#692311yescharacteristic
238C2×C4⋊C4C2×#692311yescharacteristic
239C2×C22⋊C4—1221no—
240C2×C22⋊C4—1221no—
241C2×C42—611no—
242C2×C42—611no—
243C2×C42—611no—
244C2×C42—1222no—
245C2×C42—1222no—
246C2×C42C2×#692311yescharacteristic
247C4⋊C8—614no—
248C4⋊C8—1222no—
249C4⋊C8—1222no—
250C4⋊C8—1222no—
251C22⋊C8—1224no—
252C22⋊C8—2442no—
253#97—1224no—
254#95—21no—
255#95—1221no—
256#95—1221no—
257#95—1221no—
258C24—611no—
259C2×Q8—2442no—
260C2×Q8—2441no—
261C2×Q8—2441no—
262C2×Q8—2442no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—611no—
266C22×C4—611no—
267C22×C4—611no—
268C22×C4—1221no—
269C22×C4—1221no—
270C22×C4—1222no—
271C22×C4—1221no—
272C22×C4—1222no—
273C22×C4—1222no—
274C22×C4—1221no—
275C22×C4—1222no—
276C22×C4—1222no—
277C22×C4—1222no—
278C22×C4—1221no—
279C22×C4—1221no—
280C22×C4—2441no—
281C22×C4C22⋊#692311yescharacteristic
282C22×C4C22⋊#692311yescharacteristic
283C22×C4C22×#692311yescharacteristic
284C2×C8—1224no—
285C2×C8—2442no—
286C2×C8—2442no—
287C2×C8—2442no—
288C2×C8—2442no—
289C2×C8—2442no—
290C4⋊C4—22no—
291C4⋊C4—41no—
292C4⋊C4—42no—
293C4⋊C4—1222no—
294C4⋊C4—2441no—
295C4⋊C4—2442no—
296C4⋊C4—2441no—
297C4⋊C4—2441no—
298C4⋊C4—2442no—
299C4⋊C4—2442no—
300C22⋊C4—2442no—
301C22⋊C4—2442no—
302C42—22no—
303C42—612no—
304C42—612no—
305C42—1221no—
306C42—1222no—
307C42—1221no—
308C23—612no—
309C23—1221no—
310C23—1221no—
311C23—1221no—
312C23—1221no—
313C23—1221no—
314C23—1221no—
315C23C2×C22⋊#692311yescharacteristic
316Q8—4882no—
317Q8—4882no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—611no—
333C2×C4—612no—
334C2×C4—611no—
335C2×C4—612no—
336C2×C4—611no—
337C2×C4—611no—
338C2×C4—612no—
339C2×C4—612no—
340C2×C4—1221no—
341C2×C4—1221no—
342C2×C4—1221no—
343C2×C4—2444no—
344C2×C4—2442no—
345C2×C4—2442no—
346C2×C4—2442no—
347C2×C4—2442no—
348C2×C4—2441no—
349C2×C4—2441no—
350C2×C4—2441no—
351C2×C4—2441no—
352C2×C4—2441no—
353C2×C4—2441no—
354C2×C4D4×#692312yes—
355C8—4882no—
356C8—2444no—
357C22—1222no—
358C22—1222no—
359C22—1222no—
360C22—2441no—
361C22—2441no—
362C22—2441no—
363C22—2441no—
364C22#1684071011yescharacteristic
365C22D61⋊#10111yescharacteristic
366C22#1684074811yescharacteristic
367C22#1684075211yescharacteristic
368C22#1684075311yescharacteristic
369C22#1684075411yescharacteristic
370C22#1684075511yescharacteristic
371C4—22no—
372C4—41no—
373C4—41no—
374C4—81no—
375C4—81no—
376C4—612no—
377C4—612no—
378C4—612no—
379C4—612no—
380C4—4881no—
381C4—4881no—
382C4—2441no—
383C2—2442no—
384C2#2146134111yescharacteristic
385C2#2146134411yescharacteristic
386C2#2146134911yescharacteristic
387C2#2146135011yescharacteristic
388C2#2146180711yescharacteristic
389C2#2146183411yescharacteristic
390C2#2146184711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#389C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenter#389#6923
hypocenterC61#9140
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
3. C2×#16840621C2semidirect
4. #21460697C2semidirect
220. C61#9140semidirect

Maximal subgroups

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

subgroupindex
C2×#168407852
#214616762
C2×#168406212
#214606972
#214541702
#914061