Finite Group Databasebeta

← #30595806

Subgroups — #30595806

Order 15616.

556 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 90 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#21461711C211yesmaximal, characteristic, complemented
2#21461483C211yesmaximal, characteristic, complemented
3#21460637C212yesmaximal, complemented
4#21460635C212yesmaximal, complemented
5#21454149C211yesmaximal, characteristic
6C22×#11758213—21no—
7C2×#14519834—21no—
8#16840628—21no—
9#16840628—21no—
10#16840628C2212yes—
11#16840628C2212yes—
12C2×#14519759—21no—
13C2×#14519748C411yescharacteristic
14C2×C42×D61C2211yescharacteristic
15#16839665C2211yescharacteristic
16#16839661C2211yescharacteristic
17#16839660C411yescharacteristic
18#16839656—21no—
19#16839655—21no—
20#16839638—21no—
21#16839619C411yescharacteristic
22C61×#390C411yescharacteristic
23C2×#11758213—24no—
24C2×#11758213—42no—
25C2×#11758213—42no—
26C22×#92420—22no—
27C22×#92420—22no—
28#14519834—42no—
29C22×C22⋊D61—21no—
30C22×C4×D61C2×C411yescharacteristic
31C22×C4×D61D411yescharacteristic
32C22×C4×D61D411yescharacteristic
33#14519759—44no—
34#14519748—42no—
35C2×C4⋊Dic61—21no—
36C2×C4⋊Dic61C2×C411yescharacteristic
37C2×C4⋊Dic61C2×C411yescharacteristic
38C2×#11758196—21no—
39C2×C22⋊Dic61—21no—
40C2×C22⋊Dic61—21no—
41C2×C22⋊Dic61—21no—
42C2×#11758198—21no—
43C2×C4×Dic61D411yescharacteristic
44C2×C4×Dic61D411yescharacteristic
45C2×C4×Dic61C2311yescharacteristic
46C42×D61—42no—
47#14519684—21no—
48#14519684—21no—
49#14519684—21no—
50C22×D4×C61C2×C411yescharacteristic
51C2×C4⋊C4×C61C2×C411yescharacteristic
52C2×C22⋊C4×C61—21no—
53C2×C22⋊C4×C61—21no—
54C2×C42×C61C2×C411yescharacteristic
55C24×C61—21no—
56C23×D61C22⋊C411yescharacteristic
57C2×D4×C61—22no—
58C2×D4×C61—41no—
59C2×D4×C61—41no—
60C22×C4×C61—21no—
61C22×C4×C61C22⋊C411yescharacteristic
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22×C411yescharacteristic
64C2×C22⋊D61—42no—
65C2×C22⋊D61—42no—
66C2×C22⋊D61—42no—
67C22×Dic61—21no—
68C22×Dic61—21no—
69C22×Dic61C22⋊C411yescharacteristic
70C22×Dic61C2×D411yescharacteristic
71C22×Dic61C2×D411yescharacteristic
72C2×C4×D61—22no—
73C2×C4×D61—22no—
74C2×C4×D61—21no—
75C2×C4×D61—21no—
76C2×C4×D61—22no—
77C2×C4×D61—22no—
78C2×C4×D61—22no—
79C2×C4×D61—22no—
80C2×C4×D61—41no—
81C2×C4×D61—41no—
82C2×#92420—24no—
83C2×#92420—42no—
84C2×#92420—42no—
85C2×#92420—42no—
86C2×#92420—42no—
87C2×#92420—42no—
88#11758213—48no—
89C4⋊C4×C61—22no—
90C22⋊C4×C61—42no—
91C22⋊C4×C61—42no—
92C42×C61—41no—
93C22⋊Dic61—42no—
94C22⋊Dic61—41no—
95C22⋊Dic61—42no—
96C22⋊Dic61—41no—
97#11758198—42no—
98C4⋊Dic61—42no—
99C4⋊Dic61—41no—
100C4⋊Dic61—41no—
101#11758196—42no—
102C4×Dic61—41no—
103C4×Dic61—41no—
104C4×Dic61—41no—
105C23×C61—41no—
106C23×C61—41no—
107C23×C61—41no—
108C23×C61—41no—
109C23×C61—41no—
110C23×C61—41no—
111C23×C61—41no—
112C23×C61C2×C22⋊C411yescharacteristic
113C22×D61—21no—
114C22×D61—21no—
115C22×D61—21no—
116C22×D61—21no—
117C22×D61—21no—
118C22×D61—21no—
119C22×D61#9912yes—
120D4×C61—82no—
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#10412yes—
132C2×C4×C61#13112yes—
133C22⋊D61—84no—
134C2×Dic61—21no—
135C2×Dic61—21no—
136C2×Dic61—21no—
137C2×Dic61—21no—
138C2×Dic61—21no—
139C2×Dic61—21no—
140C2×Dic61—21no—
141C2×Dic61—41no—
142C2×Dic61—41no—
143C2×Dic61—41no—
144C2×Dic61—41no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61AGL(1,ℤ/8)11yescharacteristic
148C2×Dic61AGL(1,ℤ/8)12yes—
149C2×Dic61#13711yescharacteristic
150C4×D61—24no—
151C4×D61—24no—
152C4×D61—44no—
153C4×D61—44no—
154#92420—44no—
155#92420—82no—
156#7694—611nomaximal, Sylow
157C22×C61—81no—
158C22×C61—81no—
159C22×C61—81no—
160C22×C61—81no—
161C22×C61—81no—
162C22×C61—81no—
163C22×C61—81no—
164C22×C61#40811yescharacteristic
165C22×C61C2×#9911yescharacteristic
166C22×C61#41311yescharacteristic
167C22×C61#41711yescharacteristic
168C22×C61#41811yescharacteristic
169C22×C61C2×#10411yescharacteristic
170C22×C61#42111yescharacteristic
171C2×D61—22no—
172C2×D61—22no—
173C2×D61—22no—
174C2×D61—41no—
175C2×D61—41no—
176C2×D61—41no—
177C2×D61—41no—
178C4×C61—22no—
179C4×C61—22no—
180C4×C61—41no—
181C4×C61—81no—
182Dic61—21no—
183Dic61—21no—
184Dic61—22no—
185Dic61—41no—
186Dic61—81no—
187Dic61—81no—
188#2854—611no—
189#1817—611no—
190#1728—611no—
191#1271—612no—
192#1270—612no—
193C2×C61—161no—
194C2×C61#146811yescharacteristic
195C2×C61#148411yescharacteristic
196C2×C61#149511yescharacteristic
197C2×C61#150411yescharacteristic
198C2×C61#151011yescharacteristic
199C2×C61#152911yescharacteristic
200C2×C61#155711yescharacteristic
201D61—42no—
202C22×M16—1221no—
203C2×#123—1221no—
204C2×#121—611no—
205C22×C42—611no—
206C2×C22⋊C8—1221no—
207#390—611no—
208#390#692311yescharacteristic
209#389—611no—
210#389—611no—
211#388—1221no—
212#388—1221no—
213#386—1221no—
214#380—611no—
215#336—612no—
216#336—612no—
217#336—1221no—
218#336—1221no—
219C61#769411yescharacteristic, complemented, Sylow
220C22×D4—1221no—
221C22×D4C2×#692311yescharacteristic
222C23×C4—611no—
223C23×C4—611no—
224C23×C4—611no—
225C2×M16—1224no—
226C2×M16—2442no—
227C2×M16—2442no—
228C22×C8—1222no—
229C22×C8—1222no—
230#123—2444no—
231#121—2442no—
232C2×C4⋊C4—611no—
233C2×C4⋊C4—611no—
234C2×C4⋊C4—1221no—
235C2×C4⋊C4—1221no—
236C2×C4⋊C4C2×#692311yescharacteristic
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—1221no—
240C2×C22⋊C4—1221no—
241C2×C22⋊C4—1221no—
242C2×C22⋊C4—1221no—
243C2×C42—611no—
244C2×C42—611no—
245C2×C42—611no—
246C2×C42—2442no—
247C2×C42C2×#692311yescharacteristic
248C22⋊C8—2442no—
249#95—1221no—
250#95—1221no—
251#95—1221no—
252C24—21no—
253C24—611no—
254C2×D4—22no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—2442no—
258C2×D4—2442no—
259C2×D4—2442no—
260C22×C4—21no—
261C22×C4—611no—
262C22×C4—611no—
263C22×C4—611no—
264C22×C4—1221no—
265C22×C4—1221no—
266C22×C4—1222no—
267C22×C4—1222no—
268C22×C4—1222no—
269C22×C4—1222no—
270C22×C4—1222no—
271C22×C4—1222no—
272C22×C4—1221no—
273C22×C4—1221no—
274C22×C4—2441no—
275C22×C4—2441no—
276C22×C4C22⋊#692311yescharacteristic
277C22×C4C22⋊#692311yescharacteristic
278C22×C4C22×#692311yescharacteristic
279M16—2448no—
280C2×C8—1224no—
281C2×C8—2442no—
282C2×C8—2442no—
283C2×C8—2442no—
284C2×C8—2442no—
285C2×C8—2442no—
286C4⋊C4—22no—
287C4⋊C4—2441no—
288C4⋊C4—2441no—
289C4⋊C4—2442no—
290C4⋊C4—2442no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—2442no—
294C22⋊C4—2442no—
295C22⋊C4—2441no—
296C22⋊C4—2441no—
297C22⋊C4—2442no—
298C42—41no—
299C42—2441no—
300C42—2441no—
301C42—2441no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—612no—
310C23—1221no—
311C23—1221no—
312C23—1221no—
313C23—1221no—
314C23—1221no—
315C23—1221no—
316C23C2×C22⋊#692311yescharacteristic
317D4—82no—
318D4—4884no—
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—611no—
330C2×C4—611no—
331C2×C4—612no—
332C2×C4—1224no—
333C2×C4—1224no—
334C2×C4—1221no—
335C2×C4—1221no—
336C2×C4—1221no—
337C2×C4—1221no—
338C2×C4—1221no—
339C2×C4—1221no—
340C2×C4—1221no—
341C2×C4—2444no—
342C2×C4—2444no—
343C2×C4—2441no—
344C2×C4—2441no—
345C2×C4—2441no—
346C2×C4—2441no—
347C2×C4—2441no—
348C2×C4—2441no—
349C2×C4#1451984112yes—
350C2×C4#1451986412yes—
351C8—4882no—
352C8—2444no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—1222no—
361C22—1222no—
362C22—1222no—
363C22—2441no—
364C22—2441no—
365C22—2441no—
366C22—2441no—
367C22#1684073611yescharacteristic
368C22D61⋊#9911yescharacteristic
369C22#1684074811yescharacteristic
370C22#1684075311yescharacteristic
371C22#1684075411yescharacteristic
372C22C2×#1451984111yescharacteristic
373C22#1684075911yescharacteristic
374C4—22no—
375C4—22no—
376C4—41no—
377C4—81no—
378C4—1221no—
379C4—1221no—
380C4—1222no—
381C4—4881no—
382C4—4881no—
383C4—2441no—
384C2—161no—
385C2—2442no—
386C2#2146108511yescharacteristic
387C2#2146113911yescharacteristic
388C2#2146117511yescharacteristic
389C2#2146120411yescharacteristic
390C2#2146122511yescharacteristic
391C2#2146128411yescharacteristic
392C2#2146135011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC23×C61C2×C22⋊C4
prime radicalC2×#14519748C4
cosocleC2×C4×Dic61C23
hypercenter#390#6923
hypocenterC61#7694
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #21461711C2semidirect
2. #21461483C2semidirect
3. #21460637C2semidirect
4. #21460635C2semidirect
219. C61#7694semidirect

Maximal subgroups

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

subgroupindex
#214617112
#214614832
#214606372
#214606352
#214541492
#769461