Finite Group Databasebeta

← #30605003

Subgroups — #30605003

Order 15616.

488 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 72 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×#16840642C211yesmaximal, characteristic, complemented
2#21461069C211yesmaximal, characteristic, complemented
3#21460818C212yesmaximal, complemented
4#21460801C212yesmaximal, complemented
5#21451300C211yesmaximal, characteristic
6C22×#11758217—21no—
7#16840732—21no—
8#16840642—21no—
9#16840642C2212yes—
10#16840608—21no—
11#16840608C2212yes—
12C22×#11758196C2211yescharacteristic
13#16839700C411yescharacteristic
14#16839700C2211yescharacteristic
15#16839676C411yescharacteristic
16#16839676C2211yescharacteristic
17#16839656—21no—
18#16839656—21no—
19#16839642—21no—
20#16839642—21no—
21C4×C22⋊Dic61C411yescharacteristic
22C2×#121×C61C411yescharacteristic
23C2×#11758217—24no—
24C2×#11758217—22no—
25C2×#11758217—41no—
26C2×#11758217—42no—
27C2×#11758217—41no—
28C22×#92420—41no—
29C22⋊#92420—22no—
30C22⋊#92420—41no—
31C23×Dic61D411yescharacteristic
32C23×Dic61D411yescharacteristic
33C2×#11758196—41no—
34C2×#11758196—41no—
35C2×#11758196C2×C411yescharacteristic
36C2×#11758196D411yescharacteristic
37C2×#11758196D411yescharacteristic
38C2×#11758196C2311yescharacteristic
39C2×C22⋊Dic61—21no—
40C2×C22⋊Dic61—21no—
41C2×C22⋊Dic61—21no—
42C2×C22⋊Dic61—21no—
43C2×C22⋊Dic61—21no—
44C2×C4×Dic61—21no—
45#14519684—21no—
46#14519684—21no—
47#14519684—21no—
48#14519684—21no—
49#14519684C2×C411yescharacteristic
50#14519684C2×C411yescharacteristic
51C22×D4×C61—21no—
52C22×D4×C61C2×C411yescharacteristic
53C23×C4×C61C2×C411yescharacteristic
54#121×C61—41no—
55#121×C61—41no—
56C2×C4⋊C4×C61C2×C411yescharacteristic
57C2×C22⋊C4×C61—21no—
58C24×C61—21no—
59C24×C61C22⋊C411yescharacteristic
60C2×D4×C61—41no—
61C2×D4×C61—42no—
62C2×D4×C61—42no—
63C2×D4×C61—41no—
64C2×D4×C61—41no—
65C2×D4×C61—81no—
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61—81no—
70C22×C4×C61C22⋊C411yescharacteristic
71C22×C4×C61C22×C411yescharacteristic
72C22×Dic61—21no—
73C22×Dic61—21no—
74C22×Dic61—22no—
75C22×Dic61—21no—
76C22×Dic61—21no—
77C22×Dic61—21no—
78C22×Dic61—21no—
79C22×Dic61—21no—
80C22×Dic61—21no—
81C22×Dic61—81no—
82C22×Dic61C22⋊C411yescharacteristic
83C22×Dic61C22⋊C411yescharacteristic
84C22×Dic61C2×D411yescharacteristic
85C22×Dic61C2×D411yescharacteristic
86#11758217—24no—
87#11758217—44no—
88#11758217—42no—
89#11758217—82no—
90C2×#92420—44no—
91C2×#92420—42no—
92C2×#92420—42no—
93C4⋊C4×C61—41no—
94C22⋊C4×C61—42no—
95C22⋊Dic61—42no—
96C22⋊Dic61—42no—
97C22⋊Dic61—41no—
98C22⋊Dic61—41no—
99C22⋊Dic61—81no—
100C22⋊Dic61—81no—
101#11758196—41no—
102#11758196—41no—
103#11758196—41no—
104#11758196—41no—
105C4×Dic61—81no—
106C23×C61—22no—
107C23×C61—21no—
108C23×C61—21no—
109C23×C61—41no—
110C23×C61—41no—
111C23×C61—41no—
112C23×C61—41no—
113C23×C61—41no—
114C23×C61—41no—
115C23×C61—41no—
116C23×C61—41no—
117C23×C61#9911yescharacteristic
118C23×C61#9911yescharacteristic
119C23×C61C2×C22⋊C411yescharacteristic
120D4×C61—82no—
121D4×C61—82no—
122D4×C61—82no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—41no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—81no—
131C2×C4×C61—81no—
132C2×Dic61—22no—
133C2×Dic61—22no—
134C2×Dic61—22no—
135C2×Dic61—22no—
136C2×Dic61—21no—
137C2×Dic61—21no—
138C2×Dic61—21no—
139C2×Dic61—21no—
140C2×Dic61—21no—
141C2×Dic61—21no—
142C2×Dic61—42no—
143C2×Dic61—41no—
144C2×Dic61—41no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61—41no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61—81no—
152C2×Dic61#10412yes—
153C2×Dic61AGL(1,ℤ/8)11yescharacteristic
154C2×Dic61#13711yescharacteristic
155#92420—44no—
156#92420—82no—
157#11766—611nomaximal, Sylow
158C22×C61—21no—
159C22×C61—21no—
160C22×C61—21no—
161C22×C61—21no—
162C22×C61—21no—
163C22×C61—21no—
164C22×C61—21no—
165C22×C61—42no—
166C22×C61—42no—
167C22×C61—81no—
168C22×C61—81no—
169C22×C61—81no—
170C22×C61—81no—
171C22×C61—81no—
172C22×C61—81no—
173C22×C61—81no—
174C22×C61#35512yes—
175C22×C61C2×#9911yescharacteristic
176C22×C61#41711yescharacteristic
177C22×C61C2×#10411yescharacteristic
178C4×C61—41no—
179C4×C61—41no—
180C4×C61—81no—
181Dic61—21no—
182Dic61—22no—
183Dic61—21no—
184Dic61—41no—
185Dic61—41no—
186Dic61—81no—
187Dic61—81no—
188#2591—611no—
189C2×#343—611no—
190#1463—611no—
191#1353—612no—
192#1345—612no—
193C2×C61—21no—
194C2×C61—21no—
195C2×C61—22no—
196C2×C61—41no—
197C2×C61—161no—
198C2×C61#150411yescharacteristic
199C2×C61#212111yescharacteristic
200C2×C61C2×#35511yescharacteristic
201C22×M16—1221no—
202C2×#121#692311yescharacteristic
203C22×C4⋊C4—611no—
204#407—1221no—
205#400—611no—
206#400—611no—
207#394—611no—
208#394—611no—
209#388—1221no—
210#388—1221no—
211#386—1221no—
212#386—1221no—
213C4×C22⋊C4—611no—
214#343—612no—
215#343—1221no—
216#324—612no—
217#324—1221no—
218C61#1176611yescharacteristic, complemented, Sylow
219C22×D4—21no—
220C22×D4C2×#692311yescharacteristic
221C23×C4—611no—
222C23×C4—611no—
223C23×C4C2×#692311yescharacteristic
224C2×M16—1222no—
225C2×M16—1224no—
226C2×M16—2441no—
227C2×M16—2442no—
228C2×M16—2441no—
229C22×C8—2441no—
230#121—41no—
231#121—41no—
232C2×C4⋊C4—611no—
233C2×C4⋊C4—611no—
234C2×C4⋊C4—611no—
235C2×C4⋊C4—611no—
236C2×C4⋊C4—2441no—
237C2×C4⋊C4—2441no—
238C2×C4⋊C4C2×#692311yescharacteristic
239C2×C22⋊C4—21no—
240C2×C22⋊C4—1221no—
241C2×C22⋊C4—1221no—
242C2×C22⋊C4—1221no—
243C2×C22⋊C4—1221no—
244C2×C22⋊C4—1221no—
245C2×C42—1221no—
246C22⋊C8—1222no—
247C22⋊C8—2441no—
248#95—611no—
249#95—611no—
250#95—1221no—
251#95—1221no—
252#95—1221no—
253#95—1221no—
254C24—21no—
255C24C22⋊#692311yescharacteristic
256C2×D4—41no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—81no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—81no—
266C22×C4—611no—
267C22×C4—611no—
268C22×C4—611no—
269C22×C4—611no—
270C22×C4—1222no—
271C22×C4—1221no—
272C22×C4—1221no—
273C22×C4—1221no—
274C22×C4—1221no—
275C22×C4—1221no—
276C22×C4—1221no—
277C22×C4—1221no—
278C22×C4—1221no—
279C22×C4—4881no—
280C22×C4C22⋊#692311yescharacteristic
281C22×C4C22×#692311yescharacteristic
282M16—1224no—
283M16—4882no—
284M16—2442no—
285M16—2444no—
286C2×C8—2442no—
287C2×C8—2444no—
288C2×C8—2442no—
289C4⋊C4—41no—
290C4⋊C4—2441no—
291C4⋊C4—2441no—
292C4⋊C4—2441no—
293C4⋊C4—2441no—
294C22⋊C4—42no—
295C22⋊C4—4881no—
296C22⋊C4—4881no—
297C22⋊C4—2441no—
298C22⋊C4—2441no—
299C22⋊C4—2442no—
300C22⋊C4—2442no—
301C42—4881no—
302C23—21no—
303C23—21no—
304C23—22no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23#1451983511yescharacteristic
314C23#1451983511yescharacteristic
315C23C2×C22⋊#692311yescharacteristic
316D4—82no—
317D4—82no—
318D4—82no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—611no—
329C2×C4—611no—
330C2×C4—612no—
331C2×C4—1222no—
332C2×C4—1222no—
333C2×C4—1222no—
334C2×C4—1222no—
335C2×C4—1221no—
336C2×C4—1221no—
337C2×C4—1221no—
338C2×C4—1221no—
339C2×C4—1221no—
340C2×C4—1221no—
341C2×C4—4881no—
342C2×C4—2441no—
343C2×C4—2441no—
344C2×C4—2441no—
345C2×C4—2441no—
346C2×C4—2441no—
347C2×C4—2441no—
348C2×C4—2442no—
349C2×C4—2441no—
350C2×C4—2441no—
351C8—4882no—
352C8—2444no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—42no—
361C22—42no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22#1684067012yes—
370C22C2×#1451983511yescharacteristic
371C22#1684075311yescharacteristic
372C22D61⋊#10411yescharacteristic
373C4—41no—
374C4—41no—
375C4—81no—
376C4—1221no—
377C4—1222no—
378C4—1221no—
379C4—4881no—
380C4—4881no—
381C4—2441no—
382C4—2441no—
383C2—21no—
384C2—21no—
385C2—22no—
386C2—41no—
387C2—161no—
388C2#2146120511yescharacteristic
389C2#2146207411yescharacteristic
390C2C2×#1684067011yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
Fitting, prime radicalC2×#121×C61C4
socleC22×C61C2×#99
cosocleC2×#11758196C23
hypercenterC2×#121#6923
hypocenterC61#11766
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#16840642C2semidirect
2. #21461069C2semidirect
3. #21460818C2semidirect
4. #21460801C2semidirect
218. C61#11766semidirect

Maximal subgroups

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

subgroupindex
C2×#168406422
#214610692
#214608182
#214608012
#214513002
#1176661