Finite Group Databasebeta

← #30594551

Subgroups — #30594551

Order 15616.

464 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 74 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#21461522C211yesmaximal, characteristic
2#21461004C211yesmaximal, characteristic
3#324×D61C211yesmaximal, characteristic
4C2×D61⋊M16—21no—
5C22×C4⋊#6923C411yescharacteristic
6C2×(C2×#6923)⋊C4—21no—
7C2×(C2×#6923)⋊C4—21no—
8C2×(C2×#6923)⋊C4C411yescharacteristic
9#16840642—21no—
10#16840642C412yescomplemented
11C2×C4⋊C4×D61C2211yescharacteristic
12C22⋊C8×D61—21no—
13#16839488C411yescharacteristic, complemented
14#16839487—21no—
15C61×#324C411yescharacteristic, complemented
16C23×#6923—21no—
17C23×#6923—21no—
18C2×#11758213—41no—
19D61⋊M16—22no—
20D61⋊M16—22no—
21D61⋊M16—41no—
22D61⋊M16—41no—
23C2×#11758217—22no—
24C2×D61⋊C8—41no—
25C2×C4⋊#6923—41no—
26C2×C4⋊#6923—42no—
27(C2×#6923)⋊C4—41no—
28(C2×#6923)⋊C4—42no—
29(C2×#6923)⋊C4—41no—
30(C2×#6923)⋊C4—41no—
31C22×C4×D61C2×C411yescharacteristic
32C22×C4×D61D411yescharacteristic
33C22×C4×D61D411yescharacteristic
34C2×C8×D61—41no—
35C2×C4⋊Dic61Q811yescharacteristic
36C2×#11758196C2×C411yescharacteristic
37C2×#11758196D411yescharacteristic
38C4⋊C4×D61—41no—
39C4⋊C4×D61—41no—
40C22⋊#92418—21no—
41#14519689—41no—
42C2×C4⋊C4×C61C2×C411yescharacteristic
43C22⋊C8×C61—21no—
44C23×D61C22⋊C411yescharacteristic
45C22×#6923—22no—
46C22×#6923—22no—
47C22×#6923—41no—
48C22×#6923—41no—
49C22×#6923—41no—
50C22×#6923—41no—
51C22×#6923—41no—
52C22×#6923—41no—
53C22×#6923—41no—
54C22×#6923—41no—
55C22×#6923—41no—
56C22×#6923—41no—
57C22×#6923—41no—
58C22×#6923—41no—
59C22×C4×C61C4211yescharacteristic
60C22×C4×C61C22⋊C411yescharacteristic
61C22×C4×C61C4⋊C411yescharacteristic
62C22×Dic61C22⋊C411yescharacteristic
63C22×Dic61C4⋊C411yescharacteristic
64C22×Dic61C4⋊C411yescharacteristic
65C2×C4×D61—21no—
66C2×C4×D61—21no—
67C2×C4×D61—21no—
68C2×C4×D61—21no—
69C2×C4×D61—21no—
70C2×C4×D61—21no—
71C2×C4×D61—21no—
72C2×C4×D61—21no—
73C2×C4×D61—41no—
74C2×C4×D61—41no—
75C2×C4×D61—41no—
76C2×C4×D61—41no—
77C2×C4×D61D811yescharacteristic
78C2×C4×D61SD1611yescharacteristic
79C2×C4×D61SD1611yescharacteristic
80C2×C4×D61Q1611yescharacteristic
81#11758217—24no—
82#11758217—42no—
83C2×#92420—42no—
84C4⋊#6923—42no—
85C4⋊#6923—42no—
86#11758213—42no—
87#11758213—42no—
88D61⋊C8—42no—
89D61⋊C8—42no—
90C2×C8×C61—41no—
91C4⋊C4×C61—41no—
92C4⋊Dic61—41no—
93#11758196—41no—
94#11758196—41no—
95C2×#92418—41no—
96C8×D61—81no—
97C8×D61—81no—
98C23×C61#9511yescharacteristic
99C22×D61—21no—
100C22×D61—21no—
101C22×D61—21no—
102C22×D61—21no—
103C22×D61#9911yescharacteristic
104C22×D61#9911yescharacteristic
105C22×D61#10211yescharacteristic
106C22×D61#10311yescharacteristic
107C22×D61#10411yescharacteristic
108C22×D61#10411yescharacteristic
109C2×#6923—42no—
110C2×#6923—42no—
111C2×#6923—42no—
112C2×#6923—42no—
113C2×#6923—42no—
114C2×#6923—42no—
115C2×#6923—81no—
116C2×#6923—81no—
117C2×#6923—81no—
118C2×#6923—81no—
119C2×#6923—81no—
120C2×#6923—81no—
121C2×#6923—81no—
122C2×#6923—81no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61#10611yescharacteristic
130C2×C4×C61#10711yescharacteristic
131C2×Dic61—21no—
132C2×Dic61—21no—
133C2×Dic61—21no—
134C2×Dic61—21no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61#10211yescharacteristic
138C2×Dic61#10311yescharacteristic
139C4×D61—21no—
140C4×D61—21no—
141C4×D61—21no—
142C4×D61—21no—
143C4×D61—21no—
144C4×D61—21no—
145C4×D61—21no—
146C4×D61—21no—
147C4×D61—42no—
148C4×D61—41no—
149C4×D61—41no—
150C4×D61—41no—
151C4×D61—41no—
152C4×D61—81no—
153#92420—44no—
154C8×C61—81no—
155#92418—81no—
156#7251—611nomaximal, Sylow
157C22×C61—21no—
158C22×C61—21no—
159C22×C61#33911yescharacteristic
160C22×C61#34011yescharacteristic
161C22×C61#34211yescharacteristic
162C2×D61—21no—
163C2×D61—21no—
164C2×D61—21no—
165C2×D61—21no—
166C2×D61—21no—
167C2×D61—21no—
168C2×D61—21no—
169C2×D61—21no—
170C2×D61—21no—
171C2×D61—21no—
172C2×D61#32711yescharacteristic
173C2×D61#32711yescharacteristic
174C2×D61#32811yescharacteristic
175C2×D61#32811yescharacteristic
176C2×D61AΣL(1,𝔽16)11yescharacteristic
177C2×D61#35211yescharacteristic
178C2×D61#35511yescharacteristic
179C2×D61#35611yescharacteristic
180#6923—82no—
181#6923—82no—
182C4×C61—21no—
183C4×C61—21no—
184C4×C61—41no—
185C4×C61—41no—
186Dic61—21no—
187Dic61—21no—
188Dic61—41no—
189Dic61—41no—
190C2×#343—611no—
191C2×#324—611no—
192#1441—611no—
193C2×C61—21no—
194C2×C61—21no—
195C2×C61#129411yescharacteristic
196C2×C61#139011yescharacteristic
197C2×C61#139411yescharacteristic
198D61—21no—
199D61—21no—
200D61#134511yescharacteristic
201D61#134611yescharacteristic
202D61#135311yescharacteristic
203D61#135411yescharacteristic
204C22×M16—1221no—
205C22×C4⋊C4—611no—
206C22×C4⋊C4—611no—
207C2×C22⋊C8—1221no—
208C2×#95—611no—
209C2×#95—1221no—
210C2×#95—1221no—
211#343—612no—
212#343—1221no—
213#324—611no—
214#324—1221no—
215#324#692311yescharacteristic, complemented
216C61#725111yescharacteristic, complemented, Sylow
217C23×C4—611no—
218C23×C4—611no—
219C23×C4—611no—
220C23×C4—1221no—
221C23×C4—1221no—
222C2×M16—1222no—
223C2×M16—1222no—
224C2×M16—1222no—
225C2×M16—2441no—
226C2×M16—2441no—
227C2×M16—2441no—
228C22×C8—2441no—
229C22×C8—2441no—
230C2×C4⋊C4—611no—
231C2×C4⋊C4—611no—
232C2×C4⋊C4—611no—
233C2×C4⋊C4—2441no—
234C2×C4⋊C4—2441no—
235C2×C4⋊C4—2441no—
236C2×C4⋊C4—2442no—
237C2×C4⋊C4C2×#692311yescharacteristic
238C22⋊C8—21no—
239C22⋊C8—1221no—
240C22⋊C8—2441no—
241#95—2441no—
242#95—2441no—
243#95—2441no—
244#95—2442no—
245C24—611no—
246C22×C4—611no—
247C22×C4—611no—
248C22×C4—611no—
249C22×C4—611no—
250C22×C4—611no—
251C22×C4—611no—
252C22×C4—611no—
253C22×C4—1221no—
254C22×C4—1221no—
255C22×C4—1221no—
256C22×C4—1221no—
257C22×C4—1221no—
258C22×C4—1221no—
259C22×C4—1221no—
260C22×C4—1221no—
261C22×C4—1222no—
262C22×C4—1222no—
263C22×C4—2441no—
264C22×C4—2441no—
265C22×C4—2441no—
266C22×C4—2441no—
267C22×C4—2441no—
268C22×C4—2441no—
269C22×C4—2441no—
270C22×C4—2441no—
271C22×C4—2441no—
272C22×C4—2441no—
273C22×C4—2441no—
274C22×C4—2441no—
275C22×C4—2441no—
276C22×C4—2441no—
277C22×C4—2441no—
278C22×C4—2441no—
279C22×C4C4×#692311yescharacteristic
280C22×C4C4⋊#692311yescharacteristic
281C22×C4C22⋊#692311yescharacteristic
282M16—1224no—
283M16—2442no—
284M16—2442no—
285M16—2442no—
286C2×C8—41no—
287C2×C8—4881no—
288C2×C8—4881no—
289C2×C8—2441no—
290C2×C8—2442no—
291C2×C8—2442no—
292C2×C8—2442no—
293C4⋊C4—41no—
294C4⋊C4—2441no—
295C4⋊C4—2441no—
296C4⋊C4—2441no—
297C4⋊C4—2442no—
298C4⋊C4—2442no—
299C23—611no—
300C23—611no—
301C23—611no—
302C23—611no—
303C23—611no—
304C23—611no—
305C23—1221no—
306C23—1221no—
307C23—1221no—
308C23—1221no—
309C23(C2×#6923)⋊C411yescharacteristic
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—611no—
317C2×C4—611no—
318C2×C4—1221no—
319C2×C4—1221no—
320C2×C4—1221no—
321C2×C4—1221no—
322C2×C4—1221no—
323C2×C4—1221no—
324C2×C4—1221no—
325C2×C4—1221no—
326C2×C4—1221no—
327C2×C4—1221no—
328C2×C4—1221no—
329C2×C4—1221no—
330C2×C4—4881no—
331C2×C4—4881no—
332C2×C4—4881no—
333C2×C4—4881no—
334C2×C4—4881no—
335C2×C4—4881no—
336C2×C4—4881no—
337C2×C4—4881no—
338C2×C4—4881no—
339C2×C4—2441no—
340C2×C4—2441no—
341C2×C4—2441no—
342C2×C4—2441no—
343C2×C4—2442no—
344C2×C4—2442no—
345C2×C4—2442no—
346C2×C4—2442no—
347C2×C4—2442no—
348C2×C4—2442no—
349C2×C4—2442no—
350C2×C4—2441no—
351C2×C4—2441no—
352C2×C4#1451984511yescharacteristic
353C2×C4#1451984611yescharacteristic
354C8—81no—
355C8—4881no—
356C8—2444no—
357C22—21no—
358C22—21no—
359C22—611no—
360C22—611no—
361C22—611no—
362C22—611no—
363C22—611no—
364C22—611no—
365C22—611no—
366C22—611no—
367C22—1221no—
368C22—1221no—
369C22—1221no—
370C22—1221no—
371C22—1221no—
372C22—1221no—
373C22—1221no—
374C22—1221no—
375C22—1221no—
376C22—1221no—
377C22#1684063411yescharacteristic
378C22#1684063611yescharacteristic
379C22#1684063911yescharacteristic
380C4—21no—
381C4—21no—
382C4—41no—
383C4—41no—
384C4—1221no—
385C4—1221no—
386C4—4882no—
387C4—4882no—
388C4—2441no—
389C4—2441no—
390C2—21no—
391C2—21no—
392C2—611no—
393C2—611no—
394C2—611no—
395C2—611no—
396C2—1221no—
397C2—1221no—
398C2#2146069411yescharacteristic
399C2#2146090211yescharacteristic
400C2#2146091511yescharacteristic
subgroupG/H
centerC22#16840639
commutatorC22×C4×C61C42
FrattiniC2×C4⋊C4C2×#6923
FittingC61×#324C4
socleC22×C61#342
prime radicalC23×D61C22⋊C4
cosocleC2×C4⋊C4×D61C22
hypercenter#324#6923
hypocenterC61#7251
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #324

Complements

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

NH ≅ G/Ntype
10. #16840642C4semidirect
13. #16839488C4semidirect
15. C61×#324C4semidirect
215. #324#6923semidirect
216. C61#7251semidirect

Maximal subgroups

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

subgroupindex
#214615222
#214610042
#324×D612
#725161