Finite Group Databasebeta

← #30605199

Subgroups — #30605199

Order 15616.

446 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 66 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#21461681C211yesmaximal, characteristic
2#21461077C211yesmaximal, characteristic
3#21460814C212yesmaximal, complemented
4#21460800C212yesmaximal, complemented
5#21449101C211yesmaximal, characteristic
6C2×#14519835—21no—
7#16840732—21no—
8#16840704—21no—
9#16840640C2212yes—
10#16840608—21no—
11#16840606C2212yes—
12C2×Q8×Dic61C411yescharacteristic, complemented
13C2×#14519737C2211yescharacteristic
14#16839704—21no—
15#16839689C2211yescharacteristic
16#16839687C411yescharacteristic, complemented
17#16839678—21no—
18#16839673C2211yescharacteristic
19C22⋊C4×Dic61—21no—
20#16839609—21no—
21#16839607C411yescharacteristic, complemented
22C61×#393C411yescharacteristic, complemented
23C2×#11758217—41no—
24C2×C4⋊#6923—41no—
25C2×C22⋊#6923—22no—
26#14519835—42no—
27C22⋊#92420—22no—
28C22⋊#92420—41no—
29(C2×#6923)⋊C4—41no—
30C22×C22⋊D61D411yescharacteristic
31C23×Dic61D411yescharacteristic
32C22×C4×D61D411yescharacteristic
33Q8×Dic61—41no—
34Q8×Dic61—41no—
35#14519737—41no—
36#14519737—41no—
37C2×C4⋊Dic61—21no—
38C2×C4⋊Dic61C2×C411yescharacteristic
39C2×#11758196D411yescharacteristic
40C2×C22⋊Dic61—21no—
41C2×#11758198—21no—
42C2×#11758198—21no—
43C2×#11758198C2311yescharacteristic
44C2×C4×Dic61—21no—
45C2×C4×Dic61—21no—
46C2×C4×Dic61C2×C411yescharacteristic
47#14519684—21no—
48#14519684—21no—
49#14519684—21no—
50#14519684—21no—
51#14519684C2×C411yescharacteristic
52C22×Q8×C61C2×C411yescharacteristic
53C2×C22⋊C4×C61—21no—
54C2×C22⋊C4×C61C2×C411yescharacteristic
55#95×C61—21no—
56#95×C61C2×C411yescharacteristic
57C24×C61C22⋊C411yescharacteristic
58C23×D61C2×D411yescharacteristic
59C22×#6923—42no—
60C2×Q8×C61—42no—
61C2×Q8×C61—41no—
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C22×C411yescharacteristic
66C2×C22⋊D61—21no—
67C2×C22⋊D61—21no—
68C2×C22⋊D61—41no—
69C2×C22⋊D61—41no—
70C22×Dic61—21no—
71C22×Dic61—21no—
72C22×Dic61—21no—
73C22×Dic61—21no—
74C22×Dic61—42no—
75C22×Dic61C22⋊C411yescharacteristic
76C22×Dic61C22⋊C411yescharacteristic
77C22×Dic61C2×D411yescharacteristic
78C2×C4×D61—21no—
79C2×C4×D61—21no—
80C2×C4×D61—42no—
81C22⋊#6923—42no—
82C22⋊#6923—42no—
83#11758217—82no—
84C2×#92420—42no—
85C4⋊#6923—82no—
86C22⋊C4×C61—41no—
87C22⋊C4×C61—42no—
88C22⋊Dic61—42no—
89#11758198—41no—
90#11758198—42no—
91#11758198—41no—
92#11758198—41no—
93C4⋊Dic61—42no—
94C4⋊Dic61—41no—
95#11758196—41no—
96C4×Dic61—21no—
97C4×Dic61—21no—
98C4×Dic61—42no—
99C4×Dic61—41no—
100C4×Dic61—41no—
101C23×C61—21no—
102C23×C61—21no—
103C23×C61—21no—
104C23×C61—41no—
105C23×C61—41no—
106C23×C61C2×C22⋊C411yescharacteristic
107C22×D61—21no—
108C22×D61—21no—
109C22×D61—41no—
110C22×D61—41no—
111C22×D61AGL(1,ℤ/8)11yescharacteristic
112C22×D61#13711yescharacteristic
113C2×#6923—82no—
114C2×#6923—82no—
115C2×#6923—82no—
116Q8×C61—42no—
117Q8×C61—42no—
118C2×C4×C61—22no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61—41no—
125C2×C4×C61—41no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C22⋊D61—42no—
129C22⋊D61—42no—
130C2×Dic61—22no—
131C2×Dic61—21no—
132C2×Dic61—21no—
133C2×Dic61—21no—
134C2×Dic61—21no—
135C2×Dic61—42no—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61—41no—
139C2×Dic61—41no—
140C2×Dic61—41no—
141C2×Dic61—41no—
142C2×Dic61—41no—
143C2×Dic61—81no—
144C2×Dic61#9911yescharacteristic
145C2×Dic61#9911yescharacteristic
146C4×D61—42no—
147C4×D61—81no—
148#92420—82no—
149#11823—611nomaximal, Sylow
150C22×C61—21no—
151C22×C61—21no—
152C22×C61—42no—
153C22×C61—41no—
154C22×C61—41no—
155C22×C61—41no—
156C22×C61—81no—
157C22×C61C2×#9911yescharacteristic
158C22×C61#41711yescharacteristic
159C22×C61#42111yescharacteristic
160C2×D61—21no—
161C2×D61—21no—
162C2×D61—41no—
163C2×D61—41no—
164C2×D61—41no—
165C2×D61—41no—
166C2×D61—81no—
167#6923—162no—
168C4×C61—42no—
169C4×C61—41no—
170C4×C61—41no—
171C4×C61—81no—
172Dic61—21no—
173Dic61—41no—
174Dic61—41no—
175Dic61—81no—
176Dic61—81no—
177Dic61#35412yes—
178#2398—611no—
179#1799—611no—
180#1465—611no—
181#1351—612no—
182#1344—612no—
183C2×C61—21no—
184C2×C61—21no—
185C2×C61—81no—
186C2×C61#151311yescharacteristic
187C2×C61#212011yescharacteristic
188C2×C61C2×#35411yescharacteristic
189D61—41no—
190D61—41no—
191C2×C4×Q8—611no—
192C2×C4×D4—611no—
193C2×#99—1221no—
194#407—1221no—
195#400—1221no—
196#397—611no—
197#397—611no—
198#394—1221no—
199#393—611no—
200#393#692311yescharacteristic, complemented
201#380—1221no—
202C4×C22⋊C4—1221no—
203C4×C22⋊C4—1221no—
204#376—611no—
205#342—612no—
206#324—1221no—
207#323—612no—
208C61#1182311yescharacteristic, complemented, Sylow
209C22×Q8C2×#692311yescharacteristic
210C22×D4—611no—
211C23×C4—611no—
212C23×C4—611no—
213C2×M16—2441no—
214C4×Q8—2441no—
215C4×Q8—2441no—
216C4×D4—2441no—
217C4×D4—2441no—
218C2×C4⋊C4—611no—
219C2×C4⋊C4—611no—
220C2×C4⋊C4—1221no—
221C2×C4⋊C4—2441no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—611no—
224C2×C22⋊C4—1221no—
225C2×C22⋊C4—1221no—
226C2×C22⋊C4—1221no—
227C2×C22⋊C4—1222no—
228C2×C22⋊C4C2×#692311yescharacteristic
229C2×C42—611no—
230C2×C42—1221no—
231C2×C42—1221no—
232#99—2442no—
233C22⋊C8—1222no—
234C22⋊C8—2441no—
235#95—21no—
236#95—611no—
237#95—1221no—
238#95—1221no—
239#95—1221no—
240#95—1221no—
241#95—2441no—
242#95C2×#692311yescharacteristic
243C24—611no—
244C24C22⋊#692311yescharacteristic
245C2×Q8—42no—
246C2×Q8—41no—
247C2×D4—1221no—
248C2×D4—1221no—
249C2×D4—2441no—
250C2×D4—2441no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—611no—
254C22×C4—611no—
255C22×C4—611no—
256C22×C4—1221no—
257C22×C4—1221no—
258C22×C4—1221no—
259C22×C4—1221no—
260C22×C4—1221no—
261C22×C4—1221no—
262C22×C4—2442no—
263C22×C4—2442no—
264C22×C4—2442no—
265C22×C4C22⋊#692311yescharacteristic
266C22×C4C22×#692311yescharacteristic
267M16—4882no—
268C2×C8—2442no—
269C4⋊C4—4882no—
270C4⋊C4—2441no—
271C4⋊C4—2441no—
272C4⋊C4—2442no—
273C22⋊C4—41no—
274C22⋊C4—42no—
275C22⋊C4—2441no—
276C22⋊C4—2442no—
277C22⋊C4—2441no—
278C22⋊C4—2441no—
279C22⋊C4—2442no—
280C22⋊C4—2442no—
281C22⋊C4—2442no—
282C42—1221no—
283C42—1221no—
284C42—2441no—
285C42—2441no—
286C42—2442no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—41no—
291C23—41no—
292C23—611no—
293C23—611no—
294C23—1221no—
295C23—1221no—
296C23—2441no—
297C23—2441no—
298C23C2×C22⋊#692311yescharacteristic
299Q8—42no—
300Q8—42no—
301D4—2442no—
302D4—2442no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—611no—
314C2×C4—611no—
315C2×C4—1222no—
316C2×C4—1221no—
317C2×C4—1221no—
318C2×C4—1221no—
319C2×C4—1221no—
320C2×C4—4881no—
321C2×C4—4881no—
322C2×C4—4882no—
323C2×C4—4882no—
324C2×C4—4882no—
325C2×C4—2442no—
326C2×C4—2442no—
327C2×C4—2441no—
328C2×C4—2441no—
329C2×C4—2441no—
330C2×C4—2441no—
331C2×C4—2441no—
332C2×C4—2441no—
333C2×C4—2441no—
334C8—4882no—
335C22—21no—
336C22—21no—
337C22—42no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—81no—
342C22—1221no—
343C22—1221no—
344C22—4881no—
345C22—2441no—
346C22—2441no—
347C22—2441no—
348C22—2441no—
349C22D61⋊#9911yescharacteristic
350C22#1684075311yescharacteristic
351C22#1684076111yescharacteristic
352C4—42no—
353C4—41no—
354C4—41no—
355C4—81no—
356C4—612no—
357C4—1221no—
358C4—9762no—
359C4—4881no—
360C4—4881no—
361C4—2441no—
362C4—2441no—
363C2—21no—
364C2—21no—
365C2—81no—
366C2—2441no—
367C2—2441no—
368C2#2146123711yescharacteristic
369C2#2146207111yescharacteristic
370C2D61⋊#35411yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#393C4
socleC22×C61C2×#99
prime radicalC22×C22⋊D61D4
cosocleC2×#11758198C23
hypercenter#393#6923
hypocenterC61#11823
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #393

Complements

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

NH ≅ G/Ntype
3. #21460814C2semidirect
4. #21460800C2semidirect
12. C2×Q8×Dic61C4semidirect
16. #16839687C4semidirect
21. #16839607C4semidirect
22. C61×#393C4semidirect
200. #393#6923semidirect
208. C61#11823semidirect

Maximal subgroups

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

subgroupindex
#214616812
#214610772
#214608142
#214608002
#214491012
#1182361