Finite Group Databasebeta

← #30595833

Subgroups — #30595833

Order 15616.

470 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 92 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#21461673C211yesmaximal, characteristic
2#21461545C211yesmaximal, characteristic
3#21460640C212yesmaximal, complemented
4#21460637C212yesmaximal, complemented
5#21454159C211yesmaximal, characteristic
6C2×C4⋊#92420—21no—
7C2×C22⋊#92420—21no—
8C2×C22⋊#92420—21no—
9C2×#14519832—21no—
10#16840628C2212yes—
11#16840628C2212yes—
12C2×#14519756C411yescharacteristic
13C22×C4×Dic61C2211yescharacteristic
14#16839667C411yescharacteristic
15#16839667C2211yescharacteristic
16#16839661C411yescharacteristic
17#16839661C2211yescharacteristic
18#16839647—21no—
19#16839647—21no—
20#16839637—21no—
21#16839623—21no—
22C61×#380C411yescharacteristic
23C22×#92420—22no—
24C22×#92420—22no—
25C4⋊#92420—42no—
26C22⋊#92420—24no—
27C22⋊#92420—42no—
28#14519832—42no—
29C22×Dic122C2×C411yescharacteristic
30C23×Dic61D411yescharacteristic
31C23×Dic61D411yescharacteristic
32#14519756—42no—
33C2×C4⋊Dic61—21no—
34C2×C4⋊Dic61—21no—
35C2×C4⋊Dic61C2×C411yescharacteristic
36C2×#11758196—21no—
37C2×C22⋊Dic61—21no—
38C2×C4×Dic61—42no—
39C2×C4×Dic61C2×C411yescharacteristic
40C2×C4×Dic61D411yescharacteristic
41C2×C4×Dic61D411yescharacteristic
42C2×C4×Dic61C2311yescharacteristic
43#14519684—21no—
44#14519684—21no—
45#14519684—21no—
46#14519684—21no—
47#14519684—21no—
48#14519684—21no—
49C23×C4×C61C2×C411yescharacteristic
50C2×C4⋊C4×C61C2×C411yescharacteristic
51C2×C4⋊C4×C61C2×C411yescharacteristic
52C2×C22⋊C4×C61—21no—
53#95×C61—21no—
54C24×C61C22⋊C411yescharacteristic
55C22×C4×C61—22no—
56C22×C4×C61—22no—
57C22×C4×C61—21no—
58C22×C4×C61—21no—
59C22×C4×C61—41no—
60C22×C4×C61C22⋊C411yescharacteristic
61C22×C4×C61C22×C411yescharacteristic
62C22×Dic61—22no—
63C22×Dic61—21no—
64C22×Dic61—21no—
65C22×Dic61—22no—
66C22×Dic61—21no—
67C22×Dic61—21no—
68C22×Dic61—41no—
69C22×Dic61—41no—
70C22×Dic61—41no—
71C22×Dic61C22⋊C411yescharacteristic
72C22×Dic61C22⋊C411yescharacteristic
73C22×Dic61C2×D411yescharacteristic
74C22×Dic61C2×D411yescharacteristic
75C2×Dic122—22no—
76C2×Dic122—41no—
77C2×Dic122—41no—
78C2×#92420—24no—
79C2×#92420—42no—
80C2×#92420—42no—
81C2×#92420—42no—
82C2×#92420—42no—
83C2×#92420—42no—
84C4⋊C4×C61—22no—
85C4⋊C4×C61—41no—
86C22⋊C4×C61—42no—
87C22⋊Dic61—42no—
88C4⋊Dic61—42no—
89C4⋊Dic61—41no—
90C4⋊Dic61—41no—
91C4⋊Dic61—41no—
92#11758196—42no—
93C4×Dic61—22no—
94C4×Dic61—22no—
95C4×Dic61—41no—
96C4×Dic61—41no—
97C23×C61—21no—
98C23×C61—21no—
99C23×C61—21no—
100C23×C61—21no—
101C23×C61—21no—
102C23×C61—21no—
103C23×C61#10412yes—
104C23×C61C2×C22⋊C411yescharacteristic
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—44no—
110C2×C4×C61—41no—
111C2×C4×C61—41no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61#9912yes—
117C2×C4×C61#13112yes—
118C2×Dic61—21no—
119C2×Dic61—21no—
120C2×Dic61—21no—
121C2×Dic61—21no—
122C2×Dic61—21no—
123C2×Dic61—21no—
124C2×Dic61—21no—
125C2×Dic61—21no—
126C2×Dic61—21no—
127C2×Dic61—21no—
128C2×Dic61—44no—
129C2×Dic61—42no—
130C2×Dic61—42no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61AGL(1,ℤ/8)11yescharacteristic
138C2×Dic61#13711yescharacteristic
139C2×Dic61#13712yes—
140Dic122—82no—
141#92420—44no—
142#92420—82no—
143#7701—611nomaximal, Sylow
144C22×C61—22no—
145C22×C61—22no—
146C22×C61—22no—
147C22×C61—41no—
148C22×C61—41no—
149C22×C61—41no—
150C22×C61—41no—
151C22×C61#40811yescharacteristic
152C22×C61C2×#9911yescharacteristic
153C22×C61#41311yescharacteristic
154C22×C61#41711yescharacteristic
155C22×C61#41911yescharacteristic
156C22×C61C2×#10411yescharacteristic
157C22×C61#42111yescharacteristic
158C4×C61—22no—
159C4×C61—22no—
160C4×C61—81no—
161C4×C61—81no—
162Dic61—22no—
163Dic61—21no—
164Dic61—21no—
165Dic61—41no—
166Dic61—41no—
167Dic61—81no—
168Dic61—81no—
169#2855—611no—
170#1797—611no—
171#1749—611no—
172#1272—612no—
173#1271—612no—
174C2×C61—42no—
175C2×C61#146811yescharacteristic
176C2×C61#148311yescharacteristic
177C2×C61#149711yescharacteristic
178C2×C61#150811yescharacteristic
179C2×C61#151311yescharacteristic
180C2×C61#153211yescharacteristic
181C2×C61#155911yescharacteristic
182C2×C22⋊Q8—611no—
183C22×C42—611no—
184C2×C4⋊C8—1221no—
185C2×C22⋊C8—1221no—
186C2×C22⋊C8—1221no—
187C2×#97—1221no—
188#391—611no—
189#391—611no—
190#389—611no—
191#389—611no—
192#387—1221no—
193#387—1221no—
194#385—1221no—
195#381—1221no—
196#380#692311yescharacteristic
197#336—612no—
198#336—612no—
199C61#770111yescharacteristic, complemented, Sylow
200C22×Q8—611no—
201C23×C4—611no—
202C23×C4—611no—
203C23×C4C2×#692311yescharacteristic
204C22×C8—1222no—
205C22×C8—1222no—
206C22⋊Q8—2442no—
207C2×C4⋊C4—611no—
208C2×C4⋊C4—1221no—
209C2×C4⋊C4—1221no—
210C2×C4⋊C4—1221no—
211C2×C4⋊C4C2×#692311yescharacteristic
212C2×C4⋊C4C2×#692311yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—1221no—
215C2×C42—611no—
216C2×C42—611no—
217C2×C42—611no—
218C2×C42—611no—
219C2×C42—2442no—
220C4⋊C8—2442no—
221C22⋊C8—1224no—
222C22⋊C8—2442no—
223#97—2442no—
224#95—21no—
225#95—1221no—
226#95—1221no—
227#95—1221no—
228#95—1221no—
229#95—1221no—
230#95—1221no—
231C24C22⋊#692311yescharacteristic
232C2×Q8—1222no—
233C2×Q8—2441no—
234C2×Q8—2441no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—41no—
240C22×C4—611no—
241C22×C4—611no—
242C22×C4—611no—
243C22×C4—611no—
244C22×C4—1221no—
245C22×C4—1221no—
246C22×C4—1222no—
247C22×C4—1222no—
248C22×C4—1221no—
249C22×C4—1221no—
250C22×C4—2441no—
251C22×C4—2441no—
252C22×C4—2441no—
253C22×C4C22⋊#692311yescharacteristic
254C22×C4C22×#692311yescharacteristic
255C2×C8—1224no—
256C2×C8—2442no—
257C2×C8—2442no—
258C2×C8—2442no—
259C2×C8—2442no—
260C2×C8—2442no—
261C4⋊C4—22no—
262C4⋊C4—41no—
263C4⋊C4—2441no—
264C4⋊C4—2441no—
265C4⋊C4—2441no—
266C4⋊C4—2442no—
267C4⋊C4—2442no—
268C22⋊C4—42no—
269C22⋊C4—2442no—
270C42—1222no—
271C42—1222no—
272C42—2441no—
273C42—2441no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23#1451984112yes—
281C23C2×C22⋊#692311yescharacteristic
282Q8—4882no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—44no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—611no—
295C2×C4—611no—
296C2×C4—612no—
297C2×C4—1221no—
298C2×C4—1221no—
299C2×C4—1221no—
300C2×C4—1221no—
301C2×C4—1221no—
302C2×C4—1221no—
303C2×C4—1221no—
304C2×C4—1221no—
305C2×C4—1221no—
306C2×C4—1221no—
307C2×C4—2444no—
308C2×C4—2442no—
309C2×C4—2442no—
310C2×C4—2441no—
311C2×C4—2441no—
312C2×C4—2441no—
313C2×C4—2441no—
314C2×C4—2441no—
315C2×C4—2441no—
316C2×C4#1451983612yes—
317C2×C4#1451986412yes—
318C8—4882no—
319C8—2444no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22#1684073511yescharacteristic
328C22C2×#1451983611yescharacteristic
329C22#1684074911yescharacteristic
330C22#1684075311yescharacteristic
331C22#1684075511yescharacteristic
332C22C2×#1451984111yescharacteristic
333C22#1684075911yescharacteristic
334C4—22no—
335C4—22no—
336C4—81no—
337C4—81no—
338C4—1222no—
339C4—1221no—
340C4—1221no—
341C4—4881no—
342C4—4881no—
343C4—2441no—
344C4—2441no—
345C2—42no—
346C2#2146108611yescharacteristic
347C2#2146113411yescharacteristic
348C2#2146118111yescharacteristic
349C2#2146121811yescharacteristic
350C2#2146123611yescharacteristic
351C2#2146129511yescharacteristic
352C2#2146135511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#380C4
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenter#380#6923
hypocenterC61#7701
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #21460640C2semidirect
4. #21460637C2semidirect
199. C61#7701semidirect

Maximal subgroups

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

subgroupindex
#214616732
#214615452
#214606402
#214606372
#214541592
#770161