Finite Group Databasebeta

← #29575018

Subgroups — #29575018

Order 15616.

836 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 135 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#21448417C211yesmaximal, characteristic
2#21445754C212yesmaximal, complemented
3C2×#16839486C211yesmaximal, characteristic
4#21443698C212yesmaximal, complemented
5C2×C61×#393C211yesmaximal, characteristic
6C2×#14519701—24no—
7C2×C22⋊#92418C2212yes—
8#16839655—24no—
9#16839622C2212yes—
10#16839618—21no—
11C4×C22⋊Dic61—21no—
12#16839486—22no—
13C23×Q8×C61C2211yescharacteristic
14C22×C22⋊C4×C61—21no—
15C22×C22⋊C4×C61C2211yescharacteristic
16C61×#393—22no—
17C61×#393C414yescomplemented
18C2×#95×C61—21no—
19C2×#95×C61C2211yescharacteristic
20C22×#92418—22no—
21C2×C4⋊Dic61D414yes—
22C2×C22⋊Dic61—22no—
23C2×C22⋊Dic61—22no—
24C2×C22⋊Dic61—22no—
25C2×C4×Dic61—22no—
26#14519701—48no—
27C22⋊#92418—22no—
28C22⋊#92418—22no—
29#14519684—22no—
30#14519684—22no—
31C25×C61D411yescharacteristic
32C22×Q8×C61—21no—
33C22×Q8×C61—22no—
34C22×Q8×C61—22no—
35C22×Q8×C61—44no—
36C22×Q8×C61C2×C412yes—
37C23×C4×C61—21no—
38C23×C4×C61—21no—
39C23×C4×C61D411yescharacteristic
40C23×C4×C61C2311yescharacteristic
41C2×C22⋊C4×C61—22no—
42C2×C22⋊C4×C61—22no—
43C2×C22⋊C4×C61—24no—
44C2×C22⋊C4×C61—44no—
45C2×C22⋊C4×C61C2×C412yes—
46C2×C22⋊C4×C61D412yes—
47#95×C61—24no—
48#95×C61—21no—
49#95×C61C2×C412yes—
50C24×C61—21no—
51C24×C61—22no—
52C24×C61—21no—
53C24×C61—22no—
54C24×C61—42no—
55C24×C61—42no—
56C24×C61C22⋊C412yes—
57C24×C61C2×D411yescharacteristic
58C2×Q8×C61—28no—
59C2×Q8×C61—24no—
60C2×Q8×C61—44no—
61C2×Q8×C61—44no—
62C2×Q8×C61—44no—
63C2×Q8×C61—48no—
64C2×Q8×C61—42no—
65C22×C4×C61—24no—
66C22×C4×C61—22no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61—22no—
70C22×C4×C61—42no—
71C22×C4×C61—42no—
72C22×C4×C61—41no—
73C22×C4×C61—41no—
74C22×C4×C61—42no—
75C22×C4×C61—42no—
76C22×C4×C61—42no—
77C22×C4×C61—42no—
78C22×C4×C61C22⋊C412yes—
79C22×C4×C61C22×C411yescharacteristic
80C22×C4×C61C2×D412yes—
81C22×C4×C61C2×D411yescharacteristic
82C22×C4×C61#4112yes—
83C22×Dic61—24no—
84C22⋊C4×C61—24no—
85C22⋊C4×C61—24no—
86C22⋊C4×C61—48no—
87C22⋊Dic61—42no—
88C22⋊Dic61—42no—
89C22⋊Dic61—82no—
90C22⋊Dic61—82no—
91C4⋊Dic61—44no—
92C4×Dic61—82no—
93C2×#92418—44no—
94C2×#92418—42no—
95C23×C61—22no—
96C23×C61—22no—
97C23×C61—22no—
98C23×C61—22no—
99C23×C61—22no—
100C23×C61—42no—
101C23×C61—42no—
102C23×C61—48no—
103C23×C61—42no—
104C23×C61—42no—
105C23×C61—42no—
106C23×C61—42no—
107C23×C61—81no—
108C23×C61—81no—
109C23×C61—84no—
110C23×C61C2×C22⋊C411yescharacteristic
111C23×C61#12011yescharacteristic
112C23×C61#12311yescharacteristic
113C23×C61#13512yes—
114C23×C61#13712yes—
115Q8×C61—28no—
116Q8×C61—48no—
117Q8×C61—44no—
118C2×C4×C61—24no—
119C2×C4×C61—24no—
120C2×C4×C61—22no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61—42no—
124C2×C4×C61—42no—
125C2×C4×C61—44no—
126C2×C4×C61—42no—
127C2×C4×C61—42no—
128C2×C4×C61—44no—
129C2×C4×C61—42no—
130C2×C4×C61—42no—
131C2×C4×C61—42no—
132C2×C4×C61—42no—
133C2×C4×C61—81no—
134C2×C4×C61—84no—
135C2×C4×C61C4×D412yes—
136C2×C4×C61#12112yes—
137C2×Dic61—44no—
138C2×Dic61—44no—
139C2×Dic61—44no—
140#92418—82no—
141#9016—611nomaximal, Sylow
142C22×C61—24no—
143C22×C61—24no—
144C22×C61—21no—
145C22×C61—21no—
146C22×C61—21no—
147C22×C61—21no—
148C22×C61—44no—
149C22×C61—42no—
150C22×C61—42no—
151C22×C61—42no—
152C22×C61—82no—
153C22×C61—84no—
154C22×C61—84no—
155C22×C61#38611yescharacteristic
156C22×C61#41611yescharacteristic
157C22×C61#41911yescharacteristic
158C22×C61#45212yes—
159C22×C61#45512yes—
160C22×C61B(4,𝔽2)12yes—
161C22×C61#48512yes—
162C4×C61—22no—
163C4×C61—22no—
164C4×C61—42no—
165C4×C61—44no—
166C4×C61—82no—
167Dic61—84no—
168C2×#393D6111yescharacteristic
169#2312—611no—
170#1787—612no—
171C2×#323—611no—
172#1294—612no—
173C2×C61—21no—
174C2×C61—21no—
175C2×C61—22no—
176C2×C61—82no—
177C2×C61#160614yes—
178C2×C61#187311yescharacteristic
179C2×C61#188511yescharacteristic
180C2×C61#188911yescharacteristic
181C23×Q8C2×D6111yescharacteristic
182C22×C22⋊C4—21no—
183C22×C22⋊C4C2×D6111yescharacteristic
184C2×#103—1224no—
185C2×C22⋊C8—612no—
186#393—22no—
187#393Dic6114yescomplemented
188#388—1224no—
189#380—612no—
190#379—1221no—
191C4×C22⋊C4—1221no—
192C2×#95—21no—
193C2×#95C2×D6111yescharacteristic
194#323—1222no—
195C61#901611yescharacteristic, complemented, Sylow
196C25C22⋊D6111yescharacteristic
197C22×Q8—21no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—44no—
201C22×Q8C2×Dic6112yes—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4C22⋊D6111yescharacteristic
205C23×C4C22×D6111yescharacteristic
206C22×C8—1222no—
207C2×C4⋊C4—614no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—1222no—
213C2×C22⋊C4—1222no—
214C2×C22⋊C4—1222no—
215C2×C22⋊C4C2×Dic6112yes—
216C2×C22⋊C4C22⋊D6112yes—
217C2×C42—1222no—
218#103—2448no—
219C22⋊C8—1222no—
220C22⋊C8—1222no—
221#95—24no—
222#95—21no—
223#95—1222no—
224#95—1222no—
225#95C2×Dic6112yes—
226C24—22no—
227C24—21no—
228C24—21no—
229C24—22no—
230C24—42no—
231C24—42no—
232C24C22⋊Dic6112yes—
233C24C2×C22⋊D6111yescharacteristic
234C2×Q8—28no—
235C2×Q8—24no—
236C2×Q8—44no—
237C2×Q8—44no—
238C2×Q8—44no—
239C2×Q8—48no—
240C2×Q8—42no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—1224no—
255C22×C4C22⋊Dic6112yes—
256C22×C4D4×D6112yes—
257C22×C4#1175822412yes—
258C22×C4C22×Dic6111yescharacteristic
259C22×C4C2×C22⋊D6111yescharacteristic
260C2×C8—2444no—
261C2×C8—2442no—
262C4⋊C4—2444no—
263C22⋊C4—24no—
264C22⋊C4—24no—
265C22⋊C4—48no—
266C22⋊C4—4882no—
267C22⋊C4—4882no—
268C22⋊C4—2442no—
269C22⋊C4—2442no—
270C42—4882no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—42no—
277C23—42no—
278C23—48no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—81no—
284C23—81no—
285C23—84no—
286C23C2×C22⋊Dic6111yescharacteristic
287C23#1451974511yescharacteristic
288C23#1451975811yescharacteristic
289C23#1451979712yes—
290C23#1451980812yes—
291Q8—28no—
292Q8—48no—
293Q8—44no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—22no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—44no—
302C2×C4—42no—
303C2×C4—44no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—81no—
310C2×C4—84no—
311C2×C4—2444no—
312C2×C4—2444no—
313C2×C4—2444no—
314C2×C4D4×Dic6112yes—
315C2×C4#1451974912yes—
316C8—4882no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—24no—
322C22—24no—
323C22—44no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—82no—
328C22—84no—
329C22—84no—
330C22#1683964211yescharacteristic
331C22#1683975811yescharacteristic
332C22#1683977111yescharacteristic
333C22#1683992212yes—
334C22#1683993912yes—
335C22#1683994812yes—
336C22#1684010012yes—
337C4—22no—
338C4—22no—
339C4—42no—
340C4—44no—
341C4—82no—
342C4—4884no—
343C2—21no—
344C2—21no—
345C2—22no—
346C2—82no—
347C2#2144479714yes—
348C2#2144613211yescharacteristic
349C2#2144620811yescharacteristic
350C2#2144622811yescharacteristic
subgroupG/H
centerC23#14519745
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC2×C61×#393C2
socleC23×C61#120
prime radicalC25×C61D4
cosocleC23×C4×C61C23
hypercenterC2×#393D61
hypocenterC61#9016
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
2. #21445754C2semidirect
4. #21443698C2semidirect
17. C61×#393C4semidirect
187. #393Dic61semidirect
195. C61#9016semidirect

Maximal subgroups

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

subgroupindex
#214484172
#214457542
C2×#168394862
#214436982
C2×C61×#3932
#901661