Finite Group Databasebeta

← #29652867

Subgroups — #29652867

Order 15616.

934 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 934 conjugacy classes of subgroups — too many to draw. Only its 183 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#21455914C212yesmaximal
2#21449102C211yesmaximal, characteristic
3#21444861C214yesmaximal, complemented
4#21444759C214yesmaximal
5#21444331C212yesmaximal, complemented
6#21444163C211yesmaximal, characteristic, complemented
7C61×#3502C211yesmaximal, characteristic, complemented
8C2×C4×D244C2211yescharacteristic
9#16840095C2214yes—
10#16840039—22no—
11#16839964—22no—
12#16839912C2214yes—
13#16839860—24no—
14#16839819C2212yes—
15#16839701C2212yes—
16#16839688—21no—
17#16839679C2212yes—
18#16839673—21no—
19C4×#11758198—22no—
20#16839608C2211yescharacteristic
21#16839499C2218yes—
22#16839488C2214yes—
23Q82×C61—22no—
24C61×#557—22no—
25C61×#557—24no—
26C61×#544—21no—
27C61×#544C2214yes—
28C61×#533—22no—
29C61×#533C2212yes—
30C61×#519—21no—
31C2×C4×Q8×C61C2211yescharacteristic
32C22×D244C2311yescharacteristic
33C22×C4×D61—21no—
34C2×#11758201—24no—
35C4×D244—22no—
36C4×D244—22no—
37C2×C4⋊Dic61C2311yescharacteristic
38C2×#11758198—21no—
39C2×#11758198—22no—
40C2×#11758198—22no—
41C2×C4×Dic61—21no—
42#14519710—24no—
43C4⋊#92418—22no—
44#14519701—24no—
45#14519697—24no—
46C22⋊#92418C2314yes—
47#14519688—22no—
48#14519684—22no—
49#14519684—21no—
50#14519684—22no—
51#14519684C2312yes—
52C22×Q8×C61D411yescharacteristic
53C4⋊Q8×C61—22no—
54C4⋊Q8×C61—22no—
55C4⋊Q8×C61—24no—
56C4⋊Q8×C61—28no—
57C4⋊Q8×C61D412yes—
58C4⋊Q8×C61D412yes—
59#125×C61—24no—
60#125×C61—24no—
61#125×C61—28no—
62#125×C61—22no—
63C22⋊Q8×C61—22no—
64C22⋊Q8×C61—24no—
65C22⋊Q8×C61C2314yes—
66C4×Q8×C61—24no—
67C4×Q8×C61—24no—
68C4×Q8×C61—24no—
69C4×Q8×C61—24no—
70C4×Q8×C61—22no—
71C4×Q8×C61—22no—
72C4×Q8×C61—22no—
73C4×Q8×C61D414yes—
74#117×C61—24no—
75#117×C61—22no—
76C2×C4⋊C4×C61D411yescharacteristic
77C2×C4⋊C4×C61C2312yes—
78C2×C42×C61D412yes—
79C2×C42×C61C2311yescharacteristic
80C23×D61—21no—
81C2×Q8×C61—24no—
82C2×Q8×C61—22no—
83C2×Q8×C61—22no—
84C2×Q8×C61—24no—
85C2×Q8×C61C2×D414yes—
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61C2×D411yescharacteristic
88C22×C4×C61C2×D412yes—
89C22×C4×C61C2×D412yes—
90C22×C4×C61C2411yescharacteristic
91C22×Dic61—21no—
92C22×Dic61—22no—
93C2×D244—22no—
94C2×D244—42no—
95C2×C4×D61—44no—
96C2×C4×D61—41no—
97C2×C4×D61—41no—
98C4⋊C4×C61—24no—
99C4⋊C4×C61—24no—
100C4⋊C4×C61—24no—
101C4⋊C4×C61—28no—
102C4⋊C4×C61—22no—
103C4⋊C4×C61—22no—
104C4⋊C4×C61—22no—
105C4⋊C4×C61—22no—
106C4⋊C4×C61—24no—
107C4⋊C4×C61—24no—
108C4⋊C4×C61—24no—
109C4⋊C4×C61C2×D414yes—
110C22⋊C4×C61—24no—
111C42×C61—28no—
112C42×C61—22no—
113C42×C61—22no—
114C42×C61—24no—
115C42×C61C2×D412yes—
116C42×C61C2×D412yes—
117#11758201—48no—
118#11758198—44no—
119#11758198—42no—
120#11758198—44no—
121C4⋊Dic61—22no—
122C4×Dic61—42no—
123C2×#92418—24no—
124C23×C61C22×D411yescharacteristic
125C22×D61—22no—
126C22×D61—42no—
127C22×D61—41no—
128C22×D61—41no—
129C22×D61—42no—
130Q8×C61—28no—
131Q8×C61—44no—
132Q8×C61—44no—
133Q8×C61—84no—
134C2×C4×C61—22no—
135C2×C4×C61—22no—
136C2×C4×C61—22no—
137C2×C4×C61—22no—
138C2×C4×C61—24no—
139C2×C4×C61—24no—
140C2×C4×C61—22no—
141C2×C4×C61—41no—
142C2×C4×C61—42no—
143C2×C4×C61#12012yes—
144C2×C4×C61#12011yescharacteristic
145C2×C4×C61#12011yescharacteristic
146C2×C4×C61#13714yes—
147C2×C4×C61C22×D412yes—
148C2×Dic61—42no—
149C2×Dic61—42no—
150C2×Dic61—42no—
151C2×Dic61—41no—
152C2×Dic61—42no—
153D244—24no—
154D244—42no—
155C4×D61—82no—
156C4×D61—82no—
157#92418—44no—
158#18188—611nomaximal, Sylow
159C22×C61—21no—
160C22×C61—21no—
161C22×C61C2×#12011yescharacteristic
162C22×C61C2×#13712yes—
163C2×D61—44no—
164C2×D61—42no—
165C2×D61—82no—
166C2×D61—81no—
167C2×D61—81no—
168C4×C61—22no—
169C4×C61—24no—
170C4×C61—21no—
171C4×C61—22no—
172C4×C61—44no—
173C4×C61—42no—
174C4×C61—44no—
175C4×C61#45412yes—
176Dic61—82no—
177Dic61—81no—
178#3502D6111yescharacteristic, complemented
179#3110—612no—
180#2398—611no—
181#1617—614no—
182#1599—614no—
183#1506—612no—
184#1467—611no—
185C2×C61—22no—
186C2×C61#300411yescharacteristic
187C2×C61C2×#45411yescharacteristic
188C2×C61#302511yescharacteristic
189D61—82no—
190Q82—22no—
191#557—22no—
192#557—24no—
193#544—21no—
194#544C2×D6114yes—
195#533—22no—
196#533C2×D6112yes—
197#519—21no—
198C2×C4×Q8C2×D6111yescharacteristic
199C2×C4×D4—611no—
200#484—614no—
201#474—1222no—
202#461—1222no—
203#450—614no—
204#440—1224no—
205#432—612no—
206#400—612no—
207#397—1221no—
208#394—612no—
209#393—1221no—
210C4×C22⋊C4—1222no—
211#376—611no—
212#328—618no—
213#324—614no—
214C61#1818811yescharacteristic, complemented, Sylow
215C22×Q8C22⋊D6111yescharacteristic
216C22×D4—611no—
217C23×C4—1221no—
218C2×SD16—1224no—
219C4⋊Q8—22no—
220C4⋊Q8—22no—
221C4⋊Q8—24no—
222C4⋊Q8—28no—
223C4⋊Q8C22⋊D6112yes—
224C4⋊Q8C22⋊D6112yes—
225#125—24no—
226#125—24no—
227#125—28no—
228#125—22no—
229C22⋊Q8—22no—
230C22⋊Q8—24no—
231C22⋊Q8C22×D6114yes—
232C4×Q8—24no—
233C4×Q8—24no—
234C4×Q8—24no—
235C4×Q8—24no—
236C4×Q8—22no—
237C4×Q8—22no—
238C4×Q8—22no—
239C4×Q8C22⋊D6114yes—
240C4×D4—1222no—
241C4×D4—1222no—
242#117—24no—
243#117—22no—
244C2×C4⋊C4—611no—
245C2×C4⋊C4C22⋊D6111yescharacteristic
246C2×C4⋊C4C22×D6112yes—
247C2×C22⋊C4—1221no—
248C2×C22⋊C4—1222no—
249C2×C22⋊C4—1222no—
250C2×C42—1221no—
251C2×C42C22⋊D6112yes—
252C2×C42C22×D6111yescharacteristic
253#107—1224no—
254C4⋊C8—1222no—
255#103—1224no—
256#102—1224no—
257C22⋊C8—614no—
258#97—1222no—
259#95—612no—
260#95—1221no—
261#95—1222no—
262#95—1222no—
263C24—1221no—
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8C2×C22⋊D6114yes—
269C2×D4—1222no—
270C2×D4—2442no—
271C22×C4—1222no—
272C22×C4—1221no—
273C22×C4—2444no—
274C22×C4—2441no—
275C22×C4—2441no—
276C22×C4C2×C22⋊D6112yes—
277C22×C4C2×C22⋊D6111yescharacteristic
278C22×C4C2×C22⋊D6111yescharacteristic
279C22×C4C2×C22⋊D6112yes—
280C22×C4C23×D6111yescharacteristic
281SD16—2448no—
282C2×C8—1224no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—28no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—24no—
294C4⋊C4—1222no—
295C4⋊C4C2×C22⋊D6114yes—
296C22⋊C4—24no—
297C22⋊C4—2444no—
298C22⋊C4—2444no—
299C22⋊C4—2442no—
300C42—28no—
301C42—22no—
302C42—22no—
303C42—24no—
304C42—2442no—
305C42C2×C22⋊D6112yes—
306C42C2×C22⋊D6112yes—
307C23—1222no—
308C23—2442no—
309C23—2441no—
310C23—2441no—
311C23—2442no—
312C23C22×C22⋊D6111yescharacteristic
313Q8—28no—
314Q8—44no—
315Q8—44no—
316Q8—84no—
317D4—1224no—
318D4—2442no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—22no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—4882no—
329C2×C4—4882no—
330C2×C4—2442no—
331C2×C4—2442no—
332C2×C4—2442no—
333C2×C4—2441no—
334C2×C4—2442no—
335C2×C4#1451974612yes—
336C2×C4#1451974611yescharacteristic
337C2×C4#1451974611yescharacteristic
338C2×C4#1451980814yes—
339C2×C4C22×C22⋊D6112yes—
340C8—2444no—
341C22—21no—
342C22—21no—
343C22—4882no—
344C22—4881no—
345C22—4881no—
346C22—2444no—
347C22—2442no—
348C22C2×#1451974611yescharacteristic
349C22C2×#1451980812yes—
350C4—21no—
351C4—22no—
352C4—24no—
353C4—22no—
354C4—44no—
355C4—42no—
356C4—44no—
357C4—4882no—
358C4—4881no—
359C4#1683993512yes—
360C2—22no—
361C2—4882no—
362C2#2145505311yescharacteristic
363C2C2×#1683993511yescharacteristic
364C2#2145521111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3502C2
socleC22×C61C2×#120
prime radicalC22×D244C23
hypercenter#3502D61
hypocenterC61#18188
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3502

Complements

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

NH ≅ G/Ntype
3. #21444861C2semidirect
5. #21444331C2semidirect
6. #21444163C2semidirect
7. C61×#3502C2semidirect
178. #3502D61semidirect
214. C61#18188semidirect

Maximal subgroups

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

subgroupindex
#214559142
#214491022
#214448612
#214447592
#214443312
#214441632
C61×#35022
#1818861