Finite Group Databasebeta

← #29577465

Subgroups — #29577465

Order 15616.

606 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 93 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#21450978C211yesmaximal, characteristic
2#21448414C211yesmaximal, characteristic
3#21444339C211yesmaximal, characteristic, complemented
4#21444145C211yesmaximal, characteristic
5#21443705C211yesmaximal, characteristic, complemented
6#21443704C211yesmaximal, characteristic, complemented
7C61×#1508C211yesmaximal, characteristic, complemented
8C22×#11758198C2211yescharacteristic
9#16839726C2211yescharacteristic
10#16839688C412yescomplemented
11#16839673C412yescomplemented
12#16839653—22no—
13#16839652—22no—
14#16839651—22no—
15#16839623—21no—
16#16839620—21no—
17#16839617—21no—
18#16839617—21no—
19C4×C22⋊Dic61C2211yescharacteristic
20C4×C22⋊Dic61C2211yescharacteristic
21#16839499—22no—
22#16839485—22no—
23C2×C22⋊Q8×C61C2211yescharacteristic
24C61×#407C2211yescharacteristic
25C4×C22⋊C4×C61C2211yescharacteristic
26C61×#328—22no—
27C24×D61D411yescharacteristic
28C23×Dic61D411yescharacteristic
29C2×#11758194—21no—
30C2×#11758196—22no—
31C2×C22⋊Dic61—21no—
32C2×C22⋊Dic61—21no—
33C2×#11758198—21no—
34C2×#11758198—21no—
35C2×#11758198—21no—
36C2×#11758198—21no—
37C2×#11758198—22no—
38C2×#11758198—22no—
39C2×#11758198—22no—
40C2×#11758198—22no—
41C2×#11758198C2×C412yes—
42C2×#11758198D412yes—
43C2×C4×Dic61—21no—
44C2×C4×Dic61—21no—
45C22⋊#92418—21no—
46C22⋊#92418—21no—
47#14519684—22no—
48#14519684—22no—
49#14519684—22no—
50#14519684C2×C412yes—
51#14519684D412yes—
52C22×Q8×C61C2×C411yescharacteristic
53C23×C4×C61C2311yescharacteristic
54C2×M16×C61—21no—
55C22⋊Q8×C61—21no—
56C22⋊Q8×C61—21no—
57C22⋊Q8×C61—21no—
58C22⋊Q8×C61—21no—
59C2×C4⋊C4×C61—21no—
60C2×C4⋊C4×C61C2×C411yescharacteristic
61C2×C22⋊C4×C61—21no—
62C2×C22⋊C4×C61—21no—
63C2×C42×C61—21no—
64C22⋊C8×C61—21no—
65C22⋊C8×C61—21no—
66#95×C61D412yes—
67C24×C61C2×D411yescharacteristic
68C23×D61—21no—
69C23×D61—21no—
70C23×D61—42no—
71C23×D61—42no—
72C23×D61—81no—
73C23×D61C22⋊C412yes—
74C2×Q8×C61—21no—
75C2×Q8×C61—21no—
76C2×Q8×C61—81no—
77C22×C4×C61—21no—
78C22×C4×C61—21no—
79C22×C4×C61—22no—
80C22×C4×C61—81no—
81C22×C4×C61C22×C411yescharacteristic
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61C2×D411yescharacteristic
85C22×C4×C61#4111yescharacteristic
86C22×C4×C61#4111yescharacteristic
87C22×Dic61—21no—
88C22×Dic61—21no—
89C22×Dic61—22no—
90C22×Dic61—22no—
91C22×Dic61—42no—
92C22×Dic61C22⋊C412yes—
93M16×C61—81no—
94C2×C8×C61—41no—
95C4⋊C4×C61—21no—
96C4⋊C4×C61—21no—
97C4⋊C4×C61—41no—
98C4⋊C4×C61—41no—
99C22⋊C4×C61—42no—
100C22⋊C4×C61—41no—
101C22⋊C4×C61—41no—
102C42×C61—81no—
103C22⋊Dic61—42no—
104C22⋊Dic61—42no—
105#11758198—22no—
106#11758198—22no—
107#11758198—22no—
108#11758198—22no—
109#11758198—44no—
110#11758198—82no—
111#11758198—82no—
112#11758198—82no—
113#11758196—82no—
114C4×Dic61—81no—
115C4×Dic61—81no—
116#11758194—81no—
117C2×#92418—41no—
118C23×C61—22no—
119C23×C61—22no—
120C23×C61—41no—
121C23×C61C2×C22⋊C411yescharacteristic
122C23×C61#12011yescharacteristic
123C23×C61#12311yescharacteristic
124C22×D61—22no—
125C22×D61—22no—
126C22×D61—22no—
127C22×D61—42no—
128C22×D61—42no—
129C22×D61—42no—
130C22×D61—42no—
131C22×D61—42no—
132C22×D61—42no—
133C22×D61—81no—
134C22×D61—81no—
135C22×D61—88no—
136Q8×C61—81no—
137Q8×C61—81no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—42no—
145C2×C4×C61—42no—
146C2×C4×C61—42no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61—42no—
150C2×C4×C61—42no—
151C2×C4×C61C4×D411yescharacteristic
152C2×C4×C61C4×D411yescharacteristic
153C2×C4×C61#12111yescharacteristic
154C2×C4×C61#12111yescharacteristic
155C2×Dic61—22no—
156C2×Dic61—42no—
157C2×Dic61—42no—
158C2×Dic61—42no—
159C2×Dic61—42no—
160C2×Dic61—42no—
161C2×Dic61—42no—
162C2×Dic61—42no—
163C2×Dic61—42no—
164C2×Dic61—81no—
165C8×C61—81no—
166#92418—81no—
167#9411—611nomaximal, Sylow
168C22×C61—21no—
169C22×C61—21no—
170C22×C61—21no—
171C22×C61—21no—
172C22×C61—21no—
173C22×C61—21no—
174C22×C61—21no—
175C22×C61—44no—
176C22×C61#38611yescharacteristic
177C22×C61#42112yes—
178C22×C61#45512yes—
179C2×D61—44no—
180C2×D61—42no—
181C2×D61—42no—
182C2×D61—42no—
183C2×D61—84no—
184C2×D61—84no—
185C2×D61—82no—
186C4×C61—21no—
187C4×C61—21no—
188C4×C61—21no—
189C4×C61—21no—
190C4×C61—82no—
191C4×C61—81no—
192C4×C61—81no—
193Dic61—42no—
194Dic61—82no—
195Dic61—82no—
196#2561—611no—
197#2312—611no—
198#1508—611no—
199#1508D6111yescharacteristic, complemented
200#1462—611no—
201#1297—611no—
202#1297—611no—
203C2×C61—22no—
204C2×C61—22no—
205C2×C61—41no—
206C2×C61#188511yescharacteristic
207C2×C61#189711yescharacteristic
208C2×C61#191311yescharacteristic
209D61—82no—
210C2×C22⋊Q8C2×D6111yescharacteristic
211C22×C22⋊C4—611no—
212#407—611no—
213#407C2×D6111yescharacteristic
214#397—612no—
215#393—612no—
216#388—1222no—
217#388—1222no—
218#388—1222no—
219#381—1221no—
220#380—1221no—
221#379—1221no—
222#379—1221no—
223C4×C22⋊C4—611no—
224C4×C22⋊C4—611no—
225C4×C22⋊C4C2×D6111yescharacteristic
226#328—22no—
227#328—1222no—
228#323—1222no—
229C61#941111yescharacteristic, complemented, Sylow
230C25—611no—
231C22×Q8C4×D6111yescharacteristic
232C23×C4—611no—
233C23×C4C22×D6111yescharacteristic
234C2×M16—21no—
235C2×M16—1221no—
236C22⋊Q8—21no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C22⋊Q8—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—1222no—
242C2×C4⋊C4C4×D6111yescharacteristic
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—612no—
246C2×C22⋊C4—612no—
247C2×C22⋊C4—1221no—
248C2×C22⋊C4—1221no—
249C2×C22⋊C4—1221no—
250C2×C22⋊C4—1221no—
251C2×C22⋊C4—1222no—
252C2×C22⋊C4—1222no—
253C2×C22⋊C4—1222no—
254C2×C22⋊C4—1222no—
255C2×C22⋊C4—1221no—
256C2×C22⋊C4—1221no—
257C2×C42—21no—
258C2×C42—1221no—
259C2×C42—1221no—
260C22⋊C8—21no—
261C22⋊C8—21no—
262C22⋊C8—1221no—
263C22⋊C8—1221no—
264#95—612no—
265#95—612no—
266#95—1222no—
267#95—1222no—
268#95—1222no—
269#95C22⋊D6112yes—
270C24—612no—
271C24—1221no—
272C24—1221no—
273C24—4881no—
274C24—2442no—
275C24—2442no—
276C24D4×D6111yescharacteristic
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—81no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—81no—
284C22×C4—612no—
285C22×C4—1221no—
286C22×C4—1221no—
287C22×C4—1222no—
288C22×C4—1222no—
289C22×C4—2442no—
290C22×C4C2×C4×D6111yescharacteristic
291C22×C4#1175822211yescharacteristic
292C22×C4D4×D6111yescharacteristic
293C22×C4D4×D6111yescharacteristic
294C22×C4#1175822411yescharacteristic
295C22×C4C2×C22⋊D6111yescharacteristic
296M16—81no—
297M16—4881no—
298C2×C8—41no—
299C2×C8—2441no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—4882no—
305C22⋊C4—42no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—1222no—
309C22⋊C4—1222no—
310C22⋊C4—1222no—
311C22⋊C4—1222no—
312C22⋊C4—4882no—
313C22⋊C4—4882no—
314C22⋊C4—4882no—
315C22⋊C4—2444no—
316C22⋊C4—2442no—
317C22⋊C4—2442no—
318C42—81no—
319C42—4881no—
320C42—4881no—
321C23—22no—
322C23—22no—
323C23—41no—
324C23—1222no—
325C23—1222no—
326C23—1222no—
327C23—4881no—
328C23—4881no—
329C23—4888no—
330C23—2442no—
331C23—2442no—
332C23—2442no—
333C23—2442no—
334C23—2442no—
335C23—2442no—
336C23C22⋊C4×D6111yescharacteristic
337C23#1451974511yescharacteristic
338C23#1451975911yescharacteristic
339Q8—81no—
340Q8—81no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—1222no—
355C2×C4—4881no—
356C2×C4—2442no—
357C2×C4—2442no—
358C2×C4—2442no—
359C2×C4—2442no—
360C2×C4—2442no—
361C2×C4—2442no—
362C2×C4—2442no—
363C2×C4—2442no—
364C2×C4#1451973711yescharacteristic
365C2×C4C4×C22⋊D6111yescharacteristic
366C2×C4#1451974811yescharacteristic
367C2×C4#1451975011yescharacteristic
368C8—81no—
369C8—4881no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—44no—
378C22—4882no—
379C22—4884no—
380C22—4884no—
381C22—2444no—
382C22—2442no—
383C22—2442no—
384C22—2442no—
385C22#1683963811yescharacteristic
386C22#1683977712yes—
387C22#1683993812yes—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—82no—
393C4—81no—
394C4—81no—
395C4—4882no—
396C4—4882no—
397C4—2442no—
398C2—22no—
399C2—22no—
400C2—41no—
401C2—4882no—
402C2#2144620411yescharacteristic
403C2#2144626411yescharacteristic
404C2#2144635811yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1508C2
socleC22×C61#386
prime radicalC24×D61D4
cosocleC23×C4×C61C23
hypercenter#1508D61
hypocenterC61#9411
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1508

Complements

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

NH ≅ G/Ntype
3. #21444339C2semidirect
5. #21443705C2semidirect
6. #21443704C2semidirect
7. C61×#1508C2semidirect
10. #16839688C4semidirect
11. #16839673C4semidirect
199. #1508D61semidirect
229. C61#9411semidirect

Maximal subgroups

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

subgroupindex
#214509782
#214484142
#214443392
#214441452
#214437052
#214437042
C61×#15082
#941161