Finite Group Databasebeta

← #29590901

Subgroups — #29590901

Order 15616.

506 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 91 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#21445893C211yesmaximal, characteristic
2C2×#16839486C211yesmaximal, characteristic
3#21443840C212yesmaximal, complemented
4#21443819C212yesmaximal, complemented
5C61×#2561C211yesmaximal, characteristic
6C2×C22⋊#92418—21no—
7#16839618—21no—
8#16839529—21no—
9#16839529C2212yes—
10#16839486—21no—
11#16839486C2212yes—
12C2×C22⋊Q8×C61C411yescharacteristic, complemented
13C22×C22⋊C4×C61C2211yescharacteristic
14C61×#397C411yescharacteristic, complemented
15C61×#397C2211yescharacteristic
16C61×#393C411yescharacteristic, complemented
17C61×#393C2211yescharacteristic
18C61×#388—21no—
19C61×#388—21no—
20C61×#381—21no—
21C61×#379—21no—
22C4×C22⋊C4×C61C411yescharacteristic, complemented
23C22×#92418—41no—
24C2×C22⋊Dic61—22no—
25C2×C22⋊Dic61—41no—
26C2×C22⋊Dic61—41no—
27C22⋊#92418—22no—
28C22⋊#92418—41no—
29C25×C61D411yescharacteristic
30C22×Q8×C61C2×C411yescharacteristic
31C23×C4×C61C2×C411yescharacteristic
32C23×C4×C61D411yescharacteristic
33C22⋊Q8×C61—41no—
34C22⋊Q8×C61—41no—
35C2×C4⋊C4×C61—21no—
36C2×C4⋊C4×C61C2×C411yescharacteristic
37C2×C22⋊C4×C61—21no—
38C2×C22⋊C4×C61—21no—
39C2×C22⋊C4×C61—21no—
40C2×C22⋊C4×C61—21no—
41C2×C22⋊C4×C61—21no—
42C2×C22⋊C4×C61—21no—
43C2×C22⋊C4×C61—41no—
44C2×C22⋊C4×C61—41no—
45C2×C22⋊C4×C61C2×C411yescharacteristic
46C2×C22⋊C4×C61D411yescharacteristic
47C2×C22⋊C4×C61D411yescharacteristic
48C2×C22⋊C4×C61C2311yescharacteristic
49C2×C42×C61—21no—
50#95×C61—21no—
51#95×C61—21no—
52#95×C61—21no—
53#95×C61—21no—
54#95×C61C2×C411yescharacteristic
55#95×C61C2×C411yescharacteristic
56C24×C61—21no—
57C24×C61—21no—
58C24×C61—22no—
59C24×C61—41no—
60C24×C61—42no—
61C24×C61—81no—
62C24×C61C22⋊C411yescharacteristic
63C24×C61C22⋊C411yescharacteristic
64C24×C61C2×D411yescharacteristic
65C2×Q8×C61—41no—
66C2×Q8×C61—81no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61—21no—
72C22×C4×C61—21no—
73C22×C4×C61—21no—
74C22×C4×C61—21no—
75C22×C4×C61—42no—
76C22×C4×C61—81no—
77C22×C4×C61C22⋊C411yescharacteristic
78C22×C4×C61C22⋊C411yescharacteristic
79C22×C4×C61C22×C411yescharacteristic
80C22×C4×C61C2×D411yescharacteristic
81C22×Dic61—42no—
82C4⋊C4×C61—42no—
83C4⋊C4×C61—41no—
84C22⋊C4×C61—41no—
85C22⋊C4×C61—41no—
86C22⋊C4×C61—41no—
87C22⋊C4×C61—41no—
88C22⋊C4×C61—42no—
89C22⋊C4×C61—42no—
90C22⋊C4×C61—41no—
91C22⋊C4×C61—41no—
92C22⋊C4×C61—42no—
93C22⋊C4×C61—81no—
94C22⋊C4×C61—81no—
95C42×C61—81no—
96C22⋊Dic61—42no—
97C22⋊Dic61—42no—
98C22⋊Dic61—82no—
99C22⋊Dic61—82no—
100C2×#92418—42no—
101C2×#92418—82no—
102C23×C61—22no—
103C23×C61—21no—
104C23×C61—21no—
105C23×C61—21no—
106C23×C61—21no—
107C23×C61—21no—
108C23×C61—21no—
109C23×C61—21no—
110C23×C61—41no—
111C23×C61—42no—
112C23×C61—42no—
113C23×C61—41no—
114C23×C61—41no—
115C23×C61—41no—
116C23×C61—41no—
117C23×C61—42no—
118C23×C61—42no—
119C23×C61—81no—
120C23×C61—81no—
121C23×C61—84no—
122C23×C61—84no—
123C23×C61#9911yescharacteristic
124C23×C61#9911yescharacteristic
125C23×C61C2×C22⋊C411yescharacteristic
126C23×C61#13512yes—
127Q8×C61—82no—
128C2×C4×C61—21no—
129C2×C4×C61—21no—
130C2×C4×C61—21no—
131C2×C4×C61—21no—
132C2×C4×C61—42no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
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—41no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61—81no—
150C2×C4×C61—81no—
151C2×C4×C61—81no—
152C2×Dic61—82no—
153C2×Dic61—82no—
154C2×Dic61—82no—
155#92418—82no—
156#11845—611nomaximal, Sylow
157C22×C61—21no—
158C22×C61—21no—
159C22×C61—21no—
160C22×C61—21no—
161C22×C61—21no—
162C22×C61—21no—
163C22×C61—22no—
164C22×C61—21no—
165C22×C61—42no—
166C22×C61—42no—
167C22×C61—42no—
168C22×C61—42no—
169C22×C61—41no—
170C22×C61—41no—
171C22×C61—41no—
172C22×C61—41no—
173C22×C61—41no—
174C22×C61—81no—
175C22×C61—84no—
176C22×C61—81no—
177C22×C61—84no—
178C22×C61#35412yes—
179C22×C61C2×#9911yescharacteristic
180C22×C61#41611yescharacteristic
181C22×C61#42111yescharacteristic
182C4×C61—41no—
183C4×C61—41no—
184C4×C61—41no—
185C4×C61—41no—
186C4×C61—81no—
187C4×C61—81no—
188C4×C61—81no—
189C4×C61—81no—
190Dic61—162no—
191#2561D6111yescharacteristic
192#1819—611no—
193C2×#323—611no—
194#1351—612no—
195#1344—612no—
196C2×C61—21no—
197C2×C61—21no—
198C2×C61—22no—
199C2×C61—42no—
200C2×C61—41no—
201C2×C61—81no—
202C2×C61#151611yescharacteristic
203C2×C61#212111yescharacteristic
204C2×C61C2×#35411yescharacteristic
205C2×C22⋊Q8Dic6111yescharacteristic, complemented
206C22×C22⋊C4C2×D6111yescharacteristic
207C2×C22⋊C8—1221no—
208#397Dic6111yescharacteristic, complemented
209#397C2×D6111yescharacteristic
210#393Dic6111yescharacteristic, complemented
211#393C2×D6111yescharacteristic
212#388—21no—
213#388—21no—
214#381—21no—
215#379—21no—
216#379—1221no—
217C4×C22⋊C4Dic6111yescharacteristic, complemented
218#342—612no—
219#342—1221no—
220#323—612no—
221#323—1221no—
222C61#1184511yescharacteristic, complemented, Sylow
223C25C22⋊D6111yescharacteristic
224C22×Q8C2×Dic6111yescharacteristic
225C23×C4C2×Dic6111yescharacteristic
226C23×C4C22⋊D6111yescharacteristic
227C22×C8—2441no—
228C22⋊Q8—41no—
229C22⋊Q8—41no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4C2×Dic6111yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—1222no—
241C2×C22⋊C4—2441no—
242C2×C22⋊C4—2441no—
243C2×C22⋊C4C2×Dic6111yescharacteristic
244C2×C22⋊C4C22⋊D6111yescharacteristic
245C2×C22⋊C4C22⋊D6111yescharacteristic
246C2×C22⋊C4C22×D6111yescharacteristic
247C2×C42—21no—
248C22⋊C8—1222no—
249C22⋊C8—2441no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95C2×Dic6111yescharacteristic
255#95C2×Dic6111yescharacteristic
256C24—21no—
257C24—21no—
258C24—22no—
259C24—41no—
260C24—42no—
261C24—81no—
262C24C22⋊Dic6111yescharacteristic
263C24C22⋊Dic6111yescharacteristic
264C24C2×C22⋊D6111yescharacteristic
265C2×Q8—41no—
266C2×Q8—81no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—42no—
276C22×C4—81no—
277C22×C4—2442no—
278C22×C4C22⋊Dic6111yescharacteristic
279C22×C4C22⋊Dic6111yescharacteristic
280C22×C4C22×Dic6111yescharacteristic
281C22×C4C2×C22⋊D6111yescharacteristic
282C2×C8—4882no—
283C2×C8—2442no—
284C4⋊C4—41no—
285C4⋊C4—42no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—42no—
295C22⋊C4—81no—
296C22⋊C4—81no—
297C22⋊C4—4882no—
298C22⋊C4—4882no—
299C22⋊C4—2442no—
300C22⋊C4—2442no—
301C42—81no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—22no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—42no—
312C23—42no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—42no—
318C23—42no—
319C23—81no—
320C23—81no—
321C23—84no—
322C23—84no—
323C23#1451969211yescharacteristic
324C23#1451969211yescharacteristic
325C23C2×C22⋊Dic6111yescharacteristic
326C23#1451979712yes—
327Q8—82no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—4882no—
353C2×C4—4882no—
354C2×C4—4882no—
355C8—4882no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—22no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—42no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—81no—
374C22—84no—
375C22—81no—
376C22—84no—
377C22#1683955712yes—
378C22C2×#1451969211yescharacteristic
379C22#1683975811yescharacteristic
380C22#1683977911yescharacteristic
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—81no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4—9762no—
390C2—21no—
391C2—21no—
392C2—22no—
393C2—42no—
394C2—41no—
395C2—81no—
396C2#2144437511yescharacteristic
397C2#2144746511yescharacteristic
398C2C2×#1683955711yescharacteristic
subgroupG/H
centerC22C2×#14519692
commutatorC22×C4×C61C22×C4
FrattiniC2×C22⋊C4C22×D61
FittingC61×#2561C2
socleC22×C61C2×#99
prime radicalC25×C61D4
cosocleC2×C22⋊C4×C61C23
hypercenter#2561D61
hypocenterC61#11845
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #2561

Complements

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

NH ≅ G/Ntype
3. #21443840C2semidirect
4. #21443819C2semidirect
12. C2×C22⋊Q8×C61C4semidirect
14. C61×#397C4semidirect
16. C61×#393C4semidirect
22. C4×C22⋊C4×C61C4semidirect
205. C2×C22⋊Q8Dic61semidirect
208. #397Dic61semidirect
210. #393Dic61semidirect
217. C4×C22⋊C4Dic61semidirect
222. C61#11845semidirect

Maximal subgroups

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

subgroupindex
#214458932
C2×#168394862
#214438402
#214438192
C61×#25612
#1184561