Finite Group Databasebeta

← #29590735

Subgroups — #29590735

Order 15616.

496 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 87 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
2#21444146C211yesmaximal, characteristic
3#21443843C212yesmaximal, complemented
4#21443840C212yesmaximal, complemented
5C61×#2626C211yesmaximal, characteristic
6#16839726—21no—
7#16839618—21no—
8#16839529—21no—
9#16839529C2212yes—
10#16839486—21no—
11#16839486C2212yes—
12C2×C22⋊Q8×C61C411yescharacteristic, complemented
13C22×C22⋊C4×C61C2211yescharacteristic
14C61×#399C411yescharacteristic, complemented
15C61×#399C2211yescharacteristic
16C61×#396—21no—
17C61×#396—21no—
18C61×#393C411yescharacteristic, complemented
19C61×#393C2211yescharacteristic
20C61×#385—21no—
21C61×#380C411yescharacteristic, complemented
22C61×#379—21no—
23C2×#11758194—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×C61—21no—
37C2×C4⋊C4×C61—21no—
38C2×C4⋊C4×C61—21no—
39C2×C4⋊C4×C61C2×C411yescharacteristic
40C2×C22⋊C4×C61—21no—
41C2×C22⋊C4×C61—21no—
42C2×C22⋊C4×C61—21no—
43C2×C22⋊C4×C61—21no—
44C2×C22⋊C4×C61—21no—
45C2×C22⋊C4×C61—21no—
46C2×C22⋊C4×C61—41no—
47C2×C22⋊C4×C61—41no—
48C2×C22⋊C4×C61C2×C411yescharacteristic
49C2×C22⋊C4×C61D411yescharacteristic
50C2×C22⋊C4×C61D411yescharacteristic
51C2×C22⋊C4×C61C2311yescharacteristic
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—81no—
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—81no—
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—
84C4⋊C4×C61—81no—
85C4⋊C4×C61—81no—
86C4⋊C4×C61—81no—
87C22⋊C4×C61—41no—
88C22⋊C4×C61—41no—
89C22⋊C4×C61—41no—
90C22⋊C4×C61—41no—
91C22⋊C4×C61—42no—
92C22⋊C4×C61—41no—
93C22⋊C4×C61—41no—
94C22⋊C4×C61—81no—
95C22⋊C4×C61—81no—
96C22⋊C4×C61—81no—
97C22⋊C4×C61—81no—
98C22⋊Dic61—42no—
99C22⋊Dic61—42no—
100C22⋊Dic61—82no—
101C22⋊Dic61—82no—
102#11758194—82no—
103C2×#92418—42no—
104C23×C61—21no—
105C23×C61—21no—
106C23×C61—21no—
107C23×C61—21no—
108C23×C61—21no—
109C23×C61—21no—
110C23×C61—21no—
111C23×C61—41no—
112C23×C61—41no—
113C23×C61—42no—
114C23×C61—42no—
115C23×C61—41no—
116C23×C61—41no—
117C23×C61—41no—
118C23×C61—41no—
119C23×C61—42no—
120C23×C61—42no—
121C23×C61—81no—
122C23×C61—81no—
123C23×C61—84no—
124C23×C61—84no—
125C23×C61#9911yescharacteristic
126C23×C61#9911yescharacteristic
127C23×C61C2×C22⋊C411yescharacteristic
128C23×C61#13712yes—
129Q8×C61—82no—
130C2×C4×C61—21no—
131C2×C4×C61—21no—
132C2×C4×C61—21no—
133C2×C4×C61—21no—
134C2×C4×C61—42no—
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—41no—
150C2×C4×C61—41no—
151C2×C4×C61—81no—
152C2×C4×C61—81no—
153C2×C4×C61—81no—
154C2×Dic61—82no—
155C2×Dic61—82no—
156C2×Dic61—82no—
157#92418—82no—
158#11811—611nomaximal, Sylow
159C22×C61—21no—
160C22×C61—21no—
161C22×C61—21no—
162C22×C61—21no—
163C22×C61—21no—
164C22×C61—21no—
165C22×C61—22no—
166C22×C61—21no—
167C22×C61—21no—
168C22×C61—42no—
169C22×C61—42no—
170C22×C61—42no—
171C22×C61—42no—
172C22×C61—41no—
173C22×C61—41no—
174C22×C61—41no—
175C22×C61—41no—
176C22×C61—41no—
177C22×C61—81no—
178C22×C61—81no—
179C22×C61—84no—
180C22×C61—84no—
181C22×C61C2×#9911yescharacteristic
182C22×C61#41911yescharacteristic
183C22×C61#42111yescharacteristic
184C4×C61—41no—
185C4×C61—41no—
186C4×C61—41no—
187C4×C61—41no—
188C4×C61—81no—
189C4×C61—81no—
190C4×C61—81no—
191C4×C61—81no—
192Dic61—162no—
193#2626D6111yescharacteristic
194#1819—611no—
195#1462—611no—
196#1352—612no—
197#1351—612no—
198C2×C61—21no—
199C2×C61—21no—
200C2×C61—42no—
201C2×C61—41no—
202C2×C61—41no—
203C2×C61—81no—
204C2×C61#151111yescharacteristic
205C2×C61#212111yescharacteristic
206C2×C61#212711yescharacteristic
207C2×C22⋊Q8Dic6111yescharacteristic, complemented
208C22×C22⋊C4C2×D6111yescharacteristic
209#407—1221no—
210#399Dic6111yescharacteristic, complemented
211#399C2×D6111yescharacteristic
212#396—21no—
213#396—21no—
214#393Dic6111yescharacteristic, complemented
215#393C2×D6111yescharacteristic
216#385—21no—
217#380Dic6111yescharacteristic, complemented
218#379—21no—
219#379—1221no—
220#342—612no—
221#342—1221no—
222#323—612no—
223#323—1221no—
224C61#1181111yescharacteristic, complemented, Sylow
225C25C22⋊D6111yescharacteristic
226C22×Q8C2×Dic6111yescharacteristic
227C23×C4C2×Dic6111yescharacteristic
228C23×C4C22⋊D6111yescharacteristic
229C2×M16—2441no—
230C22⋊Q8—41no—
231C22⋊Q8—41no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4C2×Dic6111yescharacteristic
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—1222no—
246C2×C22⋊C4—2441no—
247C2×C22⋊C4—2441no—
248C2×C22⋊C4C2×Dic6111yescharacteristic
249C2×C22⋊C4C22⋊D6111yescharacteristic
250C2×C22⋊C4C22⋊D6111yescharacteristic
251C2×C22⋊C4C22×D6111yescharacteristic
252C22⋊C8—1222no—
253C22⋊C8—2441no—
254#95—21no—
255#95—21no—
256#95C2×Dic6111yescharacteristic
257#95C2×Dic6111yescharacteristic
258C24—21no—
259C24—21no—
260C24—22no—
261C24—41no—
262C24—81no—
263C24—81no—
264C24C22⋊Dic6111yescharacteristic
265C24C22⋊Dic6111yescharacteristic
266C24C2×C22⋊D6111yescharacteristic
267C2×Q8—41no—
268C2×Q8—81no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—81no—
278C22×C4—81no—
279C22×C4—2442no—
280C22×C4C22⋊Dic6111yescharacteristic
281C22×C4C22⋊Dic6111yescharacteristic
282C22×C4C22×Dic6111yescharacteristic
283C22×C4C2×C22⋊D6111yescharacteristic
284M16—4882no—
285C2×C8—2442no—
286C4⋊C4—41no—
287C4⋊C4—42no—
288C4⋊C4—81no—
289C4⋊C4—81no—
290C4⋊C4—81no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—42no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—81no—
299C22⋊C4—81no—
300C22⋊C4—81no—
301C22⋊C4—81no—
302C22⋊C4—4882no—
303C22⋊C4—4882no—
304C22⋊C4—2442no—
305C22⋊C4—2442no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—41no—
314C23—41no—
315C23—42no—
316C23—42no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—42no—
322C23—42no—
323C23—81no—
324C23—81no—
325C23—84no—
326C23—84no—
327C23#1451969211yescharacteristic
328C23#1451969211yescharacteristic
329C23C2×C22⋊Dic6111yescharacteristic
330C23#1451980812yes—
331Q8—82no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
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—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—4882no—
357C2×C4—4882no—
358C2×C4—4882no—
359C8—4882no—
360C22—21no—
361C22—21no—
362C22—22no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—81no—
379C22—81no—
380C22—84no—
381C22—84no—
382C22C2×#1451969211yescharacteristic
383C22#1683977111yescharacteristic
384C22#1683977911yescharacteristic
385C4—41no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—81no—
390C4—81no—
391C4—81no—
392C4—81no—
393C4—9762no—
394C2—21no—
395C2—21no—
396C2—42no—
397C2—41no—
398C2—41no—
399C2—81no—
400C2#2144435211yescharacteristic
401C2#2144746511yescharacteristic
402C2#2144748811yescharacteristic
subgroupG/H
centerC22C2×#14519692
commutatorC22×C4×C61C22×C4
FrattiniC2×C22⋊C4C22×D61
FittingC61×#2626C2
socleC22×C61C2×#99
prime radicalC25×C61D4
cosocleC2×C22⋊C4×C61C23
hypercenter#2626D61
hypocenterC61#11811
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #2626

Complements

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

NH ≅ G/Ntype
3. #21443843C2semidirect
4. #21443840C2semidirect
12. C2×C22⋊Q8×C61C4semidirect
14. C61×#399C4semidirect
18. C61×#393C4semidirect
21. C61×#380C4semidirect
207. C2×C22⋊Q8Dic61semidirect
210. #399Dic61semidirect
214. #393Dic61semidirect
217. #380Dic61semidirect
224. C61#11811semidirect

Maximal subgroups

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

subgroupindex
#214458932
#214441462
#214438432
#214438402
C61×#26262
#1181161