Finite Group Databasebeta

← #29576911

Subgroups — #29576911

Order 15616.

612 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 97 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#21455914C211yesmaximal, characteristic
2#21448179C211yesmaximal, characteristic
3#21444331C211yesmaximal, characteristic, complemented
4#21444263C211yesmaximal, characteristic
5#21443713C211yesmaximal, characteristic, complemented
6#21443675C211yesmaximal, characteristic, complemented
7C61×#1986C211yesmaximal, characteristic, complemented
8C2×C4×D244C2211yescharacteristic
9C2×#14519739—21no—
10#16840095—21no—
11#16840039C412yescomplemented
12#16839964C412yescomplemented
13#16839912—21no—
14#16839860—22no—
15#16839819C2211yescharacteristic
16#16839819C2211yescharacteristic
17C4⋊C4×Dic61—21no—
18C4×#11758198—22no—
19#16839609—21no—
20#16839608C2211yescharacteristic
21#95×D61—21no—
22#16839499—21no—
23#16839499—21no—
24#16839491—22no—
25C61×#533C2211yescharacteristic
26C4×M16×C61C2211yescharacteristic
27C4×C4⋊C4×C61C2211yescharacteristic
28C61×#339—22no—
29C22×D244D411yescharacteristic
30C22×C4×D61—22no—
31C22×C4×D61—21no—
32C2×#11758201—22no—
33C4×D244—21no—
34C4×D244—21no—
35C4×D244C2×C412yes—
36C4×D244D412yes—
37#14519739—28no—
38C2×C4⋊Dic61D411yescharacteristic
39C2×#11758198—22no—
40C2×#11758198—21no—
41C2×C4×Dic61—21no—
42C2×C4×Dic61—22no—
43#14519710—22no—
44C4⋊#92418—21no—
45C4⋊#92418C2×C412yescomplemented
46#14519701—22no—
47#14519697—22no—
48C22⋊#92418—21no—
49C22⋊#92418—21no—
50#14519688—21no—
51#14519688—21no—
52#14519684—22no—
53#14519684—21no—
54#14519684D411yescharacteristic
55#14519684D411yescharacteristic
56C2×M16×C61—21no—
57C4⋊Q8×C61C2×C411yescharacteristic, complemented
58C4⋊Q8×C61C2×C411yescharacteristic, complemented
59#125×C61—21no—
60C22⋊Q8×C61—21no—
61C22⋊Q8×C61—21no—
62C4×Q8×C61—21no—
63C4×Q8×C61—21no—
64#117×C61—21no—
65C2×C4⋊C4×C61—21no—
66C2×C42×C61D412yes—
67C2×C42×C61C2311yescharacteristic
68#97×C61—21no—
69C4×C8×C61—21no—
70#95×C61—21no—
71C23×D61—21no—
72C2×Q8×C61—21no—
73C2×Q8×C61—21no—
74C22×C4×C61—22no—
75C22×C4×C61C2×D411yescharacteristic
76C22×C4×C61C2×D411yescharacteristic
77C22×C4×C61C2×D411yescharacteristic
78C22×Dic61—21no—
79C22×Dic61—21no—
80C22×Dic61—21no—
81C2×D244—21no—
82C2×D244—24no—
83C2×D244C22⋊C412yes—
84C2×C4×D61—28no—
85C2×C4×D61—22no—
86C2×C4×D61—22no—
87C2×C4×D61—82no—
88C2×C4×D61—82no—
89C2×C4×D61—82no—
90M16×C61—22no—
91M16×C61—41no—
92C2×C8×C61—41no—
93C4⋊C4×C61—24no—
94C4⋊C4×C61—21no—
95C4⋊C4×C61—21no—
96C4⋊C4×C61—21no—
97C4⋊C4×C61—21no—
98C4⋊C4×C61—41no—
99C4⋊C4×C61—41no—
100C22⋊C4×C61—41no—
101C42×C61—42no—
102C42×C61—41no—
103C42×C61C22×C411yescharacteristic
104C42×C61C2×D411yescharacteristic
105C42×C61#4111yescharacteristic
106C42×C61#4111yescharacteristic
107#11758201—44no—
108#11758198—28no—
109#11758198—22no—
110#11758198—22no—
111C4⋊Dic61—21no—
112C4⋊Dic61C22⋊C412yes—
113C4×Dic61—24no—
114C4×Dic61—82no—
115C2×#92418—22no—
116C2×#92418—41no—
117C23×C61#12011yescharacteristic
118C22×D61—24no—
119C22×D61—22no—
120C22×D61—42no—
121C22×D61—41no—
122C22×D61—41no—
123Q8×C61—42no—
124Q8×C61—81no—
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—22no—
128C2×C4×C61—24no—
129C2×C4×C61—41no—
130C2×C4×C61—42no—
131C2×C4×C61—42no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61C2×C22⋊C411yescharacteristic
135C2×C4×C61C4×D411yescharacteristic
136C2×C4×C61C4×D411yescharacteristic
137C2×C4×C61#12111yescharacteristic
138C2×C4×C61#12111yescharacteristic
139C2×C4×C61#12311yescharacteristic
140C2×Dic61—22no—
141C2×Dic61—22no—
142C2×Dic61—22no—
143C2×Dic61—41no—
144C2×Dic61—42no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61—41no—
148D244—24no—
149D244—22no—
150D244#10414yes—
151C4×D61—82no—
152C4×D61—82no—
153C4×D61—88no—
154C8×C61—42no—
155#92418—42no—
156#92418—81no—
157#9323—611nomaximal, Sylow
158C22×C61—21no—
159C22×C61—21no—
160C22×C61#38611yescharacteristic
161C22×C61#45511yescharacteristic
162C22×C61#45611yescharacteristic
163C2×D61—24no—
164C2×D61—44no—
165C2×D61—44no—
166C2×D61—42no—
167C2×D61—42no—
168C4×C61—21no—
169C4×C61—21no—
170C4×C61—21no—
171C4×C61—21no—
172C4×C61—21no—
173C4×C61—44no—
174C4×C61—81no—
175C4×C61—81no—
176C4×C61C2×#10412yes—
177Dic61—42no—
178Dic61—42no—
179Dic61—42no—
180#3110—611no—
181#2275—611no—
182#1986D6111yescharacteristic, complemented
183#1506—611no—
184#1490—611no—
185#1300—611no—
186#1285—611no—
187C2×C61—21no—
188C2×C61—21no—
189C2×C61#188211yescharacteristic
190C2×C61#189611yescharacteristic
191C2×C61#190411yescharacteristic
192D61—44no—
193#533C2×D6111yescharacteristic
194C2×C4×D4—611no—
195C2×C4×D4—1221no—
196#484—1221no—
197#474—612no—
198#461—612no—
199#450—1221no—
200#440—1222no—
201#432—611no—
202#432—611no—
203C4×M16C2×D6111yescharacteristic
204C4×C4⋊C4—1221no—
205C4×C4⋊C4C2×D6111yescharacteristic
206C4×C22⋊C4—1221no—
207C4×C22⋊C4—1222no—
208#376—611no—
209C2×#95—1221no—
210#339—22no—
211#328—1221no—
212#328—1221no—
213D4⋊C8—1222no—
214C61#932311yescharacteristic, complemented, Sylow
215C22×D4—611no—
216C23×C4—1222no—
217C23×C4—1221no—
218C2×SD16—1222no—
219C2×M16—21no—
220C4⋊Q8C4×D6111yescharacteristic, complemented
221C4⋊Q8C4×D6111yescharacteristic, complemented
222#125—21no—
223C22⋊Q8—21no—
224C22⋊Q8—21no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×D4—612no—
228C4×D4—612no—
229C4×D4—1228no—
230C4×D4—1221no—
231C4×D4—1221no—
232#117—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—611no—
235C2×C22⋊C4—1222no—
236C2×C22⋊C4—1221no—
237C2×C42—1221no—
238C2×C42—1222no—
239C2×C42C22⋊D6112yes—
240C2×C42C22×D6111yescharacteristic
241#107—1222no—
242C4⋊C8—612no—
243C4⋊C8—1221no—
244#103—1222no—
245#102—1222no—
246C22⋊C8—1221no—
247C22⋊C8—1221no—
248#97—21no—
249#97—1221no—
250#97—1221no—
251C4×C8—21no—
252#95—21no—
253#95—611no—
254#95—611no—
255#95—1222no—
256#95—1221no—
257C24—1221no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×D4—612no—
261C2×D4—1224no—
262C2×D4—1221no—
263C22×C4—22no—
264C22×C4—1228no—
265C22×C4—1221no—
266C22×C4—1221no—
267C22×C4—1221no—
268C22×C4—1222no—
269C22×C4—1222no—
270C22×C4—4882no—
271C22×C4—4882no—
272C22×C4—4882no—
273C22×C4D4×D6111yescharacteristic
274C22×C4D4×D6111yescharacteristic
275C22×C4C2×C22⋊D6111yescharacteristic
276SD16—2444no—
277M16—22no—
278M16—41no—
279C2×C8—41no—
280C2×C8—1222no—
281C2×C8—2441no—
282C4⋊C4—24no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—612no—
290C4⋊C4—1221no—
291C22⋊C4—41no—
292C22⋊C4—1228no—
293C22⋊C4—1222no—
294C22⋊C4—1222no—
295C42—42no—
296C42—41no—
297C42—1224no—
298C42—4882no—
299C42C2×C4×D6111yescharacteristic
300C42#1175822211yescharacteristic
301C42D4×D6111yescharacteristic
302C42#1175822411yescharacteristic
303C23—1224no—
304C23—1222no—
305C23—2442no—
306C23—2441no—
307C23—2441no—
308C23#1451974511yescharacteristic
309Q8—42no—
310Q8—81no—
311D4—614no—
312D4—1224no—
313D4—1222no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—24no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—1222no—
324C2×C4—1222no—
325C2×C4—1222no—
326C2×C4—4882no—
327C2×C4—4882no—
328C2×C4—4888no—
329C2×C4—2441no—
330C2×C4—2442no—
331C2×C4—2441no—
332C2×C4—2441no—
333C2×C4—2441no—
334C2×C4C22⋊C4×D6111yescharacteristic
335C2×C4#1451973711yescharacteristic
336C2×C4C4×C22⋊D6111yescharacteristic
337C2×C4#1451974811yescharacteristic
338C2×C4#1451975011yescharacteristic
339C2×C4#1451975911yescharacteristic
340C8—42no—
341C8—4881no—
342C8—2442no—
343C22—21no—
344C22—21no—
345C22—1224no—
346C22—2444no—
347C22—2444no—
348C22—2442no—
349C22—2442no—
350C22#1683963811yescharacteristic
351C22#1683993911yescharacteristic
352C22#1683994411yescharacteristic
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—44no—
359C4—81no—
360C4—81no—
361C4—2442no—
362C4—2442no—
363C4—2442no—
364C4#104×D6112yes—
365C2—21no—
366C2—21no—
367C2—2444no—
368C2#2144619111yescharacteristic
369C2#2144625911yescharacteristic
370C2#2144630311yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1986C2
socleC22×C61#386
prime radicalC22×D244D4
cosocleC2×C42×C61C23
hypercenter#1986D61
hypocenterC61#9323
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1986

Complements

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

NH ≅ G/Ntype
3. #21444331C2semidirect
5. #21443713C2semidirect
6. #21443675C2semidirect
7. C61×#1986C2semidirect
11. #16840039C4semidirect
12. #16839964C4semidirect
45. C4⋊#92418C2×C4semidirect
57. C4⋊Q8×C61C2×C4semidirect
58. C4⋊Q8×C61C2×C4semidirect
182. #1986D61semidirect
214. C61#9323semidirect
220. C4⋊Q8C4×D61semidirect
221. C4⋊Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214559142
#214481792
#214443312
#214442632
#214437132
#214436752
C61×#19862
#932361