Finite Group Databasebeta

← #29577689

Subgroups — #29577689

Order 15616.

518 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 95 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#21454847C211yesmaximal, characteristic
2#21450815C211yesmaximal, characteristic
3#21444545C211yesmaximal, characteristic, complemented
4#21444159C211yesmaximal, characteristic
5#21443676C211yesmaximal, characteristic, complemented
6#21443674C211yesmaximal, characteristic, complemented
7C61×#1558C211yesmaximal, characteristic, complemented
8C2×#14519766C412yes—
9C2×C4×D244C2211yescharacteristic
10#16839835—22no—
11#16839832—22no—
12#16839819C2211yescharacteristic
13#16839819C2211yescharacteristic
14#16839724—21no—
15C22⋊C8×D61—21no—
16#16839688—21no—
17#16839688—21no—
18#16839669C412yes—
19#16839619—21no—
20#16839619—21no—
21C4×C4⋊Dic61C2211yescharacteristic
22#16839511—22no—
23#16839485—21no—
24#16839485—21no—
25C2×C4⋊Q8×C61C2211yescharacteristic
26C61×#432C2211yescharacteristic
27C61×#432C2211yescharacteristic
28C61×#333—22no—
29C22×D244C2×C411yescharacteristic
30C22×C4×D61D411yescharacteristic
31C22×C4×D61D411yescharacteristic
32C2×#11758189—22no—
33C2×C8×D61—22no—
34#14519766—44no—
35C4×D244—21no—
36C4×D244—21no—
37C4×D244—21no—
38C4×D244—21no—
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61C2×C411yescharacteristic
41C2×#11758198—22no—
42C2×#11758198—22no—
43C2×#11758198D411yescharacteristic
44C2×#11758198D411yescharacteristic
45C2×C4×Dic61C2×C412yes—
46C4⋊#92418—21no—
47C4⋊#92418D412yes—
48C22⋊#92418—21no—
49C22⋊#92418—21no—
50#14519689—22no—
51#14519689—22no—
52#14519688—21no—
53#14519688—21no—
54#14519684—21no—
55#14519684—22no—
56#14519684—22no—
57C22×Q8×C61C2×C412yes—
58C4⋊Q8×C61—21no—
59C4⋊Q8×C61—21no—
60C4⋊Q8×C61—21no—
61C4⋊Q8×C61—21no—
62C2×C4⋊C4×C61—21no—
63C2×C4⋊C4×C61—21no—
64C2×C42×C61C2311yescharacteristic
65C4⋊C8×C61—21no—
66C4⋊C8×C61D412yes—
67C22⋊C8×C61—21no—
68C22⋊C8×C61—21no—
69#97×C61—21no—
70#97×C61—21no—
71C23×D61C22⋊C411yescharacteristic
72C23×D61C22⋊C411yescharacteristic
73C2×Q8×C61—22no—
74C2×Q8×C61—22no—
75C2×Q8×C61—42no—
76C2×Q8×C61—42no—
77C22×C4×C61—22no—
78C22×C4×C61C22×C411yescharacteristic
79C22×C4×C61C2×D411yescharacteristic
80C22×C4×C61C2×D411yescharacteristic
81C22×Dic61—22no—
82C22×Dic61C22⋊C411yescharacteristic
83C22×Dic61C22⋊C411yescharacteristic
84C2×D244—22no—
85C2×D244—21no—
86C2×D244—21no—
87C2×D244—41no—
88C2×C4×D61—21no—
89C2×C4×D61—21no—
90C2×C4×D61—21no—
91C2×C4×D61—21no—
92C2×C4×D61—42no—
93C2×C4×D61—42no—
94C2×C8×C61—22no—
95C2×C8×C61—41no—
96C4⋊C4×C61—41no—
97C4⋊C4×C61—41no—
98C4⋊C4×C61—41no—
99C4⋊C4×C61—41no—
100C42×C61C2×D411yescharacteristic
101C42×C61C2×D411yescharacteristic
102C42×C61#4111yescharacteristic
103C42×C61#4111yescharacteristic
104#11758198—21no—
105#11758198—21no—
106#11758198—21no—
107#11758198—21no—
108#11758198—44no—
109#11758198—44no—
110C4⋊Dic61—21no—
111C4⋊Dic61—21no—
112C4⋊Dic61—42no—
113C4×Dic61—42no—
114C2×#92418—22no—
115C2×#92418—41no—
116#11758189—82no—
117C8×D61—82no—
118C23×C61C2×C22⋊C411yescharacteristic
119C22×D61—21no—
120C22×D61—21no—
121C22×D61—21no—
122C22×D61—21no—
123C22×D61—21no—
124C22×D61—41no—
125C22×D61—41no—
126C22×D61—41no—
127C22×D61—41no—
128C22×D61#10012yes—
129Q8×C61—82no—
130Q8×C61—82no—
131C2×C4×C61—21no—
132C2×C4×C61—21no—
133C2×C4×C61—22no—
134C2×C4×C61—41no—
135C2×C4×C61—42no—
136C2×C4×C61—42no—
137C2×C4×C61—42no—
138C2×C4×C61C4×D411yescharacteristic
139C2×C4×C61C4×D411yescharacteristic
140C2×C4×C61#12011yescharacteristic
141C2×C4×C61#12111yescharacteristic
142C2×C4×C61#12111yescharacteristic
143C2×C4×C61#12311yescharacteristic
144C2×Dic61—21no—
145C2×Dic61—21no—
146C2×Dic61—41no—
147C2×Dic61—41no—
148C2×Dic61—42no—
149C2×Dic61—42no—
150C2×Dic61—42no—
151D244—42no—
152D244—42no—
153C4×D61—42no—
154C4×D61—42no—
155C4×D61—81no—
156C4×D61—81no—
157C8×C61—42no—
158C8×C61—81no—
159#92418—42no—
160#92418—81no—
161#9451—611nomaximal, Sylow
162C22×C61—21no—
163C22×C61—21no—
164C22×C61#38611yescharacteristic
165C22×C61C2×#10011yescharacteristic
166C22×C61#41311yescharacteristic
167C2×D61—22no—
168C2×D61—41no—
169C2×D61—41no—
170C2×D61—42no—
171C2×D61—42no—
172C2×D61—41no—
173C2×D61—41no—
174C2×D61—41no—
175C2×D61—81no—
176C2×D61—81no—
177C4×C61—21no—
178C4×C61—21no—
179C4×C61—21no—
180C4×C61—21no—
181C4×C61—21no—
182C4×C61—82no—
183C4×C61#45512yes—
184Dic61—41no—
185Dic61—41no—
186Dic61—82no—
187#2973—611no—
188#2547—611no—
189#1558—611no—
190#1558D6111yescharacteristic, complemented
191#1466—611no—
192#1285—611no—
193#1285—611no—
194C2×C61—21no—
195C2×C61—21no—
196C2×C61#188511yescharacteristic
197C2×C61#190011yescharacteristic
198C2×C61#190911yescharacteristic
199D61—42no—
200D61—81no—
201C2×C4⋊Q8C2×D6111yescharacteristic
202C2×#124—612no—
203C2×C4×D4—611no—
204#435—1222no—
205#435—1222no—
206#432—611no—
207#432—611no—
208#432C2×D6111yescharacteristic
209#432C2×D6111yescharacteristic
210#407—1221no—
211C2×C22⋊C8—1221no—
212#397—1221no—
213#397—1221no—
214#391—612no—
215#380—1221no—
216#380—1221no—
217C4×C4⋊C4—611no—
218#333—22no—
219#333—1222no—
220#323—1221no—
221#323—1221no—
222C61#945111yescharacteristic, complemented, Sylow
223C22×Q8C4×D6112yes—
224C22×D4—611no—
225C23×C4—611no—
226C23×C4—611no—
227C2×M16—1222no—
228C22×C8—1222no—
229C4⋊Q8—21no—
230C4⋊Q8—21no—
231C4⋊Q8—21no—
232C4⋊Q8—21no—
233#124—2444no—
234C4×D4—1221no—
235C4×D4—1221no—
236C4×D4—1221no—
237C4×D4—1221no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—611no—
241C2×C4⋊C4—1221no—
242C2×C22⋊C4—611no—
243C2×C22⋊C4—611no—
244C2×C22⋊C4—1222no—
245C2×C22⋊C4—1222no—
246C2×C42—612no—
247C2×C42C22×D6111yescharacteristic
248C4⋊C8—21no—
249C4⋊C8—612no—
250C4⋊C8—1221no—
251C4⋊C8C22⋊D6112yes—
252C22⋊C8—21no—
253C22⋊C8—21no—
254C22⋊C8—1222no—
255C22⋊C8—1222no—
256C22⋊C8—1221no—
257C22⋊C8—1221no—
258#97—21no—
259#97—21no—
260#97—1221no—
261#97—1221no—
262#95—1222no—
263#95—1222no—
264#95—1221no—
265C24—611no—
266C24—611no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×D4—1222no—
272C2×D4—1221no—
273C2×D4—1221no—
274C2×D4—2441no—
275C22×C4—22no—
276C22×C4—611no—
277C22×C4—611no—
278C22×C4—1221no—
279C22×C4—1221no—
280C22×C4—1221no—
281C22×C4—1221no—
282C22×C4—1222no—
283C22×C4—2442no—
284C22×C4—2442no—
285C22×C4C2×C4×D6111yescharacteristic
286C22×C4D4×D6111yescharacteristic
287C22×C4D4×D6111yescharacteristic
288M16—4882no—
289C2×C8—22no—
290C2×C8—41no—
291C2×C8—1222no—
292C2×C8—4882no—
293C2×C8—2441no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—1221no—
299C4⋊C4—1221no—
300C4⋊C4—2442no—
301C22⋊C4—1221no—
302C22⋊C4—1221no—
303C22⋊C4—1221no—
304C22⋊C4—1221no—
305C22⋊C4—2444no—
306C22⋊C4—2444no—
307C42—2442no—
308C42#1175822211yescharacteristic
309C42D4×D6111yescharacteristic
310C42#1175822411yescharacteristic
311C42C2×C22⋊D6111yescharacteristic
312C23—612no—
313C23—1221no—
314C23—1221no—
315C23—1221no—
316C23—1221no—
317C23—1221no—
318C23—2441no—
319C23—2441no—
320C23—2441no—
321C23—2441no—
322C23C22⋊C4×D6111yescharacteristic
323Q8—82no—
324Q8—82no—
325D4—2442no—
326D4—2442no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—1221no—
335C2×C4—1221no—
336C2×C4—4881no—
337C2×C4—4881no—
338C2×C4—2442no—
339C2×C4—2442no—
340C2×C4—2442no—
341C2×C4—2442no—
342C2×C4—2442no—
343C2×C4—2441no—
344C2×C4—2441no—
345C2×C4#1451973711yescharacteristic
346C2×C4C4×C22⋊D6111yescharacteristic
347C2×C4#1451974511yescharacteristic
348C2×C4#1451974811yescharacteristic
349C2×C4#1451975011yescharacteristic
350C2×C4#1451975911yescharacteristic
351C8—42no—
352C8—81no—
353C8—4881no—
354C8—2442no—
355C22—21no—
356C22—21no—
357C22—1222no—
358C22—4881no—
359C22—4881no—
360C22—2442no—
361C22—2442no—
362C22—2441no—
363C22—2441no—
364C22—2441no—
365C22—2441no—
366C22—2441no—
367C22#1683963811yescharacteristic
368C22#100×D6111yescharacteristic
369C22#1683974311yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—82no—
376C4—4882no—
377C4—2441no—
378C4—2441no—
379C4#1683993812yes—
380C2—21no—
381C2—21no—
382C2—4881no—
383C2—2442no—
384C2#2144620411yescharacteristic
385C2#2144627911yescharacteristic
386C2#2144633811yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1558C2
socleC22×C61#386
prime radicalC22×D244C2×C4
cosocleC2×C42×C61C23
hypercenter#1558D61
hypocenterC61#9451
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1558

Complements

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

NH ≅ G/Ntype
3. #21444545C2semidirect
5. #21443676C2semidirect
6. #21443674C2semidirect
7. C61×#1558C2semidirect
190. #1558D61semidirect
222. C61#9451semidirect

Maximal subgroups

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

subgroupindex
#214548472
#214508152
#214445452
#214441592
#214436762
#214436742
C61×#15582
#945161