Finite Group Databasebeta

← #29583248

Subgroups — #29583248

Order 15616.

414 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 75 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#21451480C211yesmaximal, characteristic
2#21445918C211yesmaximal, characteristic
3#21445896C211yesmaximal, characteristic
4#21443728C211yesmaximal, characteristic
5#21443673C211yesmaximal, characteristic
6#21443672C211yesmaximal, characteristic
7C61×#1826C211yesmaximal, characteristic
8#16839726C2211yescharacteristic
9#16839698—21no—
10#16839674—21no—
11#16839668—21no—
12#16839666—21no—
13#16839646—21no—
14#16839645—21no—
15#16839625C2211yescharacteristic
16#16839625C2211yescharacteristic
17C4×C22⋊Dic61C2211yescharacteristic
18#16839529—21no—
19#16839529—21no—
20#16839521—21no—
21#16839521—21no—
22C2×#117×C61C2211yescharacteristic
23C61×#407C2211yescharacteristic
24C4×C22⋊C4×C61C2211yescharacteristic
25C61×#339—21no—
26C61×#339—21no—
27C22×Dic122—21no—
28C2×#11758194—21no—
29C2×C4⋊Dic61—21no—
30C2×#11758196—21no—
31C2×#11758196—21no—
32C2×C22⋊Dic61—21no—
33C2×C22⋊Dic61—21no—
34C2×C22⋊Dic61D411yescharacteristic
35C2×C22⋊Dic61D411yescharacteristic
36C2×C4×Dic61D411yescharacteristic
37C2×C4×Dic61D411yescharacteristic
38C22⋊#92418—21no—
39C22⋊#92418—21no—
40#14519684—21no—
41#14519684—21no—
42#14519684—21no—
43#14519684—21no—
44#14519684—21no—
45#14519684—21no—
46#14519684—21no—
47#14519684—21no—
48#14519684—21no—
49C23×C4×C61C2311yescharacteristic
50C2×M16×C61—21no—
51#117×C61—21no—
52#117×C61—21no—
53#117×C61—21no—
54#117×C61—21no—
55C2×C4⋊C4×C61Q811yescharacteristic, complemented
56C2×C4⋊C4×C61Q811yescharacteristic, complemented
57C2×C22⋊C4×C61—21no—
58C2×C22⋊C4×C61—21no—
59C2×C42×C61—21no—
60C2×C42×C61D411yescharacteristic
61C2×C42×C61D411yescharacteristic
62C22⋊C8×C61—21no—
63C22⋊C8×C61—21no—
64#95×C61—21no—
65C24×C61C2×D411yescharacteristic
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61—41no—
71C22×C4×C61—41no—
72C22×C4×C61C2×D411yescharacteristic
73C22×C4×C61C2×D411yescharacteristic
74C22×C4×C61C2×Q811yescharacteristic
75C22×C4×C61#4111yescharacteristic
76C22×C4×C61#4111yescharacteristic
77C22×C4×C61#4111yescharacteristic
78C22×Dic61—21no—
79C22×Dic61—21no—
80C22×Dic61—21no—
81C22×Dic61—21no—
82C22×Dic61—41no—
83C2×Dic122—42no—
84C2×Dic122—42no—
85C2×Dic122—81no—
86M16×C61—41no—
87M16×C61—41no—
88C2×C8×C61—41no—
89C4⋊C4×C61—21no—
90C4⋊C4×C61—21no—
91C4⋊C4×C61—21no—
92C4⋊C4×C61—21no—
93C22⋊C4×C61—42no—
94C22⋊C4×C61—41no—
95C22⋊C4×C61—41no—
96C42×C61—41no—
97C42×C61—41no—
98C42×C61—41no—
99C42×C61—41no—
100C22⋊Dic61—42no—
101C22⋊Dic61—41no—
102C22⋊Dic61—41no—
103C22⋊Dic61—41no—
104C22⋊Dic61—41no—
105C4⋊Dic61—42no—
106#11758196—42no—
107#11758196—81no—
108C4×Dic61—41no—
109C4×Dic61—41no—
110#11758194—41no—
111#11758194—41no—
112C2×#92418—41no—
113C23×C61—21no—
114C23×C61—21no—
115C23×C61—41no—
116C23×C61—41no—
117C23×C61#12011yescharacteristic
118C23×C61C22⋊Q811yescharacteristic
119C23×C61#12411yescharacteristic
120C2×C4×C61—21no—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—22no—
125C2×C4×C61—41no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—81no—
137C2×C4×C61—81no—
138C2×C4×C61C22⋊Q811yescharacteristic
139C2×C4×C61C22⋊Q811yescharacteristic
140C2×C4×C61#12411yescharacteristic
141C2×C4×C61#12411yescharacteristic
142C2×Dic61—22no—
143C2×Dic61—41no—
144C2×Dic61—41no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61—41no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61—41no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61—81no—
155C2×Dic61—81no—
156C2×Dic61—81no—
157Dic122—84no—
158C8×C61—81no—
159#92418—81no—
160#10387—611nomaximal, Sylow
161C22×C61—21no—
162C22×C61—21no—
163C22×C61—21no—
164C22×C61—21no—
165C22×C61—21no—
166C22×C61—21no—
167C22×C61—21no—
168C22×C61—21no—
169C22×C61—81no—
170C22×C61—81no—
171C22×C61#39311yescharacteristic
172C22×C61#45811yescharacteristic
173C22×C61#45811yescharacteristic
174C4×C61—21no—
175C4×C61—21no—
176C4×C61—21no—
177C4×C61—21no—
178C4×C61—42no—
179C4×C61—81no—
180C4×C61—81no—
181C4×C61—81no—
182Dic61—42no—
183Dic61—81no—
184Dic61—81no—
185Dic61—81no—
186Dic61—161no—
187#2607—611no—
188#1826—611no—
189#1826D6111yescharacteristic
190#1820—611no—
191#1306—611no—
192#1284—611no—
193#1284—611no—
194C2×C61—21no—
195C2×C61—21no—
196C2×C61—41no—
197C2×C61—41no—
198C2×C61#203011yescharacteristic
199C2×C61#203111yescharacteristic
200C2×C61#203111yescharacteristic
201C2×#117C2×D6111yescharacteristic
202#407—611no—
203#407C2×D6111yescharacteristic
204#399—1221no—
205#393—1221no—
206#391—1221no—
207#391—1221no—
208#387—1221no—
209#387—1221no—
210#381—611no—
211#381—611no—
212C4×C22⋊C4—611no—
213C4×C22⋊C4C2×D6111yescharacteristic
214#342—1221no—
215#342—1221no—
216#339—21no—
217#339—21no—
218#339—1221no—
219#339—1221no—
220C61#1038711yescharacteristic, complemented, Sylow
221C22×Q8—1221no—
222C23×C4C22×D6111yescharacteristic
223C2×M16—21no—
224C2×M16—1221no—
225#117—21no—
226#117—21no—
227#117—21no—
228#117—21no—
229C2×C4⋊C4—1221no—
230C2×C4⋊C4—1221no—
231C2×C4⋊C4—1221no—
232C2×C4⋊C4Dic12211yescharacteristic, complemented
233C2×C4⋊C4Dic12211yescharacteristic, complemented
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—611no—
237C2×C22⋊C4—611no—
238C2×C22⋊C4—1221no—
239C2×C22⋊C4—1221no—
240C2×C42—21no—
241C2×C42—611no—
242C2×C42—611no—
243C2×C42C22⋊D6111yescharacteristic
244C2×C42C22⋊D6111yescharacteristic
245C22⋊C8—21no—
246C22⋊C8—21no—
247C22⋊C8—1221no—
248C22⋊C8—1221no—
249#95—21no—
250#95—1221no—
251#95—1221no—
252#95—1221no—
253#95—1221no—
254#95—1221no—
255#95—1221no—
256#95—1221no—
257#95—1221no—
258#95—1221no—
259C24D4×D6111yescharacteristic
260C2×Q8—4881no—
261C2×Q8—2442no—
262C2×Q8—2442no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—1221no—
270C22×C4—1221no—
271C22×C4—1221no—
272C22×C4—1221no—
273C22×C4—2441no—
274C22×C4C2×Dic12211yescharacteristic
275C22×C4#1175822211yescharacteristic
276C22×C4D4×D6111yescharacteristic
277C22×C4#1175822411yescharacteristic
278C22×C4#1175822411yescharacteristic
279C22×C4C2×C22⋊D6111yescharacteristic
280M16—41no—
281M16—41no—
282M16—2441no—
283M16—2441no—
284C2×C8—41no—
285C2×C8—2441no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—4881no—
291C4⋊C4—2442no—
292C4⋊C4—2442no—
293C22⋊C4—42no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—2442no—
297C22⋊C4—2441no—
298C22⋊C4—2441no—
299C22⋊C4—2441no—
300C22⋊C4—2441no—
301C42—41no—
302C42—41no—
303C42—41no—
304C42—41no—
305C42—2441no—
306C42—2441no—
307C23—21no—
308C23—21no—
309C23—41no—
310C23—41no—
311C23#1451974511yescharacteristic
312C23#1451975411yescharacteristic
313C23#1451976311yescharacteristic
314Q8—4884no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—1222no—
334C2×C4—4881no—
335C2×C4—4881no—
336C2×C4—4881no—
337C2×C4—2441no—
338C2×C4—2441no—
339C2×C4—2441no—
340C2×C4—2441no—
341C2×C4—2441no—
342C2×C4—2441no—
343C2×C4—2441no—
344C2×C4—2441no—
345C2×C4—2441no—
346C2×C4—2441no—
347C2×C4—2441no—
348C2×C4#1451975411yescharacteristic
349C2×C4#1451975611yescharacteristic
350C2×C4#1451976311yescharacteristic
351C2×C4#1451976411yescharacteristic
352C8—81no—
353C8—4881no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—81no—
363C22—81no—
364C22#1683967411yescharacteristic
365C22#1683995111yescharacteristic
366C22#1683995111yescharacteristic
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—42no—
372C4—81no—
373C4—81no—
374C4—81no—
375C4—9761no—
376C4—4881no—
377C4—4881no—
378C4—4881no—
379C4—2442no—
380C2—21no—
381C2—21no—
382C2—41no—
383C2—41no—
384C2#2144698611yescharacteristic
385C2#2144698911yescharacteristic
386C2#2144699111yescharacteristic
subgroupG/H
centerC22#16839674
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#1826C2
socleC22×C61#393
prime radicalC24×C61C2×D4
hypercenter#1826D61
hypocenterC61#10387
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1826

Complements

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

NH ≅ G/Ntype
55. C2×C4⋊C4×C61Q8semidirect
56. C2×C4⋊C4×C61Q8semidirect
220. C61#10387semidirect
232. C2×C4⋊C4Dic122semidirect
233. C2×C4⋊C4Dic122semidirect

Maximal subgroups

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

subgroupindex
#214514802
#214459182
#214458962
#214437282
#214436732
#214436722
C61×#18262
#1038761