Finite Group Databasebeta

← #29577699

Subgroups — #29577699

Order 15616.

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

normal subgroups only

This group has 498 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#21450993C211yesmaximal, characteristic
3#21444536C211yesmaximal, characteristic, complemented
4#21444159C211yesmaximal, characteristic
5#21443676C211yesmaximal, characteristic, complemented
6#21443674C211yesmaximal, characteristic, complemented
7C61×#1556C211yesmaximal, characteristic, complemented
8C2×C4×D244C2211yescharacteristic
9#16839835—22no—
10#16839832—22no—
11#16839819C2211yescharacteristic
12#16839819C2211yescharacteristic
13#16839724—21no—
14C22⋊C8×D61—21no—
15#16839673—21no—
16#16839673—21no—
17#16839663C412yes—
18#16839627C412yes—
19#16839619—21no—
20#16839619—21no—
21C4×C4⋊Dic61C2211yescharacteristic
22#16839508—22no—
23#16839485—21no—
24#16839485—21no—
25C2×C4⋊Q8×C61C2211yescharacteristic
26C61×#432C2211yescharacteristic
27C61×#432C2211yescharacteristic
28C61×#332—22no—
29C22×D244C2×C411yescharacteristic
30C22×C4×D61D411yescharacteristic
31C22×C4×D61D411yescharacteristic
32C2×#11758189—22no—
33C2×C8×D61—22no—
34C4×D244—21no—
35C4×D244—21no—
36C4×D244—21no—
37C4×D244—21no—
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61C2×C411yescharacteristic
40C2×#11758198—22no—
41C2×#11758198—22no—
42C2×#11758198D411yescharacteristic
43C2×#11758198D411yescharacteristic
44C2×C4×Dic61C2×C412yes—
45C4⋊#92418—21no—
46C4⋊#92418D412yes—
47C22⋊#92418—21no—
48C22⋊#92418—21no—
49#14519689—22no—
50#14519689—22no—
51#14519688—21no—
52#14519688—21no—
53#14519684—21no—
54#14519684—22no—
55#14519684—22no—
56C22×Q8×C61—21no—
57C4⋊Q8×C61—21no—
58C4⋊Q8×C61—21no—
59C4⋊Q8×C61—21no—
60C4⋊Q8×C61—21no—
61C2×C4⋊C4×C61—21no—
62C2×C4⋊C4×C61C2×C412yes—
63C2×C42×C61C2311yescharacteristic
64C4⋊C8×C61—21no—
65C4⋊C8×C61D412yes—
66C22⋊C8×C61—21no—
67C22⋊C8×C61—21no—
68#97×C61—21no—
69#97×C61—21no—
70C23×D61C22⋊C411yescharacteristic
71C23×D61C22⋊C411yescharacteristic
72C2×Q8×C61—41no—
73C2×Q8×C61—41no—
74C2×Q8×C61—82no—
75C22×C4×C61—22no—
76C22×C4×C61C22×C411yescharacteristic
77C22×C4×C61C2×D411yescharacteristic
78C22×C4×C61C2×D411yescharacteristic
79C22×Dic61—22no—
80C22×Dic61C22⋊C411yescharacteristic
81C22×Dic61C22⋊C411yescharacteristic
82C2×D244—22no—
83C2×D244—21no—
84C2×D244—21no—
85C2×D244—41no—
86C2×C4×D61—21no—
87C2×C4×D61—21no—
88C2×C4×D61—21no—
89C2×C4×D61—21no—
90C2×C4×D61—42no—
91C2×C4×D61—42no—
92C2×C8×C61—22no—
93C2×C8×C61—41no—
94C4⋊C4×C61—22no—
95C4⋊C4×C61—22no—
96C4⋊C4×C61—41no—
97C4⋊C4×C61—41no—
98C42×C61C2×D411yescharacteristic
99C42×C61C2×D411yescharacteristic
100C42×C61#4111yescharacteristic
101C42×C61#4111yescharacteristic
102#11758198—21no—
103#11758198—21no—
104#11758198—21no—
105#11758198—21no—
106#11758198—82no—
107#11758198—82no—
108C4⋊Dic61—21no—
109C4⋊Dic61—21no—
110C4⋊Dic61—42no—
111C4×Dic61—42no—
112C2×#92418—22no—
113C2×#92418—41no—
114#11758189—82no—
115C8×D61—82no—
116C23×C61C2×C22⋊C411yescharacteristic
117C22×D61—21no—
118C22×D61—21no—
119C22×D61—21no—
120C22×D61—21no—
121C22×D61—21no—
122C22×D61—41no—
123C22×D61—41no—
124C22×D61—41no—
125C22×D61—41no—
126C22×D61#10112yes—
127Q8×C61—82no—
128Q8×C61—82no—
129C2×C4×C61—21no—
130C2×C4×C61—21no—
131C2×C4×C61—22no—
132C2×C4×C61—41no—
133C2×C4×C61—42no—
134C2×C4×C61—42no—
135C2×C4×C61—42no—
136C2×C4×C61C4×D411yescharacteristic
137C2×C4×C61C4×D411yescharacteristic
138C2×C4×C61#12011yescharacteristic
139C2×C4×C61#12111yescharacteristic
140C2×C4×C61#12111yescharacteristic
141C2×C4×C61#12311yescharacteristic
142C2×Dic61—21no—
143C2×Dic61—21no—
144C2×Dic61—41no—
145C2×Dic61—41no—
146C2×Dic61—42no—
147C2×Dic61—42no—
148C2×Dic61—42no—
149D244—42no—
150D244—42no—
151C4×D61—42no—
152C4×D61—42no—
153C4×D61—81no—
154C4×D61—81no—
155C8×C61—42no—
156C8×C61—81no—
157#92418—42no—
158#92418—81no—
159#9453—611nomaximal, Sylow
160C22×C61—21no—
161C22×C61—21no—
162C22×C61#38611yescharacteristic
163C22×C61C2×#10111yescharacteristic
164C22×C61#41311yescharacteristic
165C2×D61—22no—
166C2×D61—41no—
167C2×D61—41no—
168C2×D61—42no—
169C2×D61—42no—
170C2×D61—41no—
171C2×D61—41no—
172C2×D61—41no—
173C2×D61—81no—
174C2×D61—81no—
175C4×C61—21no—
176C4×C61—21no—
177C4×C61—21no—
178C4×C61—21no—
179C4×C61—21no—
180C4×C61—82no—
181C4×C61#45512yes—
182Dic61—41no—
183Dic61—41no—
184Dic61—82no—
185#2973—611no—
186#2563—611no—
187#1556—611no—
188#1556D6111yescharacteristic, complemented
189#1466—611no—
190#1285—611no—
191#1285—611no—
192C2×C61—21no—
193C2×C61—21no—
194C2×C61#188511yescharacteristic
195C2×C61#190111yescharacteristic
196C2×C61#191011yescharacteristic
197D61—42no—
198D61—81no—
199C2×C4⋊Q8C2×D6111yescharacteristic
200C2×C4×D4—611no—
201#435—1222no—
202#435—1222no—
203#432—611no—
204#432—611no—
205#432C2×D6111yescharacteristic
206#432C2×D6111yescharacteristic
207#407—1221no—
208C2×C22⋊C8—1221no—
209#393—1221no—
210#393—1221no—
211#390—612no—
212#382—612no—
213#380—1221no—
214#380—1221no—
215C4×C4⋊C4—611no—
216#332—22no—
217#332—1222no—
218#323—1221no—
219#323—1221no—
220C61#945311yescharacteristic, complemented, Sylow
221C22×Q8—21no—
222C22×D4—611no—
223C23×C4—611no—
224C23×C4—611no—
225C2×M16—1222no—
226C22×C8—1222no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8—21no—
230C4⋊Q8—21no—
231C4×D4—1221no—
232C4×D4—1221no—
233C4×D4—1221no—
234C4×D4—1221no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—611no—
237C2×C4⋊C4—1221no—
238C2×C4⋊C4C4×D6112yes—
239C2×C22⋊C4—611no—
240C2×C22⋊C4—611no—
241C2×C22⋊C4—1222no—
242C2×C22⋊C4—1222no—
243C2×C42—612no—
244C2×C42C22×D6111yescharacteristic
245C4⋊C8—21no—
246C4⋊C8—612no—
247C4⋊C8—1221no—
248C4⋊C8C22⋊D6112yes—
249C22⋊C8—21no—
250C22⋊C8—21no—
251C22⋊C8—1222no—
252C22⋊C8—1222no—
253C22⋊C8—1221no—
254C22⋊C8—1221no—
255#97—21no—
256#97—21no—
257#97—1221no—
258#97—1221no—
259#95—1222no—
260#95—1222no—
261#95—1221no—
262C24—611no—
263C24—611no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—82no—
267C2×D4—1222no—
268C2×D4—1221no—
269C2×D4—1221no—
270C2×D4—2441no—
271C22×C4—22no—
272C22×C4—611no—
273C22×C4—611no—
274C22×C4—1221no—
275C22×C4—1221no—
276C22×C4—1221no—
277C22×C4—1221no—
278C22×C4—1222no—
279C22×C4—2442no—
280C22×C4—2442no—
281C22×C4C2×C4×D6111yescharacteristic
282C22×C4D4×D6111yescharacteristic
283C22×C4D4×D6111yescharacteristic
284M16—4882no—
285C2×C8—22no—
286C2×C8—41no—
287C2×C8—1222no—
288C2×C8—4882no—
289C2×C8—2441no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—1221no—
295C4⋊C4—1221no—
296C4⋊C4—2442no—
297C22⋊C4—1221no—
298C22⋊C4—1221no—
299C22⋊C4—1221no—
300C22⋊C4—1221no—
301C22⋊C4—4882no—
302C22⋊C4—4882no—
303C42—2442no—
304C42#1175822211yescharacteristic
305C42D4×D6111yescharacteristic
306C42#1175822411yescharacteristic
307C42C2×C22⋊D6111yescharacteristic
308C23—612no—
309C23—1221no—
310C23—1221no—
311C23—1221no—
312C23—1221no—
313C23—1221no—
314C23—2441no—
315C23—2441no—
316C23—2441no—
317C23—2441no—
318C23C22⋊C4×D6111yescharacteristic
319Q8—82no—
320Q8—82no—
321D4—2442no—
322D4—2442no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—1221no—
331C2×C4—1221no—
332C2×C4—4881no—
333C2×C4—4881no—
334C2×C4—2442no—
335C2×C4—2442no—
336C2×C4—2442no—
337C2×C4—2442no—
338C2×C4—2442no—
339C2×C4—2441no—
340C2×C4—2441no—
341C2×C4#1451973711yescharacteristic
342C2×C4C4×C22⋊D6111yescharacteristic
343C2×C4#1451974511yescharacteristic
344C2×C4#1451974811yescharacteristic
345C2×C4#1451975011yescharacteristic
346C2×C4#1451975911yescharacteristic
347C8—42no—
348C8—81no—
349C8—4881no—
350C8—2442no—
351C22—21no—
352C22—21no—
353C22—1222no—
354C22—4881no—
355C22—4881no—
356C22—2442no—
357C22—2442no—
358C22—2441no—
359C22—2441no—
360C22—2441no—
361C22—2441no—
362C22—2441no—
363C22#1683963811yescharacteristic
364C22#101×D6111yescharacteristic
365C22#1683974511yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—82no—
372C4—4882no—
373C4—2441no—
374C4—2441no—
375C4#1683993812yes—
376C2—21no—
377C2—21no—
378C2—4881no—
379C2—2442no—
380C2#2144620411yescharacteristic
381C2#2144628411yescharacteristic
382C2#2144634311yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1556C2
socleC22×C61#386
prime radicalC22×D244C2×C4
cosocleC2×C42×C61C23
hypercenter#1556D61
hypocenterC61#9453
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1556

Complements

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

NH ≅ G/Ntype
3. #21444536C2semidirect
5. #21443676C2semidirect
6. #21443674C2semidirect
7. C61×#1556C2semidirect
188. #1556D61semidirect
220. C61#9453semidirect

Maximal subgroups

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

subgroupindex
#214548472
#214509932
#214445362
#214441592
#214436762
#214436742
C61×#15562
#945361