Finite Group Databasebeta

← #29577701

Subgroups — #29577701

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#21450988C211yesmaximal, characteristic
3#21444537C211yesmaximal, characteristic, complemented
4#21444535C211yesmaximal, characteristic, complemented
5#21444159C211yesmaximal, characteristic
6#21443675C211yesmaximal, characteristic, complemented
7C61×#1285C211yesmaximal, characteristic, complemented
8C2×#14519777C2211yescharacteristic
9C2×C4×D244C2211yescharacteristic
10#16839835—22no—
11#16839832—22no—
12#16839819C2211yescharacteristic
13#16839819C2211yescharacteristic
14#16839724—21no—
15C22⋊C8×D61—21no—
16#16839675—21no—
17#16839675—21no—
18#16839664C412yes—
19#16839628C412yes—
20#16839619—21no—
21#16839619—21no—
22#16839509—22no—
23#16839507—22no—
24#16839485—21no—
25#16839485—21no—
26C61×#432C2211yescharacteristic
27C61×#432C2211yescharacteristic
28C4×C4⋊C4×C61C2211yescharacteristic
29C22×Dic122—21no—
30C22×D244C2×C411yescharacteristic
31C22×C4×D61D411yescharacteristic
32C22×C4×D61D411yescharacteristic
33C2×#11758189—22no—
34C2×C8×D61—22no—
35#14519777—21no—
36#14519777—21no—
37#14519777—21no—
38#14519777—21no—
39C4×D244—21no—
40C4×D244—21no—
41C4×D244—21no—
42C4×D244—21no—
43C2×C4⋊Dic61—21no—
44C2×C4⋊Dic61C2×C412yes—
45C2×C4⋊Dic61C2×C411yescharacteristic
46C2×#11758198—22no—
47C2×#11758198—22no—
48C2×#11758198D411yescharacteristic
49C2×#11758198D411yescharacteristic
50C4⋊#92418—21no—
51C4⋊#92418D412yes—
52C22⋊#92418—21no—
53C22⋊#92418—21no—
54#14519689—22no—
55#14519689—22no—
56#14519688—21no—
57#14519688—21no—
58#14519684—22no—
59#14519684—22no—
60C2×C4⋊C4×C61—21no—
61C2×C42×C61C2×C412yes—
62C2×C42×C61C2311yescharacteristic
63C4⋊C8×C61—21no—
64C4⋊C8×C61D412yes—
65C22⋊C8×C61—21no—
66C22⋊C8×C61—21no—
67#97×C61—21no—
68#97×C61—21no—
69#95×C61—21no—
70C23×D61C22⋊C411yescharacteristic
71C23×D61C22⋊C411yescharacteristic
72C22×C4×C61—22no—
73C22×C4×C61C22×C411yescharacteristic
74C22×C4×C61C2×D411yescharacteristic
75C22×C4×C61C2×D411yescharacteristic
76C22×Dic61—22no—
77C22×Dic61C22⋊C411yescharacteristic
78C22×Dic61C22⋊C411yescharacteristic
79C2×D244—22no—
80C2×D244—21no—
81C2×D244—21no—
82C2×D244—41no—
83C2×C4×D61—21no—
84C2×C4×D61—21no—
85C2×C4×D61—21no—
86C2×C4×D61—21no—
87C2×C4×D61—42no—
88C2×C4×D61—42no—
89C2×Dic122—41no—
90C2×Dic122—41no—
91C2×Dic122—82no—
92C2×C8×C61—22no—
93C2×C8×C61—41no—
94C4⋊C4×C61—42no—
95C42×C61—42no—
96C42×C61C2×D411yescharacteristic
97C42×C61C2×D411yescharacteristic
98C42×C61#4111yescharacteristic
99C42×C61#4111yescharacteristic
100#11758198—21no—
101#11758198—21no—
102#11758198—21no—
103#11758198—21no—
104#11758198—82no—
105#11758198—82no—
106C4⋊Dic61—21no—
107C4⋊Dic61—21no—
108C4⋊Dic61—22no—
109C4⋊Dic61—22no—
110C4⋊Dic61—41no—
111C4⋊Dic61—41no—
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—
127C2×C4×C61—21no—
128C2×C4×C61—21no—
129C2×C4×C61—22no—
130C2×C4×C61—41no—
131C2×C4×C61—42no—
132C2×C4×C61—42no—
133C2×C4×C61—42no—
134C2×C4×C61C4×D411yescharacteristic
135C2×C4×C61C4×D411yescharacteristic
136C2×C4×C61#12011yescharacteristic
137C2×C4×C61#12111yescharacteristic
138C2×C4×C61#12111yescharacteristic
139C2×C4×C61#12311yescharacteristic
140C2×Dic61—21no—
141C2×Dic61—21no—
142C2×Dic61—41no—
143C2×Dic61—41no—
144C2×Dic61—42no—
145C2×Dic61—42no—
146C2×Dic61—42no—
147D244—42no—
148D244—42no—
149C4×D61—42no—
150C4×D61—42no—
151C4×D61—81no—
152C4×D61—81no—
153Dic122—82no—
154Dic122—82no—
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#1556—611no—
189#1466—611no—
190#1285—611no—
191#1285D6111yescharacteristic, complemented
192C2×C61—21no—
193C2×C61—21no—
194C2×C61#188511yescharacteristic
195C2×C61#190111yescharacteristic
196C2×C61#191011yescharacteristic
197D61—42no—
198D61—81no—
199C2×C4⋊Q8—611no—
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⋊C4C2×D6111yescharacteristic
216#332—1222no—
217#332—1222no—
218#323—1221no—
219#323—1221no—
220C61#945311yescharacteristic, complemented, Sylow
221C22×Q8—1221no—
222C22×D4—611no—
223C23×C4—611no—
224C23×C4—611no—
225C2×M16—1222no—
226C22×C8—1222no—
227C4⋊Q8—1221no—
228C4⋊Q8—1221no—
229C4⋊Q8—1221no—
230C4⋊Q8—1221no—
231C4×D4—1221no—
232C4×D4—1221no—
233C4×D4—1221no—
234C4×D4—1221no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—611no—
237C2×C4⋊C4—612no—
238C2×C4⋊C4—1221no—
239C2×C22⋊C4—611no—
240C2×C22⋊C4—611no—
241C2×C22⋊C4—1222no—
242C2×C22⋊C4—1222no—
243C2×C42C4×D6112yes—
244C2×C42C22×D6111yescharacteristic
245C4⋊C8—21no—
246C4⋊C8—612no—
247C4⋊C8—1221no—
248C4⋊C8D24412yes—
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—21no—
260#95—1222no—
261#95—1222no—
262C24—611no—
263C24—611no—
264C2×Q8—4882no—
265C2×Q8—2441no—
266C2×Q8—2441no—
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—42no—
291C4⋊C4—1221no—
292C4⋊C4—1221no—
293C4⋊C4—1222no—
294C4⋊C4—1222no—
295C4⋊C4—2441no—
296C4⋊C4—2441no—
297C22⋊C4—1221no—
298C22⋊C4—1221no—
299C22⋊C4—1221no—
300C22⋊C4—1221no—
301C22⋊C4—4882no—
302C22⋊C4—4882no—
303C42—42no—
304C42C2×D24411yescharacteristic
305C42#1175822211yescharacteristic
306C42D4×D6111yescharacteristic
307C42#1175822611yescharacteristic
308C23—612no—
309C23—1221no—
310C23—1221no—
311C23—1221no—
312C23—1221no—
313C23—1221no—
314C23—2441no—
315C23—2441no—
316C23—2441no—
317C23—2441no—
318C23C22⋊C4×D6111yescharacteristic
319Q8—4882no—
320Q8—4882no—
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#1451973911yescharacteristic
342C2×C4C4×D24411yescharacteristic
343C2×C4#1451974711yescharacteristic
344C2×C4#1451975011yescharacteristic
345C2×C4#1451975211yescharacteristic
346C2×C4#1451976211yescharacteristic
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#1683964011yescharacteristic
364C22#101×D6111yescharacteristic
365C22#1683974311yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—82no—
372C4—4882no—
373C4—2441no—
374C4—2441no—
375C4#1683994112yes—
376C2—21no—
377C2—21no—
378C2—4881no—
379C2—2442no—
380C2#2144620611yescharacteristic
381C2#2144628611yescharacteristic
382C2#2144634511yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1285C2
socleC22×C61#386
prime radicalC22×D244C2×C4
cosocleC2×C42×C61C23
hypercenter#1285D61
hypocenterC61#9453
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1285

Complements

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

NH ≅ G/Ntype
3. #21444537C2semidirect
4. #21444535C2semidirect
6. #21443675C2semidirect
7. C61×#1285C2semidirect
191. #1285D61semidirect
220. C61#9453semidirect

Maximal subgroups

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

subgroupindex
#214548472
#214509882
#214445372
#214445352
#214441592
#214436752
C61×#12852
#945361