Finite Group Databasebeta

← #29571724

Subgroups — #29571724

Order 15616.

1050 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 1050 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C2×#16840433C211yesmaximal, characteristic
2#21445811C211yesmaximal, characteristic, complemented
3#21444486C212yesmaximal, complemented
4#21444249C212yesmaximal, complemented
5C61×#1460C211yesmaximal, characteristic, complemented
6C22×#11758222C412yes—
7#16840433—24no—
8#16840433—24no—
9C2×#14519774C2211yescharacteristic
10C2×#14519773—21no—
11C2×#14519751C412yes—
12C2×#14519750—22no—
13C2×C4×Dic122C2211yescharacteristic
14C2×#14519739—21no—
15#16839786C2212yes—
16C2×#14519694—21no—
17#16839531—21no—
18#16839505—24no—
19#16839494—24no—
20C2×C4×D4×C61C2211yescharacteristic
21C61×#431C2212yes—
22C61×#323—22no—
23C2×#11758222—24no—
24C2×#11758222—24no—
25C2×#11758222—48no—
26C22×Dic122C2×C411yescharacteristic
27C22×D244—21no—
28C22×D244—21no—
29C22×D244C2×C412yes—
30C22×C22⋊D61—22no—
31C22×C22⋊D61—22no—
32C22×C4×D61—22no—
33C2×#11758194—22no—
34#14519774—22no—
35#14519774D414yes—
36#14519773—28no—
37#14519751—24no—
38#14519751—24no—
39#14519750—216no—
40C4×Dic122—22no—
41C4×Dic122D414yes—
42#14519739—28no—
43C2×C4⋊Dic61C2×C411yescharacteristic
44C2×#11758196—21no—
45C2×#11758198—22no—
46C2×C4×Dic61—21no—
47C4⋊#92418—22no—
48C4⋊#92418—22no—
49#14519694—42no—
50C22×D4×C61D411yescharacteristic
51C23×C4×C61C2×C412yes—
52C4×D4×C61—22no—
53C4×D4×C61—22no—
54C2×C4⋊C4×C61D411yescharacteristic
55C2×C22⋊C4×C61D412yes—
56C2×C42×C61C2311yescharacteristic
57C4⋊C8×C61—22no—
58C22⋊C8×C61—22no—
59C4×C8×C61—22no—
60C24×C61C22⋊C412yes—
61C23×D61—22no—
62C2×D4×C61—22no—
63C2×D4×C61—22no—
64C2×D4×C61—21no—
65C2×D4×C61—21no—
66C22×C4×C61—22no—
67C22×C4×C61—22no—
68C22×C4×C61—44no—
69C22×C4×C61C22⋊C412yes—
70C22×C4×C61C22×C411yescharacteristic
71C22×C4×C61C2×D411yescharacteristic
72C22×C4×C61C2×D411yescharacteristic
73C2×C22⋊D61—28no—
74C2×C22⋊D61—28no—
75C2×C22⋊D61—48no—
76C2×C22⋊D61—84no—
77C22×Dic61—21no—
78C22×Dic61—21no—
79#11758222—416no—
80#11758222—48no—
81#11758222—48no—
82C2×D244—22no—
83C2×D244—24no—
84C2×D244—24no—
85C2×D244—42no—
86C2×D244—41no—
87C2×D244—41no—
88C2×D244—44no—
89C2×D244—82no—
90C2×D244C22⋊C414yes—
91C2×C4×D61—28no—
92C2×C4×D61—48no—
93C2×Dic122—21no—
94C2×Dic122—24no—
95C2×Dic122C22⋊C412yes—
96C2×C8×C61—42no—
97C4⋊C4×C61—22no—
98C22⋊C4×C61—24no—
99C42×C61C2×D412yes—
100C42×C61C2×D412yes—
101#11758198—28no—
102C4⋊Dic61—21no—
103C4⋊Dic61C22⋊C412yes—
104#11758196—24no—
105C4×Dic61—24no—
106#11758194—42no—
107#11758194—42no—
108C2×#92418—42no—
109C23×C61—22no—
110C23×C61—22no—
111C23×C61—22no—
112C23×C61—42no—
113C23×C61—42no—
114C23×C61C2×C22⋊C411yescharacteristic
115C22×D61—24no—
116C22×D61—44no—
117C22×D61—44no—
118C22×D61—42no—
119C22×D61—42no—
120D4×C61—42no—
121D4×C61—42no—
122C2×C4×C61—22no—
123C2×C4×C61—22no—
124C2×C4×C61—41no—
125C2×C4×C61—44no—
126C2×C4×C61—44no—
127C2×C4×C61—42no—
128C2×C4×C61C2×C22⋊C411yescharacteristic
129C2×C4×C61C2×C22⋊C411yescharacteristic
130C2×C4×C61#12012yes—
131C2×C4×C61#12012yes—
132C2×C4×C61#13512yes—
133C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C61#13711yescharacteristic
135C22⋊D61—416no—
136C22⋊D61—88no—
137C2×Dic61—24no—
138C2×Dic61—22no—
139C2×Dic61—22no—
140C2×Dic61—42no—
141D244—44no—
142D244—44no—
143D244—44no—
144D244—44no—
145D244—84no—
146C4×D61—416no—
147Dic122—24no—
148Dic122—22no—
149Dic122#10014yescomplemented
150C8×C61—82no—
151#92418—82no—
152#8385—611nomaximal, Sylow
153C22×C61—21no—
154C22×C61—21no—
155C22×C61—42no—
156C22×C61—42no—
157C22×C61—42no—
158C22×C61—82no—
159C22×C61—82no—
160C22×C61#37911yescharacteristic
161C22×C61#41611yescharacteristic
162C22×C61#41711yescharacteristic
163C2×D61—44no—
164C2×D61—44no—
165C2×D61—44no—
166C2×D61—84no—
167C2×D61—84no—
168C4×C61—22no—
169C4×C61—21no—
170C4×C61—42no—
171C4×C61C2×#10012yes—
172C4×C61#44912yes—
173C4×C61#45212yes—
174Dic61—24no—
175Dic61—42no—
176C2×#550—611no—
177#1800—611no—
178#1545—612no—
179#1487—612no—
180#1460D6111yescharacteristic, complemented
181C2×C61—21no—
182C2×C61—21no—
183C2×C61—82no—
184C2×C61#178311yescharacteristic
185C2×C61#178811yescharacteristic
186C2×C61#179511yescharacteristic
187D61—84no—
188C22×#41—612no—
189#550—1224no—
190#550—1224no—
191C2×#127—611no—
192C2×#127—1221no—
193C2×#121—612no—
194C2×#121—1222no—
195C2×C4×Q8—611no—
196C2×C4×D4—1221no—
197C2×C4×D4C2×D6111yescharacteristic
198#431C2×D6112yes—
199#423—612no—
200C2×#100—1221no—
201#343—1221no—
202#331—1224no—
203Q8⋊C8—1224no—
204#323—22no—
205C61#838511yescharacteristic, complemented, Sylow
206C2×#41—1224no—
207C2×#41—1224no—
208C2×#41—2448no—
209C22×Q8—611no—
210C22×D4—612no—
211C22×D4—1222no—
212C22×D4—1222no—
213C22×D4—1221no—
214C22×D4—1221no—
215C22×D4D24411yescharacteristic
216C23×C4—1222no—
217C23×C4C4×D6112yes—
218C2×M16—1222no—
219#127—614no—
220#127—1228no—
221#127—1222no—
222#121—12216no—
223#121—1224no—
224#121—1224no—
225C4×Q8—614no—
226C4×Q8—1222no—
227C4×D4—22no—
228C4×D4—22no—
229C4×D4—1228no—
230C2×C4⋊C4—611no—
231C2×C4⋊C4—1221no—
232C2×C4⋊C4D24411yescharacteristic
233C2×C22⋊C4—1222no—
234C2×C22⋊C4C22⋊D6112yes—
235C2×C42—1221no—
236C2×C42C22×D6111yescharacteristic
237C4⋊C8—22no—
238C4⋊C8—1222no—
239C4⋊C8—1222no—
240#100—2442no—
241C22⋊C8—22no—
242C4×C8—22no—
243C24—1222no—
244C24#1175819812yes—
245#41—2448no—
246#41—24416no—
247#41—2448no—
248C2×Q8—612no—
249C2×Q8—1224no—
250C2×Q8—1221no—
251C2×D4—21no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—21no—
255C2×D4—614no—
256C2×D4—1228no—
257C2×D4—1228no—
258C2×D4—1222no—
259C2×D4—1224no—
260C2×D4—1224no—
261C2×D4—4882no—
262C2×D4—4884no—
263C2×D4—2444no—
264C2×D4—2448no—
265C2×D4—2441no—
266C2×D4—2442no—
267C2×D4—2441no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—44no—
271C22×C4—1228no—
272C22×C4—1221no—
273C22×C4—1221no—
274C22×C4—2448no—
275C22×C4#1175819812yes—
276C22×C4C2×C4×D6111yescharacteristic
277C22×C4C2×D24411yescharacteristic
278C22×C4C2×C22⋊D6111yescharacteristic
279M16—2442no—
280M16—2442no—
281C2×C8—42no—
282C2×C8—2442no—
283C4⋊C4—22no—
284C4⋊C4—612no—
285C4⋊C4—1221no—
286C4⋊C4—1224no—
287C22⋊C4—24no—
288C22⋊C4—1228no—
289C42—1224no—
290C42D4×D6112yes—
291C42D4×D6112yes—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—42no—
296C23—42no—
297C23—1224no—
298C23—2444no—
299C23—2444no—
300C23—2442no—
301C23—2442no—
302C23C2×#1175819811yescharacteristic
303Q8—614no—
304Q8—1224no—
305Q8—1222no—
306D4—42no—
307D4—42no—
308D4—4884no—
309D4—4888no—
310D4—2444no—
311D4—24416no—
312D4—2444no—
313D4—2444no—
314D4—2444no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—41no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—42no—
321C2×C4—1224no—
322C2×C4—1222no—
323C2×C4—1222no—
324C2×C4—24416no—
325C2×C4—2442no—
326C2×C4C22⋊C4×D6111yescharacteristic
327C2×C4C22⋊C4×D6111yescharacteristic
328C2×C4#1451974512yes—
329C2×C4#1451974712yes—
330C2×C4#1451979912yes—
331C2×C4#1451980511yescharacteristic
332C2×C4#1451981011yescharacteristic
333C8—82no—
334C8—4882no—
335C22—21no—
336C22—21no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—82no—
341C22—82no—
342C22—4884no—
343C22—4884no—
344C22—2444no—
345C22—2444no—
346C22—2444no—
347C22#1683961711yescharacteristic
348C22#1683975911yescharacteristic
349C22#1683976411yescharacteristic
350C4—21no—
351C4—22no—
352C4—42no—
353C4—1224no—
354C4—2442no—
355C4#100×D6112yes—
356C4#1683990912yes—
357C4#1683992612yes—
358C2—21no—
359C2—21no—
360C2—82no—
361C2—4884no—
362C2#2144573811yescharacteristic
363C2#2144576011yescharacteristic
364C2#2144578711yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1460C2
socleC22×C61#379
prime radicalC2×#16840433C2
cosocleC2×C42×C61C23
hypercenter#1460D61
hypocenterC61#8385
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1460

Complements

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

NH ≅ G/Ntype
2. #21445811C2semidirect
3. #21444486C2semidirect
4. #21444249C2semidirect
5. C61×#1460C2semidirect
149. Dic122#100semidirect
180. #1460D61semidirect
205. C61#8385semidirect

Maximal subgroups

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

subgroupindex
C2×#168404332
#214458112
#214444862
#214442492
C61×#14602
#838561