Finite Group Databasebeta

← #30599626

Subgroups — #30599626

Order 15616.

532 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 102 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#21462672C211yesmaximal, characteristic
2#21461794C211yesmaximal, characteristic
3#21461762C212yesmaximal, complemented
4#21460662C212yesmaximal, complemented
5#21445606C211yesmaximal, characteristic
6C23×#92420C2211yescharacteristic
7#16840791—22no—
8#16840791—22no—
9#16840783—21no—
10C2×C4⋊#92420—22no—
11#16840733—21no—
12C2×#14519834—21no—
13C2×C22⋊#92420C2211yescharacteristic
14C2×C22⋊#92420C2212yes—
15#16840628—21no—
16#16840628—22no—
17C2×#14519737C2211yescharacteristic
18C2×#14519691C412yes—
19C22⋊C4×Dic61C2212yes—
20#16839529C411yescharacteristic
21C61×#342C411yescharacteristic
22C2×#11758213—21no—
23C2×#11758213—21no—
24C22×#92420—21no—
25C22×#92420—22no—
26C22×#92420—21no—
27C22×#92420—42no—
28C22×#92420D412yes—
29C22×#92420D412yes—
30C4⋊#92420—21no—
31C4⋊#92420—21no—
32C4⋊#92420—44no—
33#14519834—22no—
34#14519834—42no—
35C22⋊#92420—22no—
36C22⋊#92420—22no—
37C22⋊#92420—22no—
38#14519832—22no—
39C22×C22⋊D61C2×C411yescharacteristic
40C23×Dic61C2311yescharacteristic
41C22×C4×D61D411yescharacteristic
42#14519737—22no—
43#14519737—22no—
44C2×#11758196Q811yescharacteristic
45C2×C22⋊Dic61C2×C412yes—
46C2×#11758198Q811yescharacteristic
47C2×C4×Dic61—22no—
48C2×C4×Dic61D411yescharacteristic
49#14519691—24no—
50#14519684—22no—
51C2×C22⋊C4×C61C2×C411yescharacteristic
52C2×C22⋊C4×C61C2×C412yes—
53C24×C61C22×C411yescharacteristic
54C23×D61C4⋊C411yescharacteristic
55C22×C4×C61—22no—
56C22×C4×C61C4⋊C411yescharacteristic
57C22×C4×C61C4⋊C411yescharacteristic
58C2×C22⋊D61—22no—
59C2×C22⋊D61—22no—
60C2×C22⋊D61—21no—
61C2×C22⋊D61—21no—
62C22×Dic61—22no—
63C22×Dic61—41no—
64C22×Dic61—41no—
65C22×Dic61C4⋊C411yescharacteristic
66C22×Dic61C2×D412yes—
67C22×Dic61C2×D411yescharacteristic
68C22×Dic61C2×Q811yescharacteristic
69C22×Dic61#4112yes—
70C2×C4×D61—21no—
71C2×C4×D61—21no—
72C2×C4×D61—41no—
73C2×C4×D61—41no—
74C2×#92420—22no—
75C2×#92420—24no—
76C2×#92420—22no—
77C2×#92420—22no—
78C2×#92420—42no—
79C2×#92420—41no—
80C2×#92420—41no—
81C2×#92420—42no—
82C2×#92420—42no—
83C2×#92420—41no—
84C2×#92420—41no—
85C2×#92420—41no—
86#11758213—81no—
87#11758213—81no—
88C22⋊C4×C61—22no—
89C22⋊C4×C61—22no—
90C22⋊C4×C61M1614yescomplemented
91C22⋊Dic61—22no—
92C22⋊Dic61—22no—
93#11758198—22no—
94#11758196—22no—
95C4×Dic61—21no—
96C4×Dic61—21no—
97C4×Dic61—42no—
98C4×Dic61—42no—
99C23×C61—41no—
100C23×C61—41no—
101C23×C61C2×C4⋊C411yescharacteristic
102C23×C61C4×D412yes—
103C23×C61C2×M1612yes—
104C23×C61#13111yescharacteristic
105C23×C61#13111yescharacteristic
106C22×D61—21no—
107C22×D61—21no—
108C22×D61—21no—
109C22×D61—41no—
110C22×D61—41no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—24no—
114C2×C4×C61—42no—
115C2×C4×C61—42no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C22⋊D61—42no—
119C22⋊D61—42no—
120C2×Dic61—22no—
121C2×Dic61—21no—
122C2×Dic61—22no—
123C2×Dic61—21no—
124C2×Dic61—21no—
125C2×Dic61—21no—
126C2×Dic61—21no—
127C2×Dic61—41no—
128C2×Dic61—42no—
129C2×Dic61—42no—
130C2×Dic61—42no—
131C2×Dic61—42no—
132C2×Dic61#12011yescharacteristic
133C2×Dic61C22⋊Q812yes—
134C2×Dic61#12311yescharacteristic
135C4×D61—41no—
136C4×D61—41no—
137C4×D61—81no—
138#92420—42no—
139#92420—42no—
140#92420—42no—
141#92420—81no—
142#8852—611nomaximal, Sylow
143C22×C61—22no—
144C22×C61—21no—
145C22×C61—22no—
146C22×C61—21no—
147C22×C61—42no—
148C22×C61—41no—
149C22×C61—41no—
150C22×C61#38511yescharacteristic
151C22×C61#42311yescharacteristic
152C22×C61#42411yescharacteristic
153C22×C61#43512yes—
154C22×C61#43612yes—
155C2×D61—42no—
156C2×D61—41no—
157C2×D61—41no—
158C2×D61—41no—
159C2×D61—81no—
160C4×C61—44no—
161C4×C61—41no—
162C4×C61—41no—
163Dic61—21no—
164Dic61—22no—
165Dic61—41no—
166Dic61—82no—
167Dic61B(4,𝔽2)11yescharacteristic
168Dic61#45811yescharacteristic
169#2973—611no—
170#1869—611no—
171#1850—612no—
172C4×#99—611no—
173#1282—612no—
174C2×C61—22no—
175C2×C61—22no—
176C2×C61—21no—
177C2×C61—21no—
178C2×C61#185111yescharacteristic
179C2×C61#186711yescharacteristic
180C2×C61#186811yescharacteristic
181D61—81no—
182C23×C8—611no—
183C2×C4×D4—611no—
184#435—1222no—
185#435—1222no—
186#432—1221no—
187C2×C4⋊C8—1222no—
188C2×#99—612no—
189#407—1221no—
190C2×C22⋊C8—612no—
191C2×C22⋊C8—611no—
192C2×C22⋊C8—1221no—
193C4×C22⋊C4—612no—
194#342—611no—
195#342#692311yescharacteristic
196#336—1221no—
197#336—1222no—
198C61#885211yescharacteristic, complemented, Sylow
199C22×D4—611no—
200C23×C4—611no—
201C23×C4—611no—
202C2×M16—1221no—
203C2×M16—1221no—
204C22×C8—612no—
205C22×C8—612no—
206C22×C8—1221no—
207C22×C8—1222no—
208C22×C8—1221no—
209C22×C8—2442no—
210C4×D4—1222no—
211C4×D4—1222no—
212C2×C4⋊C4—611no—
213C2×C22⋊C4—611no—
214C2×C22⋊C4—612no—
215C2×C22⋊C4C2×#692311yescharacteristic
216C2×C22⋊C4C2×#692312yes—
217C2×C42—611no—
218C2×C42—1222no—
219C4⋊C8—1221no—
220C4⋊C8—1221no—
221C4⋊C8—2444no—
222#99—1224no—
223C22⋊C8—1222no—
224C22⋊C8—1222no—
225C22⋊C8—1222no—
226C22⋊C8—1222no—
227C22⋊C8—2442no—
228#97—1222no—
229#95—1222no—
230C24—611no—
231C24C22×#692311yescharacteristic
232C2×D4—1222no—
233C2×D4—1222no—
234C2×D4—1221no—
235C2×D4—1221no—
236C22×C4—22no—
237C22×C4—611no—
238C22×C4—612no—
239C22×C4—611no—
240C22×C4—612no—
241C22×C4—611no—
242C22×C4—1221no—
243C22×C4—1221no—
244C22×C4—1222no—
245C22×C4—2441no—
246C22×C4—2441no—
247C22×C4—2441no—
248C22×C4—2441no—
249C22×C4C4⋊#692311yescharacteristic
250C22×C4C4⋊#692311yescharacteristic
251M16—4881no—
252M16—4881no—
253C2×C8—1222no—
254C2×C8—1222no—
255C2×C8—1222no—
256C2×C8—1224no—
257C2×C8—2441no—
258C2×C8—2441no—
259C2×C8—2441no—
260C2×C8—2441no—
261C2×C8—2442no—
262C2×C8—2441no—
263C2×C8—2442no—
264C2×C8—2442no—
265C4⋊C4—1222no—
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—1222no—
269C22⋊C4—1222no—
270C22⋊C4—1222no—
271C22⋊C4#1175821314yescomplemented
272C42—1221no—
273C42—1221no—
274C42—2442no—
275C42—2442no—
276C23—41no—
277C23—41no—
278C23—1221no—
279C23—1221no—
280C23—1221no—
281C23—2441no—
282C23—2441no—
283C23C2×C4⋊#692311yescharacteristic
284C23D4×#692312yes—
285C23C2×#1175821312yes—
286C23#1451986411yescharacteristic
287C23#1451986511yescharacteristic
288D4—2442no—
289D4—2442no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—24no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—611no—
298C2×C4—612no—
299C2×C4—611no—
300C2×C4—1222no—
301C2×C4—1222no—
302C2×C4—1221no—
303C2×C4—1221no—
304C2×C4—1221no—
305C2×C4—1221no—
306C2×C4—1221no—
307C2×C4—4881no—
308C2×C4—2441no—
309C2×C4—2441no—
310C2×C4—2442no—
311C2×C4—2441no—
312C2×C4—2442no—
313C2×C4—2442no—
314C2×C4—2442no—
315C8—4881no—
316C8—2442no—
317C8—2442no—
318C8—2442no—
319C22—22no—
320C22—21no—
321C22—22no—
322C22—21no—
323C22—42no—
324C22—41no—
325C22—41no—
326C22—4881no—
327C22—2442no—
328C22—2441no—
329C22—2441no—
330C22—2441no—
331C22#1684070911yescharacteristic
332C22#1684076611yescharacteristic
333C22#1684076911yescharacteristic
334C22#1684079112yes—
335C22#1684079612yes—
336C4—44no—
337C4—41no—
338C4—41no—
339C4—611no—
340C4—611no—
341C4—1221no—
342C4—1222no—
343C4—4882no—
344C4—2441no—
345C2—22no—
346C2—22no—
347C2—21no—
348C2—21no—
349C2—4881no—
350C2#2146176511yescharacteristic
351C2#2146179011yescharacteristic
352C2#2146179111yescharacteristic
subgroupG/H
centerC22#16840709
commutatorC24×C61C22×C4
FrattiniC24C22×#6923
FittingC61×#342C4
socleC22×C61#385
prime radicalC22×C22⋊D61C2×C4
cosocleC23×Dic61C23
hypercenter#342#6923
hypocenterC61#8852
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
3. #21461762C2semidirect
4. #21460662C2semidirect
90. C22⋊C4×C61M16semidirect
198. C61#8852semidirect
271. C22⋊C4#11758213semidirect

Maximal subgroups

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

subgroupindex
#214626722
#214617942
#214617622
#214606622
#214456062
#885261