Finite Group Databasebeta

← #29596807

Subgroups — #29596807

Order 15616.

672 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 99 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×#16839555C211yesmaximal, characteristic
2#21446218C211yesmaximal, characteristic
3#21446099C212yesmaximal, complemented
4#21443937C212yesmaximal, complemented
5C61×#2366C211yesmaximal, characteristic
6#16840237C2211yescharacteristic
7#16839767—21no—
8C2×#14519694C2211yescharacteristic
9C2×#14519692C2212yes—
10#16839635—22no—
11#16839555C414yescomplemented
12#16839528—22no—
13#16839522—21no—
14C2×#127×C61C2211yescharacteristic
15C2×C4×D4×C61—22no—
16C61×#390—22no—
17C61×#386—21no—
18C61×#386C2212yes—
19C4×C4⋊C4×C61—21no—
20C2×#11758194—21no—
21D4×Dic61—41no—
22D4×Dic61—41no—
23#14519736—21no—
24C2×#11758196—42no—
25C2×C22⋊Dic61—22no—
26C2×C22⋊Dic61—22no—
27#14519698—41no—
28#14519698—41no—
29#14519694—22no—
30#14519692—22no—
31#14519692C2×C414yescomplemented
32#14519684—42no—
33C22×D4×C61—22no—
34C22×D4×C61D411yescharacteristic
35C22×D4×C61C2311yescharacteristic
36C23×C4×C61—21no—
37C23×C4×C61D412yes—
38#127×C61—22no—
39#127×C61C2×C412yes—
40#127×C61Q812yes—
41C4×D4×C61—48no—
42C2×C4⋊C4×C61—21no—
43C2×C4⋊C4×C61—21no—
44C2×C22⋊C4×C61—22no—
45C2×C22⋊C4×C61—22no—
46C2×C22⋊C4×C61—21no—
47C2×C22⋊C4×C61D412yes—
48C2×C42×C61—22no—
49C2×C42×C61D411yescharacteristic
50#95×C61—22no—
51C24×C61—21no—
52C24×C61C2×D412yes—
53C2×D4×C61—24no—
54C2×D4×C61—24no—
55C2×D4×C61—21no—
56C2×D4×C61—41no—
57C2×D4×C61—41no—
58C2×D4×C61—44no—
59C2×D4×C61—44no—
60C2×D4×C61—41no—
61C2×D4×C61—41no—
62C2×D4×C61C4⋊C412yescomplemented
63C2×D4×C61C22×C411yescharacteristic
64C2×D4×C61C2×Q811yescharacteristic
65C2×D4×C61#4112yes—
66C22×C4×C61—21no—
67C22×C4×C61—22no—
68C22×C4×C61—22no—
69C22×C4×C61—44no—
70C22×C4×C61—44no—
71C22×C4×C61—42no—
72C22×C4×C61—81no—
73C22×C4×C61C2×D411yescharacteristic
74C22×Dic61—22no—
75C22×Dic61—22no—
76C22×Dic61—42no—
77C4⋊C4×C61—24no—
78C4⋊C4×C61—42no—
79C22⋊C4×C61—44no—
80C22⋊C4×C61—44no—
81C22⋊C4×C61—42no—
82C22⋊C4×C61—81no—
83C42×C61—21no—
84C42×C61—44no—
85C42×C61C4⋊C412yescomplemented
86C22⋊Dic61—24no—
87C22⋊Dic61—42no—
88C22⋊Dic61—42no—
89C22⋊Dic61—41no—
90C4⋊Dic61—21no—
91C4⋊Dic61—21no—
92#11758196—84no—
93C4×Dic61—41no—
94#11758194—41no—
95#11758194—41no—
96C2×#92418—41no—
97C23×C61—22no—
98C23×C61—22no—
99C23×C61—22no—
100C23×C61—42no—
101C23×C61—42no—
102C23×C61—42no—
103C23×C61—42no—
104C23×C61—41no—
105C23×C61—41no—
106C23×C61C4×D412yes—
107C23×C61#12011yescharacteristic
108C23×C61C22⋊Q812yes—
109D4×C61—41no—
110D4×C61—41no—
111D4×C61—42no—
112D4×C61—42no—
113D4×C61—48no—
114D4×C61—48no—
115D4×C61—81no—
116C2×C4×C61—22no—
117C2×C4×C61—22no—
118C2×C4×C61—22no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—24no—
122C2×C4×C61—42no—
123C2×C4×C61—42no—
124C2×C4×C61—44no—
125C2×C4×C61—42no—
126C2×C4×C61—42no—
127C2×C4×C61—42no—
128C2×C4×C61—82no—
129C2×C4×C61—84no—
130C2×C4×C61C2×C4⋊C411yescharacteristic
131C2×C4×C61#12311yescharacteristic
132C2×Dic61—44no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—82no—
136C2×Dic61—82no—
137C2×Dic61—82no—
138C2×Dic61—82no—
139#92418—81no—
140#12964—611nomaximal, Sylow
141C22×C61—21no—
142C22×C61—21no—
143C22×C61—22no—
144C22×C61—22no—
145C22×C61—42no—
146C22×C61—42no—
147C22×C61—44no—
148C22×C61—42no—
149C22×C61—42no—
150C22×C61—42no—
151C22×C61—82no—
152C22×C61—82no—
153C22×C61—81no—
154C22×C61—81no—
155C22×C61#38511yescharacteristic
156C22×C61B(4,𝔽2)11yescharacteristic
157C22×C61#45811yescharacteristic
158C4×C61—24no—
159C4×C61—21no—
160C4×C61—21no—
161C4×C61—44no—
162C4×C61—82no—
163Dic61—84no—
164Dic61—81no—
165Dic61—81no—
166#2366D6111yescharacteristic
167C2×#353—611no—
168#1887—611no—
169#1867—612no—
170#1390—612no—
171C2×C61—21no—
172C2×C61—21no—
173C2×C61—42no—
174C2×C61—42no—
175C2×C61—82no—
176C2×C61#186711yescharacteristic
177C2×C61#219611yescharacteristic
178C2×C61#219811yescharacteristic
179C2×#127C2×D6111yescharacteristic
180#518—611no—
181C2×C4×D4—22no—
182#418—1221no—
183C2×#100—611no—
184C2×#99—612no—
185#390—22no—
186#386—21no—
187#386C2×D6112yes—
188#385—1222no—
189C4×C4⋊C4—21no—
190#353—614no—
191#342—1222no—
192#339—1221no—
193C61#1296411yescharacteristic, complemented, Sylow
194C22×D4—22no—
195C22×D4C22⋊D6111yescharacteristic
196C22×D4C22×D6111yescharacteristic
197C23×C4—21no—
198C23×C4C22⋊D6112yes—
199C2×M16—1221no—
200#127—22no—
201#127Dic12212yes—
202#127C4×D6112yes—
203C4×D4—48no—
204C4×D4—2441no—
205C4×D4—2441no—
206#117—1221no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—2442no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—1222no—
214C2×C22⋊C4—1222no—
215C2×C22⋊C4C22⋊D6112yes—
216C2×C42—22no—
217C2×C42C22⋊D6111yescharacteristic
218#102—2441no—
219#102—2441no—
220#100—1222no—
221#99—614no—
222#99—1222no—
223#95—22no—
224#95—2442no—
225C24—21no—
226C24C2×C22⋊D6112yes—
227C2×D4—24no—
228C2×D4—24no—
229C2×D4—21no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—44no—
233C2×D4—44no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4#1175819612yescomplemented
237C2×D4C2×Dic12211yescharacteristic
238C2×D4C2×C4×D6111yescharacteristic
239C2×D4#1175822212yes—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—42no—
246C22×C4—81no—
247C22×C4—1222no—
248C22×C4—1222no—
249C22×C4—2442no—
250C22×C4C2×C22⋊D6111yescharacteristic
251M16—2441no—
252M16—2441no—
253C2×C8—2441no—
254C4⋊C4—24no—
255C4⋊C4—42no—
256C4⋊C4—1221no—
257C4⋊C4—1221no—
258C4⋊C4—4884no—
259C22⋊C4—44no—
260C22⋊C4—44no—
261C22⋊C4—42no—
262C22⋊C4—81no—
263C22⋊C4—1224no—
264C22⋊C4—2442no—
265C22⋊C4—2442no—
266C22⋊C4—2441no—
267C42—21no—
268C42—44no—
269C42—2441no—
270C42#1175819612yescomplemented
271C23—22no—
272C23—22no—
273C23—22no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—41no—
279C23—41no—
280C23C4×C22⋊D6112yes—
281C23#1451974611yescharacteristic
282C23#1451975612yes—
283D4—41no—
284D4—41no—
285D4—42no—
286D4—42no—
287D4—48no—
288D4—48no—
289D4—81no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—82no—
303C2×C4—84no—
304C2×C4—4882no—
305C2×C4—4882no—
306C2×C4—4882no—
307C2×C4—4882no—
308C2×C4—2444no—
309C2×C4—2441no—
310C2×C4—2441no—
311C2×C4C2×#1175819611yescharacteristic
312C2×C4#1451976011yescharacteristic
313C8—4881no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—22no—
318C22—42no—
319C22—42no—
320C22—44no—
321C22—42no—
322C22—42no—
323C22—42no—
324C22—82no—
325C22—82no—
326C22—81no—
327C22—81no—
328C22#1683963511yescharacteristic
329C22#1683994911yescharacteristic
330C22#1683995211yescharacteristic
331C4—24no—
332C4—21no—
333C4—21no—
334C4—44no—
335C4—82no—
336C4—4884no—
337C4—4881no—
338C4—4881no—
339C2—21no—
340C2—21no—
341C2—42no—
342C2—42no—
343C2—82no—
344C2#2144610011yescharacteristic
345C2#2144780311yescharacteristic
346C2#2144781511yescharacteristic
subgroupG/H
centerC22#16839949
commutatorC2×D4×C61C22×C4
FrattiniC22×D4C22×D61
FittingC61×#2366C2
socleC22×C61B(4,𝔽2)
prime radicalC2×#127×C61C22
cosocleC22×D4×C61C23
hypercenter#2366D61
hypocenterC61#12964
LCSG ▷ C2×D4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #2366

Complements

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

NH ≅ G/Ntype
3. #21446099C2semidirect
4. #21443937C2semidirect
11. #16839555C4semidirect
31. #14519692C2×C4semidirect
62. C2×D4×C61C4⋊C4semidirect
85. C42×C61C4⋊C4semidirect
193. C61#12964semidirect
236. C2×D4#11758196semidirect
270. C42#11758196semidirect

Maximal subgroups

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

subgroupindex
C2×#168395552
#214462182
#214460992
#214439372
C61×#23662
#1296461