Finite Group Databasebeta

← #29578112

Subgroups — #29578112

Order 15616.

428 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 89 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#21455918C211yesmaximal, characteristic
2#21454916C211yesmaximal, characteristic, complemented
3#21444550C211yesmaximal, characteristic, complemented
4#21444244C211yesmaximal, characteristic
5#21443729C211yesmaximal, characteristic, complemented
6#21443680C211yesmaximal, characteristic, complemented
7C61×#1466C211yesmaximal, characteristic
8#16840319C2211yescharacteristic
9#16840097C412yes—
10#16840042—21no—
11#16839967—21no—
12#16839913C412yes—
13#16839890—22no—
14#16839889—21no—
15#16839886—21no—
16#16839862—22no—
17#16839820—22no—
18#16839819C2211yescharacteristic
19#16839815C2211yescharacteristic
20#16839786C2211yescharacteristic
21#16839510—21no—
22#16839507—21no—
23D4⋊#92418—21no—
24D4⋊#92418—21no—
25C2×C4×D4×C61C2211yescharacteristic
26C61×#432C2211yescharacteristic
27C61×#432C2211yescharacteristic
28C61×#323—22no—
29C2×#11758200—22no—
30C2×#11758194—21no—
31#14519777D411yescharacteristic
32#14519777D411yescharacteristic
33#14519769—21no—
34#14519756—21no—
35#14519756C2×C412yes—
36C4×Dic122—21no—
37C4×Dic122D412yes—
38#14519736—21no—
39#14519710—22no—
40C4⋊#92418—21no—
41C4⋊#92418—21no—
42C4⋊#92418—21no—
43C4⋊#92418—21no—
44#14519706—22no—
45#14519706—22no—
46#14519700—22no—
47#14519698—22no—
48C22⋊#92418—21no—
49C22⋊#92418C2×C412yes—
50#14519688—21no—
51#14519687—22no—
52C4×#92418D411yescharacteristic
53C4×#92418D411yescharacteristic
54C8×Dic61—22no—
55C22×D4×C61C2×C411yescharacteristic
56C23×C4×C61—21no—
57C4×D4×C61—21no—
58C4×D4×C61—21no—
59C4×D4×C61—21no—
60C4×D4×C61—21no—
61C2×C4⋊C4×C61C2×C411yescharacteristic
62C2×C22⋊C4×C61—21no—
63C2×C42×C61C2311yescharacteristic
64C4⋊C8×C61—21no—
65C4⋊C8×C61—21no—
66C22⋊C8×C61—21no—
67C22⋊C8×C61D412yes—
68#97×C61—21no—
69#97×C61—21no—
70C24×C61—21no—
71C2×D4×C61—21no—
72C2×D4×C61—21no—
73C2×D4×C61—42no—
74C22×C4×C61—21no—
75C22×C4×C61—41no—
76C22×C4×C61—41no—
77C22×C4×C61—81no—
78C22×C4×C61—81no—
79C22×C4×C61C22×C411yescharacteristic
80C22×C4×C61C2×D411yescharacteristic
81C22×C4×C61#4111yescharacteristic
82C2×Dic122—21no—
83C2×Dic122C22⋊C412yes—
84C2×C8×C61—22no—
85C2×C8×C61—41no—
86C4⋊C4×C61—21no—
87C4⋊C4×C61—21no—
88C22⋊C4×C61—41no—
89C22⋊C4×C61—41no—
90C42×C61C2×D411yescharacteristic
91C42×C61C2×D411yescharacteristic
92C42×C61C2×D411yescharacteristic
93C42×C61#4111yescharacteristic
94C22⋊Dic61—22no—
95#11758200—44no—
96C4⋊Dic61—21no—
97C4⋊Dic61—21no—
98C4⋊Dic61—21no—
99C4⋊Dic61C22⋊C412yes—
100#11758196—22no—
101#11758196—22no—
102C4×Dic61—22no—
103#11758194—81no—
104C2×#92418—22no—
105C2×#92418—21no—
106C2×#92418—21no—
107C2×#92418—41no—
108C23×C61—22no—
109C23×C61—41no—
110C23×C61—41no—
111C23×C61—81no—
112C23×C61—81no—
113C23×C61C4×D411yescharacteristic
114D4×C61—22no—
115D4×C61—22no—
116D4×C61—41no—
117D4×C61—41no—
118C2×C4×C61—21no—
119C2×C4×C61—21no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61—82no—
124C2×C4×C61—81no—
125C2×C4×C61—81no—
126C2×C4×C61—81no—
127C2×C4×C61C2×C22⋊C411yescharacteristic
128C2×C4×C61C4×D411yescharacteristic
129C2×C4×C61#12011yescharacteristic
130C2×C4×C61#12111yescharacteristic
131C2×C4×C61#12111yescharacteristic
132C2×C4×C61#12311yescharacteristic
133C2×Dic61—22no—
134C2×Dic61—22no—
135Dic122—41no—
136Dic122—41no—
137Dic122—42no—
138C8×C61—42no—
139C8×C61—81no—
140#92418—41no—
141#92418—41no—
142#92418—42no—
143#92418—81no—
144#9530—611nomaximal, Sylow
145C22×C61—21no—
146C22×C61—21no—
147C22×C61—21no—
148C22×C61—42no—
149C22×C61—82no—
150C22×C61—82no—
151C22×C61—81no—
152C22×C61—81no—
153C22×C61#38611yescharacteristic
154C4×C61—21no—
155C4×C61—21no—
156C4×C61—21no—
157C4×C61—21no—
158C4×C61—21no—
159C4×C61—81no—
160C4×C61#47111yescharacteristic
161C4×C61#47211yescharacteristic
162Dic61—42no—
163Dic61—42no—
164#3110—611no—
165#2985—611no—
166#1559—611no—
167#1486—611no—
168#1466D6111yescharacteristic
169#1307—611no—
170#1287—611no—
171C2×C61—41no—
172C2×C61—82no—
173C2×C61#189111yescharacteristic
174C2×C61#190511yescharacteristic
175C2×C61#190711yescharacteristic
176#533—611no—
177C2×C4×D4C2×D6111yescharacteristic
178#484—612no—
179#474—1221no—
180#461—1221no—
181#450—612no—
182#446—1221no—
183#446—1222no—
184C8×Q8—1221no—
185#440—1222no—
186#433—1222no—
187#432—611no—
188#432C2×D6111yescharacteristic
189#432C2×D6111yescharacteristic
190#431—611no—
191#423—611no—
192#333—1221no—
193#332—1221no—
194D4⋊C8—1221no—
195D4⋊C8—1221no—
196#323—22no—
197C61#953011yescharacteristic, complemented, Sylow
198C22×D4C4×D6111yescharacteristic
199C23×C4—21no—
200C2×SD16—1222no—
201C2×M16—1221no—
202C4⋊Q8—611no—
203C4⋊Q8—611no—
204#125—1221no—
205C22⋊Q8—612no—
206C22⋊Q8—1221no—
207C4×Q8—612no—
208C4×Q8—1221no—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4—21no—
212C4×D4—21no—
213#117—1221no—
214C2×C4⋊C4C4×D6111yescharacteristic
215C2×C22⋊C4—21no—
216C2×C42C22×D6111yescharacteristic
217#107—1222no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—1222no—
221C4⋊C8—1222no—
222C4⋊C8—1221no—
223C4⋊C8—1221no—
224C4⋊C8—1221no—
225C4⋊C8—1221no—
226#103—1222no—
227#102—1222no—
228C22⋊C8—21no—
229C22⋊C8—612no—
230C22⋊C8—1221no—
231C22⋊C8C22⋊D6112yes—
232#97—21no—
233#97—21no—
234#97—1222no—
235#97—1221no—
236C4×C8—611no—
237C4×C8—611no—
238C4×C8—1222no—
239C24—21no—
240C2×Q8—612no—
241C2×Q8—1221no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—42no—
245C22×C4—21no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—81no—
249C22×C4—81no—
250C22×C4C2×C4×D6111yescharacteristic
251C22×C4#1175822211yescharacteristic
252C22×C4C2×C22⋊D6111yescharacteristic
253SD16—2444no—
254M16—4881no—
255C2×C8—22no—
256C2×C8—41no—
257C2×C8—1222no—
258C2×C8—1221no—
259C2×C8—1221no—
260C2×C8—2441no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—612no—
264C4⋊C4—1222no—
265C4⋊C4—1221no—
266C4⋊C4—1221no—
267C4⋊C4—1221no—
268C4⋊C4—1222no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—1222no—
272C42—1222no—
273C42D4×D6111yescharacteristic
274C42D4×D6111yescharacteristic
275C42D4×D6111yescharacteristic
276C42#1175822411yescharacteristic
277C23—22no—
278C23—41no—
279C23—41no—
280C23—81no—
281C23—81no—
282C23C4×C22⋊D6111yescharacteristic
283Q8—2442no—
284Q8—2441no—
285Q8—2441no—
286D4—22no—
287D4—22no—
288D4—41no—
289D4—41no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—82no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—1222no—
300C2×C4—1222no—
301C2×C4C22⋊C4×D6111yescharacteristic
302C2×C4#1451973711yescharacteristic
303C2×C4#1451974511yescharacteristic
304C2×C4#1451974811yescharacteristic
305C2×C4#1451975011yescharacteristic
306C2×C4#1451975911yescharacteristic
307C8—42no—
308C8—81no—
309C8—4881no—
310C8—2441no—
311C8—2441no—
312C8—2442no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—42no—
317C22—82no—
318C22—82no—
319C22—81no—
320C22—81no—
321C22#1683963811yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—81no—
328C4—2442no—
329C4—2442no—
330C4#1684002411yescharacteristic
331C4#1684003111yescharacteristic
332C2—41no—
333C2—82no—
334C2#2144623411yescharacteristic
335C2#2144631611yescharacteristic
336C2#2144632411yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1466C2
socleC22×C61#386
prime radicalC22×D4×C61C2×C4
cosocleC2×C42×C61C23
hypercenter#1466D61
hypocenterC61#9530
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1466

Complements

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

NH ≅ G/Ntype
2. #21454916C2semidirect
3. #21444550C2semidirect
5. #21443729C2semidirect
6. #21443680C2semidirect
197. C61#9530semidirect

Maximal subgroups

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

subgroupindex
#214559182
#214549162
#214445502
#214442442
#214437292
#214436802
C61×#14662
#953061