Finite Group Databasebeta

← #29663942

Subgroups — #29663942

Order 15616.

702 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840167C211yesmaximal, characteristic, complemented
2#21455306C212yesmaximal
3#21445213C214yesmaximal, complemented
4#21445208C214yesmaximal, complemented
5#21444578C211yesmaximal, characteristic, complemented
6#21444490C212yesmaximal, complemented
7C8×#41×C61C211yesmaximal, characteristic
8C2×#14519807—22no—
9C2×#14519777C2212yes—
10#16840167—21no—
11#16840167—21no—
12#16840167C2214yes—
13#16839972—22no—
14#16839972C2214yes—
15#16839967—22no—
16#16839967C2214yes—
17C2×#14519711—22no—
18#16839786C2212yes—
19#16839770—22no—
20#16839515C2214yes—
21#16839507C2218yes—
22C4×#41×C61C2211yescharacteristic
23C8×Q8×C61C2212yes—
24C8×D4×C61—22no—
25C8×D4×C61C2212yes—
26C61×#431—21no—
27C61×#431C2211yescharacteristic
28C2×C4×C8×C61—21no—
29C2×C4×C8×C61C2211yescharacteristic
30C22×Dic122—22no—
31#14519807—48no—
32C2×#11758202—24no—
33C2×#11758200—24no—
34C2×#11758194—22no—
35#14519777—22no—
36#14519777—22no—
37#14519777—22no—
38#14519777C2314yes—
39C2×C4⋊Dic61—22no—
40C2×C4⋊Dic61—22no—
41#14519711—24no—
42#14519711—24no—
43C4⋊#92418—22no—
44C4⋊#92418C2314yes—
45#14519700—24no—
46#14519700—24no—
47C2×#41×C61D411yescharacteristic
48C22×C8×C61—22no—
49C22×C8×C61D412yes—
50C4×Q8×C61D411yescharacteristic
51C4×Q8×C61C2311yescharacteristic
52C4×D4×C61—21no—
53C4×D4×C61—21no—
54C4×D4×C61D411yescharacteristic
55C4×D4×C61C2311yescharacteristic
56#117×C61—21no—
57#117×C61D411yescharacteristic
58C2×C42×C61—21no—
59C2×C42×C61C2311yescharacteristic
60C4⋊C8×C61—22no—
61C4⋊C8×C61C2312yes—
62C22⋊C8×C61—21no—
63C22⋊C8×C61—22no—
64C4×C8×C61—22no—
65C4×C8×C61—21no—
66C4×C8×C61D411yescharacteristic
67C4×C8×C61D411yescharacteristic
68C4×C8×C61C2312yes—
69#41×C61—22no—
70#41×C61D814yes—
71C2×Q8×C61C2×D411yescharacteristic
72C2×D4×C61—21no—
73C2×D4×C61C2×D411yescharacteristic
74C22×C4×C61—21no—
75C22×C4×C61—21no—
76C22×C4×C61—21no—
77C22×C4×C61C2×D411yescharacteristic
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61#4111yescharacteristic
80C22×Dic61—42no—
81C2×Dic122—24no—
82C2×Dic122—44no—
83C2×Dic122—44no—
84C2×Dic122—42no—
85C2×C8×C61—24no—
86C2×C8×C61—22no—
87C2×C8×C61—44no—
88C2×C8×C61C2×D412yes—
89C2×C8×C61C2×D412yes—
90C4⋊C4×C61—21no—
91C4⋊C4×C61—21no—
92C4⋊C4×C61C2×D411yescharacteristic
93C4⋊C4×C61C2×D411yescharacteristic
94C22⋊C4×C61—21no—
95C22⋊C4×C61—21no—
96C22⋊C4×C61—21no—
97C42×C61—21no—
98C42×C61—21no—
99C42×C61—21no—
100C42×C61C2×D411yescharacteristic
101C42×C61C2×D411yescharacteristic
102C42×C61#4111yescharacteristic
103C42×C61C2411yescharacteristic
104#11758202—48no—
105#11758200—48no—
106C4⋊Dic61—24no—
107C4⋊Dic61—24no—
108C4⋊Dic61—42no—
109C4⋊Dic61—42no—
110#11758194—44no—
111C2×#92418—24no—
112C23×C61—21no—
113C23×C61#12111yescharacteristic
114Q8×C61—21no—
115Q8×C61C2×D812yes—
116D4×C61—24no—
117D4×C61—21no—
118D4×C61C2×D812yes—
119C2×C4×C61—21no—
120C2×C4×C61—22no—
121C2×C4×C61—24no—
122C2×C4×C61—22no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—42no—
126C2×C4×C61—42no—
127C2×C4×C61#12111yescharacteristic
128C2×C4×C61#12111yescharacteristic
129C2×C4×C61#12111yescharacteristic
130C2×C4×C61C2×D812yes—
131C2×C4×C61#13512yes—
132C2×C4×C61C22×D411yescharacteristic
133C2×C4×C61C22×D411yescharacteristic
134C2×C4×C61C2×#4111yescharacteristic
135C2×Dic61—44no—
136C2×Dic61—84no—
137Dic122—44no—
138Dic122—44no—
139Dic122—84no—
140C8×C61—22no—
141C8×C61—42no—
142C8×C61#13714yescomplemented
143#92418—44no—
144#19271—611nomaximal, Sylow
145C22×C61—21no—
146C22×C61—21no—
147C22×C61—22no—
148C22×C61—41no—
149C22×C61—41no—
150C22×C61#46612yes—
151C22×C61C2×#12111yescharacteristic
152C4×C61—21no—
153C4×C61—22no—
154C4×C61—21no—
155C4×C61—41no—
156C4×C61—41no—
157C4×C61#46612yes—
158C4×C61#47312yes—
159C4×C61C22×D811yescharacteristic
160C4×C61C2×#13511yescharacteristic
161C4×C61C2×#13712yes—
162Dic61—84no—
163C2×#500—611no—
164#3039—612no—
165C8×#41D6111yescharacteristic
166#1674—614no—
167#1673—614no—
168#1567—611no—
169#1546—612no—
170C2×C61—22no—
171C2×C61—42no—
172C2×C61#303911yescharacteristic
173C2×C61C2×#46611yescharacteristic
174C2×C61C2×#47311yescharacteristic
175C2×#137—1222no—
176C2×C4⋊Q8—612no—
177C4×#41C2×D6111yescharacteristic
178#500—614no—
179#500—1221no—
180#500—1221no—
181#462—614no—
182#462—1222no—
183#461—614no—
184#461—1222no—
185C8×Q8C2×D6112yes—
186C8×D4—22no—
187C8×D4C2×D6112yes—
188#431—21no—
189#431C2×D6111yescharacteristic
190C2×#107—1222no—
191#423—612no—
192#419—1222no—
193C2×C4×C8—21no—
194C2×C4×C8C2×D6111yescharacteristic
195#335—614no—
196#332—618no—
197C61#1927111yescharacteristic, complemented, Sylow
198C2×#41D24411yescharacteristic
199C22×Q8—1222no—
200#137—2448no—
201C2×Q16—1224no—
202C2×SD16—1224no—
203C2×M16—1222no—
204C22×C8—22no—
205C22×C8C22⋊D6112yes—
206C4⋊Q8—614no—
207C4⋊Q8—1222no—
208C4⋊Q8—1222no—
209C4⋊Q8—1222no—
210C4×Q8D24411yescharacteristic
211C4×Q8C22×D6111yescharacteristic
212C4×D4—21no—
213C4×D4—21no—
214C4×D4D24411yescharacteristic
215C4×D4C22×D6111yescharacteristic
216#117—21no—
217#117D24411yescharacteristic
218C2×C4⋊C4—1222no—
219C2×C4⋊C4—1222no—
220C2×C42—21no—
221C2×C42C22×D6111yescharacteristic
222#107—1224no—
223#107—1224no—
224C4⋊C8—22no—
225C4⋊C8—614no—
226C4⋊C8—1222no—
227C4⋊C8C22×D6112yes—
228#103—1224no—
229#103—1224no—
230C22⋊C8—21no—
231C22⋊C8—22no—
232C4×C8—22no—
233C4×C8—21no—
234C4×C8C22⋊D6111yescharacteristic
235C4×C8C22⋊D6111yescharacteristic
236C4×C8C22×D6112yes—
237#41—22no—
238#41D48814yes—
239C2×Q8—1224no—
240C2×Q8—2444no—
241C2×Q8—2444no—
242C2×Q8—2442no—
243C2×Q8C2×D24411yescharacteristic
244C2×D4—21no—
245C2×D4C2×D24411yescharacteristic
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—2442no—
250C22×C4C2×D24411yescharacteristic
251C22×C4#1175822211yescharacteristic
252C22×C4C2×C22⋊D6111yescharacteristic
253Q16—2448no—
254SD16—2448no—
255M16—2444no—
256C2×C8—24no—
257C2×C8—22no—
258C2×C8—44no—
259C2×C8—1224no—
260C2×C8C2×C22⋊D6112yes—
261C2×C8C2×C22⋊D6112yes—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—1224no—
265C4⋊C4—1224no—
266C4⋊C4—2442no—
267C4⋊C4—2442no—
268C4⋊C4C2×D24411yescharacteristic
269C4⋊C4C2×D24411yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—21no—
273C42—21no—
274C42—21no—
275C42—21no—
276C42C2×D24411yescharacteristic
277C42#1175822211yescharacteristic
278C42C2×C22⋊D6111yescharacteristic
279C42C23×D6111yescharacteristic
280C23—21no—
281C23#1451975111yescharacteristic
282Q8—21no—
283Q8—4884no—
284Q8—2444no—
285Q8—2444no—
286Q8C2×D48812yes—
287D4—24no—
288D4—21no—
289D4C2×D48812yes—
290C2×C4—21no—
291C2×C4—24no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—4884no—
299C2×C4—2444no—
300C2×C4#1451975111yescharacteristic
301C2×C4#1451975111yescharacteristic
302C2×C4#1451975111yescharacteristic
303C2×C4C2×D48812yes—
304C2×C4#1451979912yes—
305C2×C4C22×C22⋊D6111yescharacteristic
306C2×C4C22×D24411yescharacteristic
307C2×C4C2×#1175822211yescharacteristic
308C8—22no—
309C8—42no—
310C8—2444no—
311C8#1451980714yescomplemented
312C22—21no—
313C22—21no—
314C22—22no—
315C22—41no—
316C22—41no—
317C22#1683999812yes—
318C22C2×#1451975111yescharacteristic
319C4—22no—
320C4—21no—
321C4—21no—
322C4—41no—
323C4—41no—
324C4—4884no—
325C4#1683999812yes—
326C4#1684003712yes—
327C4C22×D48811yescharacteristic
328C4C2×#1451979911yescharacteristic
329C4C2×#1451980712yes—
330C2—22no—
331C2—42no—
332C2#2145530611yescharacteristic
333C2C2×#1683999811yescharacteristic
334C2C2×#1684003711yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC8×#41×C61C2
socleC22×C61C2×#121
prime radicalC2×#41×C61D4
hypercenterC8×#41D61
hypocenterC61#19271
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×#41

Complements

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

NH ≅ G/Ntype
1. C2×#16840167C2semidirect
3. #21445213C2semidirect
4. #21445208C2semidirect
5. #21444578C2semidirect
6. #21444490C2semidirect
142. C8×C61#137semidirect
197. C61#19271semidirect
311. C8#14519807semidirect

Maximal subgroups

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

subgroupindex
C2×#168401672
#214553062
#214452132
#214452082
#214445782
#214444902
C8×#41×C612
#1927161