Finite Group Databasebeta

← #29836900

Subgroups — #29836900

Order 15616.

548 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458212C211yesmaximal, characteristic
2D4×Dic244C211yesmaximal, characteristic
3#21448047C211yesmaximal, characteristic, complemented
4#21448022C212yesmaximal, complemented
5#21447972C211yesmaximal, characteristic, complemented
6#21447910C212yesmaximal, complemented
7#21447870C211yesmaximal, characteristic, complemented
8#21447755C212yesmaximal, complemented
9C4×Dic488C211yesmaximal, characteristic, complemented
10C2×#16839567C212yesmaximal, complemented
11D4×C16×C61C211yesmaximal, characteristic
12C22×Dic244C2212yes—
13#16840469—21no—
14D4×Dic122—21no—
15C2×Dic488C2211yescharacteristic
16C2×Dic488C2212yes—
17#16840167C2211yescharacteristic
18#16840142C2211yescharacteristic
19#16840098—22no—
20#16840058—21no—
21#16840003C2212yes—
22#16840003C2212yes—
23#16839973—21no—
24#16839920—22no—
25C4×Dic244C2211yescharacteristic
26C4×Dic244C2211yescharacteristic
27C2×#14519711C2212yes—
28#16839588C2212yes—
29#16839588C2211yescharacteristic
30#16839567C2212yes—
31#16839567C2212yes—
32#16839567C2212yes—
33#16839567C2211yescharacteristic
34#16839567C2212yes—
35#16839567C2211yescharacteristic
36C22×C16×C61C2212yes—
37C8×D4×C61C2211yescharacteristic
38C61×C4⋊C16C2211yescharacteristic
39C61×C22⋊C16C2212yes—
40C4×C16×C61C2211yescharacteristic
41C22×Dic122—22no—
42C2×Dic244—24no—
43C2×Dic244—22no—
44C2×Dic244—21no—
45C2×Dic244—42no—
46C2×Dic244C2311yescharacteristic
47C2×Dic244C2312yes—
48C2×Dic244C2311yescharacteristic
49#14519777—21no—
50#14519777—21no—
51#14519776—41no—
52#14519768—41no—
53#14519756—22no—
54#14519756—22no—
55#14519754—42no—
56#14519754—42no—
57C4×Dic122—21no—
58C4×Dic122—21no—
59D4×Dic61—41no—
60D4×Dic61—41no—
61C2×C4⋊Dic61—22no—
62Dic488—22no—
63Dic488—42no—
64#14519711—22no—
65#14519711—21no—
66#14519711C2311yescharacteristic
67#14519711C2312yes—
68#14519711C2311yescharacteristic
69#14519702—22no—
70#14519702—22no—
71#14519702—21no—
72#14519702—21no—
73C22×C8×C61C2312yes—
74C4×D4×C61D411yescharacteristic
75C2×C16×C61—42no—
76C2×C16×C61C2311yescharacteristic
77C2×C16×C61C2311yescharacteristic
78C2×C16×C61C2312yes—
79C4⋊C8×C61D411yescharacteristic
80C22⋊C8×C61D412yes—
81C4×C8×C61C2311yescharacteristic
82C2×D4×C61D811yescharacteristic
83C22×C4×C61C2×D412yes—
84C22×Dic61—42no—
85C22×Dic61—42no—
86C2×Dic122—21no—
87C2×Dic122—22no—
88C2×Dic122—21no—
89C2×Dic122—44no—
90C2×Dic122—42no—
91C2×Dic122—41no—
92C2×Dic122—82no—
93C2×C8×C61—22no—
94C2×C8×C61—22no—
95C2×C8×C61C2×D411yescharacteristic
96C2×C8×C61C2×D412yes—
97C2×C8×C61C2411yescharacteristic
98C4⋊C4×C61D811yescharacteristic
99C22⋊C4×C61D812yes—
100C42×C61C2×D411yescharacteristic
101C22⋊Dic61—42no—
102C22⋊Dic61—42no—
103C4⋊Dic61—21no—
104C4⋊Dic61—21no—
105C4⋊Dic61—22no—
106C4⋊Dic61—42no—
107C4⋊Dic61—41no—
108C4⋊Dic61—41no—
109C4⋊Dic61—41no—
110#11758196—41no—
111#11758196—41no—
112#11758196—42no—
113#11758196—42no—
114C4×Dic61—41no—
115C4×Dic61—41no—
116Dic244—21no—
117Dic244—22no—
118Dic244—21no—
119Dic244—21no—
120Dic244—22no—
121Dic244—44no—
122Dic244—81no—
123Dic244#4112yes—
124C16×C61—21no—
125C16×C61—21no—
126C16×C61—22no—
127C23×C61C2×D812yes—
128D4×C61Q3214yescomplemented
129C2×C4×C61—22no—
130C2×C4×C61—22no—
131C2×C4×C61C2×D811yescharacteristic
132C2×C4×C61C2×D812yes—
133C2×C4×C61C2×D811yescharacteristic
134C2×C4×C61C22×D411yescharacteristic
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61—41no—
139C2×Dic61—42no—
140C2×Dic61—42no—
141C2×Dic61—84no—
142C2×Dic61—162no—
143Dic122—22no—
144Dic122—41no—
145Dic122—42no—
146Dic122—41no—
147Dic122—42no—
148Dic122—41no—
149Dic122—84no—
150Dic122—161no—
151C8×C61—22no—
152C8×C61—21no—
153C8×C61C2×#4111yescharacteristic
154C8×C61H(5,𝔽2)11yescharacteristic
155#33866—611nomaximal, Sylow
156C22×C61—22no—
157C22×C61—22no—
158C22×C61C2×Q3214yes—
159C22×C61C22×D811yescharacteristic
160C4×C61—21no—
161C4×C61—22no—
162C4×C61C2×Q3212yes—
163C4×C61#54611yescharacteristic
164C4×C61#57711yescharacteristic
165Dic61—42no—
166Dic61—82no—
167Dic61—81no—
168Dic61—81no—
169Dic61—82no—
170Dic61—81no—
171#3338—611no—
172D4×Q16—611no—
173#2241—611no—
174#2236—612no—
175#2227—611no—
176#2216—612no—
177#2209—611no—
178#2188—612no—
179C4×Q32—611no—
180D4×C16D6111yescharacteristic
181C2×#358—612no—
182C2×C61—24no—
183C2×C61#329411yescharacteristic
184C2×C61C22×Q3211yescharacteristic
185C2×C61#341611yescharacteristic
186C22×Q16—612no—
187#554—1221no—
188D4×Q8—1221no—
189C2×Q32—611no—
190C2×Q32—612no—
191C22×C16C2×D6112yes—
192#500—611no—
193#494—611no—
194#484—1222no—
195#477—1221no—
196#467—612no—
197#467—612no—
198#462—1221no—
199#451—1222no—
200C4×Q16—611no—
201C4×Q16—611no—
202C8×D4C2×D6111yescharacteristic
203C2×#107—612no—
204#366—611no—
205#366—612no—
206C4⋊C16C2×D6111yescharacteristic
207#358—611no—
208#358—612no—
209#358—612no—
210#358—612no—
211#358—611no—
212#358—612no—
213C22⋊C16C2×D6112yes—
214C4×C16C2×D6111yescharacteristic
215C61#3386611yescharacteristic, complemented, Sylow
216C22×Q8—1222no—
217C2×Q16—611no—
218C2×Q16—612no—
219C2×Q16—611no—
220C2×Q16—1224no—
221C2×Q16—1221no—
222C2×Q16—1222no—
223C2×Q16—2442no—
224C22×C8C22×D6112yes—
225C4⋊Q8—1221no—
226C4⋊Q8—1221no—
227C4⋊Q8—2441no—
228#125—2441no—
229C22⋊Q8—1222no—
230C22⋊Q8—1222no—
231C22⋊Q8—2442no—
232C22⋊Q8—2442no—
233C4×Q8—1221no—
234C4×Q8—1221no—
235C4×D4—2441no—
236C4×D4—2441no—
237C4×D4D24411yescharacteristic
238C2×C4⋊C4—1222no—
239Q32—1222no—
240Q32—2442no—
241C2×C16—42no—
242C2×C16C22×D6111yescharacteristic
243C2×C16C22×D6111yescharacteristic
244C2×C16C22×D6112yes—
245#107—611no—
246#107—612no—
247#107—611no—
248#107—1222no—
249#107—1221no—
250C4⋊C8D24411yescharacteristic
251#103—1221no—
252#103—1221no—
253#103—1222no—
254#103—1222no—
255C22⋊C8D24412yes—
256C4×C8C22×D6111yescharacteristic
257C2×Q8—1221no—
258C2×Q8—1221no—
259C2×Q8—1222no—
260C2×Q8—4882no—
261C2×Q8—2444no—
262C2×Q8—2441no—
263C2×Q8—2442no—
264C2×D4D48811yescharacteristic
265C22×C4—2442no—
266C22×C4—2442no—
267C22×C4C2×D24412yes—
268Q16—612no—
269Q16—1221no—
270Q16—1222no—
271Q16—1221no—
272Q16—1221no—
273Q16—1222no—
274Q16—4881no—
275Q16—2444no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8C2×D24411yescharacteristic
279C2×C8C2×D24412yes—
280C2×C8C23×D6111yescharacteristic
281C4⋊C4—1221no—
282C4⋊C4—1221no—
283C4⋊C4—1222no—
284C4⋊C4—2442no—
285C4⋊C4—2441no—
286C4⋊C4—2441no—
287C4⋊C4—2442no—
288C4⋊C4—2442no—
289C4⋊C4—2441no—
290C4⋊C4—2441no—
291C4⋊C4—2441no—
292C4⋊C4D48811yescharacteristic
293C22⋊C4—2442no—
294C22⋊C4—2442no—
295C22⋊C4D48812yes—
296C42—2441no—
297C42—2441no—
298C42C2×D24411yescharacteristic
299C16—21no—
300C16—21no—
301C16—22no—
302C23C2×D48812yes—
303Q8—1222no—
304Q8—9761no—
305Q8—4884no—
306Q8—2442no—
307Q8—2441no—
308Q8—2442no—
309Q8—2441no—
310Q8—2441no—
311D4Dic48814yescomplemented
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—9762no—
315C2×C4—4884no—
316C2×C4—2441no—
317C2×C4—2441no—
318C2×C4—2441no—
319C2×C4—2442no—
320C2×C4—2442no—
321C2×C4—2441no—
322C2×C4C2×D48811yescharacteristic
323C2×C4C2×D48812yes—
324C2×C4C2×D48811yescharacteristic
325C2×C4C22×D24411yescharacteristic
326C8—21no—
327C8—22no—
328C8#41×D6111yescharacteristic
329C8#1451982211yescharacteristic
330C22—22no—
331C22—22no—
332C22C2×Dic48814yes—
333C22C22×D48811yescharacteristic
334C4—21no—
335C4—22no—
336C4—4881no—
337C4—4882no—
338C4—4881no—
339C4—4882no—
340C4—4881no—
341C4—2442no—
342C4C2×Dic48812yes—
343C4#1684040711yescharacteristic
344C4#1684057411yescharacteristic
345C2—24no—
346C2#2145763311yescharacteristic
347C2C22×Dic48811yescharacteristic
348C2#2145896711yescharacteristic
subgroupG/H
centerC22C22×D488
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingD4×C16×C61C2
socleC22×C61C22×D8
prime radicalC2×D4×C61D8
hypercenterD4×C16D61
hypocenterC61#33866
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×C16

Complements

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

NH ≅ G/Ntype
3. #21448047C2semidirect
4. #21448022C2semidirect
5. #21447972C2semidirect
6. #21447910C2semidirect
7. #21447870C2semidirect
8. #21447755C2semidirect
9. C4×Dic488C2semidirect
10. C2×#16839567C2semidirect
128. D4×C61Q32semidirect
215. C61#33866semidirect
311. D4Dic488semidirect

Maximal subgroups

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

subgroupindex
#214582122
D4×Dic2442
#214480472
#214480222
#214479722
#214479102
#214478702
#214477552
C4×Dic4882
C2×#168395672
D4×C16×C612
#3386661