Finite Group Databasebeta

← #29597075

Subgroups — #29597075

Order 15616.

526 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519720C211yesmaximal, characteristic
2C2×Q16×Dic61C211yesmaximal, characteristic, complemented
3C2×#16839564C211yesmaximal, characteristic
4#21446683C211yesmaximal, characteristic
5#21443910C211yesmaximal, characteristic, complemented
6#21443896C211yesmaximal, characteristic, complemented
7C2×C61×#358C211yesmaximal, characteristic, complemented
8C22×D488C2211yescharacteristic
9C2×Q8×Dic61—21no—
10C2×#14519720—22no—
11C2×#14519720—22no—
12C2×#14519720C414yescomplemented
13C22×#11758185C2211yescharacteristic
14Q16×Dic61—22no—
15Q16×Dic61—22no—
16C2×#14519711C2211yescharacteristic
17C2×#14519701—21no—
18C2×#14519697—21no—
19C2×C8×Dic61C2211yescharacteristic
20#16839663—21no—
21#16839564—22no—
22C22×Q16×C61C2211yescharacteristic
23C22×C16×C61C2211yescharacteristic
24C2×#107×C61C2211yescharacteristic
25C61×#358—22no—
26C22×D244—21no—
27C2×D488—21no—
28C2×D488—21no—
29C2×D488—21no—
30C2×D488—21no—
31C2×D488—21no—
32C2×D488C2×C412yes—
33C22×#92418D411yescharacteristic
34Q8×Dic61—24no—
35Q8×Dic61—42no—
36C2×C4⋊Dic61—21no—
37C2×C4⋊Dic61—41no—
38C2×#11758198—41no—
39C2×#11758198—41no—
40C2×C4×Dic61—41no—
41C2×C4×Dic61D411yescharacteristic
42#14519720—28no—
43C2×#11758185—21no—
44C2×#11758185—21no—
45C2×#11758185C2×C412yes—
46#14519711—21no—
47#14519711D412yes—
48#14519701—24no—
49#14519697—42no—
50C8×Dic61—21no—
51C8×Dic61—21no—
52C22×Q8×C61—21no—
53C2×Q16×C61—21no—
54C2×Q16×C61—21no—
55C2×Q16×C61—21no—
56C2×Q16×C61C2×C412yes—
57C2×Q16×C61D412yes—
58C2×Q16×C61D412yes—
59C22×C8×C61C2311yescharacteristic
60C2×C4⋊C4×C61—21no—
61C2×C16×C61—21no—
62C2×C16×C61—21no—
63C2×C16×C61—21no—
64#107×C61—21no—
65#107×C61—21no—
66C23×D61—41no—
67C2×Q8×C61—22no—
68C2×Q8×C61—22no—
69C2×Q8×C61—22no—
70C2×Q8×C61—41no—
71C2×Q8×C61—41no—
72C2×Q8×C61—41no—
73C22×C4×C61—41no—
74C22×C4×C61—41no—
75C22×C4×C61C2×D411yescharacteristic
76C22×Dic61—41no—
77C22×Dic61D811yescharacteristic
78C22×Dic61D811yescharacteristic
79C2×D244—22no—
80C2×D244—41no—
81C2×D244—41no—
82C2×D244—41no—
83C2×D244—41no—
84C2×D244—41no—
85Q16×C61—22no—
86Q16×C61—22no—
87Q16×C61—22no—
88Q16×C61C22⋊C414yescomplemented
89C2×C8×C61C22×C411yescharacteristic
90C2×C8×C61C2×D411yescharacteristic
91C2×C8×C61C2×D411yescharacteristic
92C2×C8×C61C2×D411yescharacteristic
93C2×C8×C61#4111yescharacteristic
94C2×C8×C61#4111yescharacteristic
95C4⋊C4×C61—41no—
96C4⋊C4×C61—41no—
97#11758198—82no—
98#11758198—82no—
99C4⋊Dic61—22no—
100C4⋊Dic61—44no—
101C4⋊Dic61—41no—
102C4×Dic61—21no—
103C4×Dic61—21no—
104C4×Dic61—44no—
105C2×#92418—21no—
106C2×#92418—21no—
107C2×#92418—21no—
108D488—22no—
109D488—22no—
110D488—41no—
111D488—41no—
112C16×C61—41no—
113#11758185—22no—
114C23×C61C2×D811yescharacteristic
115C22×D61—42no—
116C22×D61—81no—
117C22×D61—81no—
118C22×D61—81no—
119C22×D61—81no—
120C22×D61—81no—
121C22×D61—81no—
122Q8×C61—24no—
123Q8×C61—42no—
124Q8×C61—42no—
125Q8×C61—42no—
126C2×C4×C61—42no—
127C2×C4×C61—42no—
128C2×C4×C61—42no—
129C2×C4×C61—81no—
130C2×C4×C61—81no—
131C2×C4×C61—81no—
132C2×C4×C61C4×D411yescharacteristic
133C2×C4×C61#12011yescharacteristic
134C2×C4×C61#12111yescharacteristic
135C2×C4×C61#13511yescharacteristic
136C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
138C2×Dic61—21no—
139C2×Dic61—21no—
140C2×Dic61—21no—
141C2×Dic61—21no—
142C2×Dic61—21no—
143C2×Dic61—42no—
144C2×Dic61—81no—
145C2×Dic61—81no—
146C2×Dic61SD3212yescomplemented
147D244—42no—
148D244—42no—
149D244—81no—
150D244—81no—
151C8×C61C2×C22⋊C411yescharacteristic
152C8×C61C4×D411yescharacteristic
153C8×C61#12111yescharacteristic
154C8×C61#12311yescharacteristic
155#92418—41no—
156#13013—611nomaximal, Sylow
157C22×C61C4×D811yescharacteristic
158C22×C61#44711yescharacteristic
159C22×C61#45911yescharacteristic
160C22×C61C2×SD3211yescharacteristic
161C22×C61#50811yescharacteristic
162C22×C61#50911yescharacteristic
163C22×C61#51011yescharacteristic
164C2×D61—82no—
165C2×D61—82no—
166C2×D61—82no—
167C2×D61—161no—
168C2×D61—161no—
169C2×D61—161no—
170C2×D61—161no—
171C4×C61—44no—
172C4×C61—161no—
173C4×C61#38611yescharacteristic
174C4×C61#44211yescharacteristic
175C4×C61#44911yescharacteristic
176C4×C61#46311yescharacteristic
177Dic61—22no—
178Dic61—41no—
179Dic61—82no—
180C22×SD32—611no—
181C2×C4×Q16—611no—
182C2×#358D6111yescharacteristic, complemented
183C2×#357—611no—
184#1970—611no—
185#1380—611no—
186#1374—611no—
187C2×C61#187911yescharacteristic
188C2×C61C4×SD3211yescharacteristic
189C2×C61#217511yescharacteristic
190C2×C61#218511yescharacteristic
191C2×C61#218611yescharacteristic
192C2×C61#220711yescharacteristic
193C2×C61#221111yescharacteristic
194D61—162no—
195C22×Q16C2×D6111yescharacteristic
196C22×D8—611no—
197C2×C4×Q8—1221no—
198C2×SD32—614no—
199C2×SD32—1222no—
200C2×SD32—1222no—
201C22×C16—611no—
202C22×C16C2×D6111yescharacteristic
203C4×Q16—1222no—
204C4×Q16—1222no—
205C2×#107—611no—
206C2×#107C2×D6111yescharacteristic
207C2×#103—1221no—
208C2×#102—1221no—
209C2×C4×C8—611no—
210#390—1221no—
211#358—22no—
212#357—1222no—
213C61#1301311yescharacteristic, complemented, Sylow
214C22×Q8—21no—
215C22×D4—1221no—
216C2×Q16—21no—
217C2×Q16—21no—
218C2×Q16—21no—
219C2×Q16C4×D6112yes—
220C2×Q16D24412yes—
221C2×Q16C22⋊D6112yes—
222C2×D8—612no—
223C2×D8—1221no—
224C2×D8—1221no—
225C2×D8—1221no—
226C2×D8—1221no—
227C2×D8—1221no—
228C22×C8—611no—
229C22×C8C22×D6111yescharacteristic
230C4×Q8—1224no—
231C4×Q8—2442no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—1221no—
234C2×C4⋊C4—2441no—
235C2×C22⋊C4—2441no—
236C2×C22⋊C4—2441no—
237C2×C42—611no—
238C2×C42—2441no—
239SD32—1228no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—21no—
243C2×C16—612no—
244C2×C16—1221no—
245C2×C16—1221no—
246#107—21no—
247#107—21no—
248#107—612no—
249#107—1221no—
250#103—1224no—
251#102—2442no—
252C4×C8—1221no—
253C4×C8—1221no—
254C24—2441no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×D4—1222no—
262C2×D4—2441no—
263C2×D4—2441no—
264C2×D4—2441no—
265C2×D4—2441no—
266C2×D4—2441no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—611no—
270C22×C4—611no—
271C22×C4—2441no—
272C22×C4D4×D6111yescharacteristic
273Q16—22no—
274Q16—22no—
275Q16—22no—
276Q16#1175819814yescomplemented
277D8—1222no—
278D8—1222no—
279D8—2441no—
280D8—2441no—
281C2×C8—1221no—
282C2×C8—1221no—
283C2×C8—1221no—
284C2×C8C2×C4×D6111yescharacteristic
285C2×C8C2×D24411yescharacteristic
286C2×C8D4×D6111yescharacteristic
287C2×C8#1175822411yescharacteristic
288C2×C8#1175822411yescharacteristic
289C2×C8C2×C22⋊D6111yescharacteristic
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—1222no—
293C4⋊C4—2444no—
294C4⋊C4—2441no—
295C22⋊C4—4882no—
296C22⋊C4—4882no—
297C42—1221no—
298C42—1221no—
299C42—2444no—
300C16—41no—
301C16—1222no—
302C23—4881no—
303C23—4881no—
304C23—4881no—
305C23—4881no—
306C23—4881no—
307C23—4881no—
308C23—2442no—
309C23D8×D6111yescharacteristic
310Q8—24no—
311Q8—42no—
312Q8—42no—
313Q8—42no—
314D4—4881no—
315D4—4881no—
316D4—2442no—
317D4—2442no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—612no—
325C2×C4—1221no—
326C2×C4—1221no—
327C2×C4—1221no—
328C2×C4—1221no—
329C2×C4—1221no—
330C2×C4—4881no—
331C2×C4—4881no—
332C2×C4—2442no—
333C2×C4#1451973711yescharacteristic
334C2×C4#1451974711yescharacteristic
335C2×C4#1451974911yescharacteristic
336C2×C4#1451979811yescharacteristic
337C2×C4#1451980111yescharacteristic
338C2×C4#1451980411yescharacteristic
339C8—2441no—
340C8C2×#1175819811yescharacteristic
341C8#1451973711yescharacteristic
342C8#1451974911yescharacteristic
343C8#1451976111yescharacteristic
344C22—9761no—
345C22—9761no—
346C22—9761no—
347C22—9761no—
348C22—4882no—
349C22—4882no—
350C22—4882no—
351C22#1683984211yescharacteristic
352C22#1683989711yescharacteristic
353C22#1683995511yescharacteristic
354C22SD32×D6111yescharacteristic
355C22#1684019511yescharacteristic
356C22#1684019911yescharacteristic
357C22#1684020411yescharacteristic
358C4—44no—
359C4—161no—
360C4—1222no—
361C4—4882no—
362C4—2441no—
363C4#1683963911yescharacteristic
364C4#1683987111yescharacteristic
365C4#1683990811yescharacteristic
366C4#1683997911yescharacteristic
367C2—9762no—
368C2#2144616511yescharacteristic
369C2#2144766811yescharacteristic
370C2#2144768611yescharacteristic
371C2#2144773711yescharacteristic
372C2#2144774511yescharacteristic
373C2#2144785711yescharacteristic
374C2#2144788311yescharacteristic
subgroupG/H
centerC23D8×D61
commutatorC2×C8×C61C22×C4
FrattiniC22×C8C22×D61
FittingC2×C61×#358C2
socleC23×C61C2×D8
prime radicalC22×D488C22
cosocleC22×C8×C61C23
hypercenterC2×#358D61
hypocenterC61#13013
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#358

Complements

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

NH ≅ G/Ntype
2. C2×Q16×Dic61C2semidirect
5. #21443910C2semidirect
6. #21443896C2semidirect
7. C2×C61×#358C2semidirect
12. C2×#14519720C4semidirect
88. Q16×C61C22⋊C4semidirect
146. C2×Dic61SD32semidirect
182. C2×#358D61semidirect
213. C61#13013semidirect
276. Q16#11758198semidirect

Maximal subgroups

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

subgroupindex
C22×#145197202
C2×Q16×Dic612
C2×#168395642
#214466832
#214439102
#214438962
C2×C61×#3582
#1301361