Finite Group Databasebeta

← #29665365

Subgroups — #29665365

Order 15616.

676 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456942C211yesmaximal, characteristic
2#21455291C211yesmaximal, characteristic
3C2×#16839972C211yesmaximal, characteristic, complemented
4#21445263C214yesmaximal, complemented
5#21445213C214yesmaximal, complemented
6#21444594C211yesmaximal, characteristic, complemented
7#21444490C211yesmaximal, characteristic, complemented
8#21444466C211yesmaximal, characteristic
9C61×#2982C211yesmaximal, characteristic, complemented
10C2×#14519808—21no—
11C22×#11758202—21no—
12#16840312C2211yescharacteristic
13C2×#14519777C2211yescharacteristic
14#16840167C2212yes—
15#16840109C2212yes—
16#16840076—22no—
17#16840043—22no—
18#16839984—22no—
19#16839972—22no—
20#16839972C2212yes—
21#16839972C2214yes—
22#16839962C2212yes—
23#16839786C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25#16839761—21no—
26C2×#14519700—21no—
27#16839515C2214yes—
28#16839507C2212yes—
29#16839507C2214yes—
30#16839505C2212yes—
31C2×C4×Q8×C61C2211yescharacteristic
32C8×Q8×C61—22no—
33C8×Q8×C61C2214yes—
34C61×#431—22no—
35C61×#431C2211yescharacteristic
36C61×#431C2211yescharacteristic
37C2×#11758222—21no—
38C22×Dic122—21no—
39#14519808—44no—
40C2×#11758202—24no—
41C2×#11758202—22no—
42C2×#11758202—44no—
43C2×#11758201—22no—
44C2×#11758194—21no—
45C22×#92418—21no—
46#14519777—21no—
47#14519777—21no—
48#14519777—21no—
49#14519777C2312yes—
50#14519777C2311yescharacteristic
51#14519774C2311yescharacteristic
52#14519765—21no—
53#14519751—21no—
54#14519751—21no—
55C4×D244—21no—
56C4×D244—21no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61—21no—
59#14519711—24no—
60C4⋊#92418—21no—
61C4⋊#92418—21no—
62C4⋊#92418C2312yes—
63C4⋊#92418C2312yes—
64#14519700—24no—
65#14519700—22no—
66#14519699—24no—
67#14519697—22no—
68C22×Q8×C61D411yescharacteristic
69C4×Q8×C61—21no—
70C4×Q8×C61—21no—
71C4×Q8×C61D412yes—
72C4×Q8×C61C2312yes—
73C2×C4⋊C4×C61—21no—
74C2×C4⋊C4×C61D411yescharacteristic
75C2×C42×C61—21no—
76C2×C42×C61C2311yescharacteristic
77C4⋊C8×C61—24no—
78C4⋊C8×C61D412yes—
79C4⋊C8×C61C2312yes—
80C22⋊C8×C61—21no—
81C22⋊C8×C61D412yes—
82C4×C8×C61—24no—
83C4×C8×C61—21no—
84C4×C8×C61C2312yes—
85C2×Q8×C61—21no—
86C2×Q8×C61—22no—
87C2×Q8×C61D814yes—
88C2×Q8×C61C2×D412yes—
89C22×C4×C61—21no—
90C22×C4×C61—21no—
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61#4111yescharacteristic
94C2×C22⋊D61—41no—
95C22×Dic61—41no—
96#11758222—42no—
97#11758222—42no—
98C2×D244—21no—
99C2×D244—21no—
100C2×D244—21no—
101C2×C4×D61—41no—
102C2×Dic122—22no—
103C2×Dic122—21no—
104C2×Dic122—42no—
105C2×Dic122—42no—
106C2×Dic122—41no—
107C2×C8×C61—22no—
108C2×C8×C61C2×D414yes—
109C4⋊C4×C61—22no—
110C4⋊C4×C61—21no—
111C4⋊C4×C61—22no—
112C4⋊C4×C61C2×D412yes—
113C42×C61—22no—
114C42×C61—22no—
115C42×C61C2×D411yescharacteristic
116C42×C61C2×D411yescharacteristic
117C42×C61#4111yescharacteristic
118C42×C61C2411yescharacteristic
119#11758202—44no—
120#11758202—48no—
121#11758201—44no—
122#11758198—41no—
123#11758198—41no—
124C4⋊Dic61—22no—
125C4⋊Dic61—21no—
126C4⋊Dic61—22no—
127C4⋊Dic61—21no—
128C4⋊Dic61—41no—
129C4⋊Dic61—41no—
130#11758194—42no—
131C2×#92418—22no—
132C2×#92418—22no—
133C2×#92418—42no—
134C23×C61#12111yescharacteristic
135C22×D61—41no—
136Q8×C61—22no—
137Q8×C61—24no—
138Q8×C61C2×D814yes—
139C2×C4×C61—22no—
140C2×C4×C61—24no—
141C2×C4×C61—22no—
142C2×C4×C61—22no—
143C2×C4×C61—41no—
144C2×C4×C61—81no—
145C2×C4×C61#12111yescharacteristic
146C2×C4×C61#12111yescharacteristic
147C2×C4×C61#12111yescharacteristic
148C2×C4×C61C2×D812yes—
149C2×C4×C61#13512yes—
150C2×C4×C61C22×D411yescharacteristic
151C2×C4×C61C22×D411yescharacteristic
152C2×C4×C61C2×#4111yescharacteristic
153C22⋊D61—82no—
154C2×Dic61—42no—
155C2×Dic61—41no—
156C2×Dic61—82no—
157D244—41no—
158D244—41no—
159D244—42no—
160D244—81no—
161C4×D61—82no—
162Dic122—42no—
163Dic122—41no—
164Dic122—42no—
165Dic122—41no—
166Dic122—82no—
167C8×C61—24no—
168C8×C61—42no—
169#92418—42no—
170#92418—42no—
171#19404—611nomaximal, Sylow
172C22×C61—21no—
173C22×C61—21no—
174C22×C61#46612yes—
175C22×C61C2×#12111yescharacteristic
176C2×D61—81no—
177C2×D61—81no—
178C2×D61—81no—
179C4×C61—21no—
180C4×C61—24no—
181C4×C61—21no—
182C4×C61—21no—
183C4×C61—42no—
184C4×C61#46612yes—
185C4×C61C22×D811yescharacteristic
186C4×C61C2×#13511yescharacteristic
187C4×C61#57812yes—
188Dic61—82no—
189Dic61—81no—
190#3226—611no—
191#3037—611no—
192C2×#462—611no—
193#2982D6111yescharacteristic, complemented
194#1682—614no—
195#1674—614no—
196#1571—611no—
197#1546—611no—
198#1540—611no—
199C2×C61—22no—
200C2×C61#304411yescharacteristic
201C2×C61C2×#46611yescharacteristic
202C2×C61#306911yescharacteristic
203D61—161no—
204C2×#137—1221no—
205C22×Q16—1221no—
206#532—611no—
207C2×C4⋊Q8—611no—
208C2×C4×Q8C2×D6111yescharacteristic
209#500—612no—
210#486—612no—
211#480—1222no—
212#474—1222no—
213#464—1222no—
214#462—612no—
215#462—614no—
216#462—1222no—
217#460—612no—
218C8×Q8—22no—
219C8×Q8C2×D6114yes—
220#431—22no—
221#431C2×D6111yescharacteristic
222#431C2×D6111yescharacteristic
223#423—611no—
224C2×C4⋊C8—611no—
225#417—1221no—
226C2×#103—1221no—
227#335—614no—
228#332—612no—
229#332—614no—
230#331—612no—
231C61#1940411yescharacteristic, complemented, Sylow
232C2×#41—1221no—
233C22×Q8—1221no—
234C22×Q8D24411yescharacteristic
235#137—2444no—
236C2×Q16—1224no—
237C2×Q16—1222no—
238C2×Q16—2444no—
239C2×SD16—1222no—
240C2×M16—1221no—
241C22×C8—1221no—
242C4⋊Q8—612no—
243C4⋊Q8—611no—
244C4⋊Q8—1221no—
245C4⋊Q8—1221no—
246C4⋊Q8—1221no—
247#127—611no—
248#124—1221no—
249#121—1221no—
250#121—1221no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8D24412yes—
254C4×Q8C22×D6112yes—
255C4×D4—1221no—
256C4×D4—1221no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—1221no—
259C2×C4⋊C4—1221no—
260C2×C4⋊C4D24411yescharacteristic
261C2×C42—21no—
262C2×C42C22×D6111yescharacteristic
263#107—1224no—
264C4⋊C8—24no—
265C4⋊C8—612no—
266C4⋊C8—612no—
267C4⋊C8—1221no—
268C4⋊C8—1221no—
269C4⋊C8C22⋊D6112yes—
270C4⋊C8C22×D6112yes—
271#103—1224no—
272#103—1222no—
273#102—1224no—
274#102—1222no—
275C22⋊C8—21no—
276C22⋊C8C22⋊D6112yes—
277C4×C8—24no—
278C4×C8—21no—
279C4×C8C22×D6112yes—
280#41—2442no—
281#41—2442no—
282C2×Q8—21no—
283C2×Q8—22no—
284C2×Q8—1221no—
285C2×Q8—1222no—
286C2×Q8—2442no—
287C2×Q8—2442no—
288C2×Q8—2441no—
289C2×Q8D48814yes—
290C2×Q8C2×D24412yes—
291C2×D4—1221no—
292C2×D4—1221no—
293C2×D4—1221no—
294C2×D4—2441no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—2441no—
298C22×C4—2441no—
299C22×C4C2×D24411yescharacteristic
300C22×C4#1175822211yescharacteristic
301C22×C4C2×C22⋊D6111yescharacteristic
302Q16—2448no—
303Q16—2444no—
304SD16—2444no—
305M16—2442no—
306C2×C8—22no—
307C2×C8—1222no—
308C2×C8—1222no—
309C2×C8—2442no—
310C2×C8C2×C22⋊D6114yes—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—21no—
314C4⋊C4—1222no—
315C4⋊C4—1221no—
316C4⋊C4—1222no—
317C4⋊C4—1221no—
318C4⋊C4—2441no—
319C4⋊C4—2441no—
320C4⋊C4C2×D24412yes—
321C22⋊C4—2441no—
322C22⋊C4—2441no—
323C42—22no—
324C42—22no—
325C42C2×D24411yescharacteristic
326C42#1175822211yescharacteristic
327C42C2×C22⋊D6111yescharacteristic
328C42C23×D6111yescharacteristic
329C23—2441no—
330C23#1451975111yescharacteristic
331Q8—24no—
332Q8—22no—
333Q8—4882no—
334Q8—2442no—
335Q8—2441no—
336Q8—2442no—
337Q8—2441no—
338Q8C2×D48814yes—
339D4—4881no—
340D4—4882no—
341D4—2441no—
342D4—2441no—
343D4—2442no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—81no—
350C2×C4—4882no—
351C2×C4—4882no—
352C2×C4—2441no—
353C2×C4—2442no—
354C2×C4#1451975111yescharacteristic
355C2×C4#1451975111yescharacteristic
356C2×C4#1451975111yescharacteristic
357C2×C4C2×D48812yes—
358C2×C4#1451979912yes—
359C2×C4C22×C22⋊D6111yescharacteristic
360C2×C4C22×D24411yescharacteristic
361C2×C4C2×#1175822211yescharacteristic
362C8—24no—
363C8—42no—
364C8—2442no—
365C8—2442no—
366C22—21no—
367C22—21no—
368C22—4881no—
369C22—4881no—
370C22—4881no—
371C22#1683999812yes—
372C22C2×#1451975111yescharacteristic
373C4—24no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—42no—
378C4—4881no—
379C4—4882no—
380C4#1683999812yes—
381C4C22×D48811yescharacteristic
382C4C2×#1451979911yescharacteristic
383C4#1684057712yes—
384C2—22no—
385C2—9761no—
386C2#2145534911yescharacteristic
387C2C2×#1683999811yescharacteristic
388C2#2145555611yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#2982C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenter#2982D61
hypocenterC61#19404
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
3. C2×#16839972C2semidirect
4. #21445263C2semidirect
5. #21445213C2semidirect
6. #21444594C2semidirect
7. #21444490C2semidirect
9. C61×#2982C2semidirect
193. #2982D61semidirect
231. C61#19404semidirect

Maximal subgroups

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

subgroupindex
#214569422
#214552912
C2×#168399722
#214452632
#214452132
#214445942
#214444902
#214444662
C61×#29822
#1940461