Finite Group Databasebeta

← #29573330

Subgroups — #29573330

Order 15616.

384 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454795C211yesmaximal, characteristic
2C4×#14519710C211yesmaximal, characteristic, complemented
3#21444580C211yesmaximal, characteristic, complemented
4#21444344C211yesmaximal, characteristic
5#21443675C211yesmaximal, characteristic, complemented
6#21443674C211yesmaximal, characteristic, complemented
7C61×#1509C211yesmaximal, characteristic
8#16839893C411yescharacteristic, complemented
9Q8×#92418C411yescharacteristic, complemented
10#16839840C411yescharacteristic, complemented
11#16839834—21no—
12D4×#92418—21no—
13D4×#92418C411yescharacteristic, complemented
14#16839823—21no—
15#16839819C2211yescharacteristic
16#16839819C2211yescharacteristic
17C2×#14519710—21no—
18#16839718—21no—
19C2×C4×#92418C2211yescharacteristic
20C4×C4⋊Dic61C2211yescharacteristic
21#16839524—21no—
22#16839514—21no—
23#16839512—21no—
24#16839499—21no—
25#16839496—21no—
26C61×#529C2211yescharacteristic
27C61×#432C2211yescharacteristic
28C4×C4⋊C4×C61C2211yescharacteristic
29C61×#328—21no—
30C61×#327—21no—
31C2×#11758194—21no—
32C22×#92418—21no—
33C22×#92418D411yescharacteristic
34C22×#92418D411yescharacteristic
35C2×C4⋊Dic61—21no—
36C2×C4×Dic61—21no—
37#14519710—24no—
38C4⋊#92418—21no—
39C4⋊#92418—21no—
40C4⋊#92418—21no—
41C4⋊#92418C2×C411yescharacteristic
42C4⋊#92418C2×C411yescharacteristic
43C22⋊#92418—21no—
44C22⋊#92418—21no—
45C22⋊#92418—21no—
46C22⋊#92418—21no—
47#14519688—21no—
48#14519688—21no—
49#14519688—21no—
50C4×#92418—21no—
51C4×#92418C2×C411yescharacteristic
52C4×#92418C2×C411yescharacteristic
53C4×#92418D411yescharacteristic
54C4×#92418D411yescharacteristic
55#14519684—21no—
56#126×C61—21no—
57#125×C61Q811yescharacteristic
58#124×C61Q811yescharacteristic
59#123×C61—21no—
60C22⋊Q8×C61D411yescharacteristic
61#121×C61D411yescharacteristic
62C4×Q8×C61C2×C411yescharacteristic
63C4×D4×C61—21no—
64C4×D4×C61C2×C411yescharacteristic
65#117×C61—21no—
66C2×C4⋊C4×C61—21no—
67C2×C42×C61—21no—
68C2×C42×C61C2311yescharacteristic
69C4⋊C8×C61—21no—
70C22⋊C8×C61—21no—
71#97×C61—21no—
72#95×C61—21no—
73C2×Q8×C61C4⋊C411yescharacteristic
74C2×D4×C61—21no—
75C2×D4×C61C4⋊C411yescharacteristic
76C22×C4×C61—21no—
77C22×C4×C61—41no—
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61C2×D411yescharacteristic
80C22×C4×C61#4111yescharacteristic
81C22×Dic61—41no—
82C2×C8×C61—41no—
83C4⋊C4×C61—21no—
84C4⋊C4×C61—21no—
85C4⋊C4×C61—21no—
86C4⋊C4×C61—21no—
87C4⋊C4×C61—22no—
88C4⋊C4×C61—41no—
89C4⋊C4×C61C4⋊C411yescharacteristic
90C4⋊C4×C61C4⋊C411yescharacteristic
91C22⋊C4×C61—21no—
92C22⋊C4×C61—21no—
93C22⋊C4×C61—21no—
94C22⋊C4×C61—21no—
95C22⋊C4×C61—21no—
96C42×C61—21no—
97C42×C61—81no—
98C42×C61C22×C411yescharacteristic
99C42×C61C2×D411yescharacteristic
100C42×C61C2×Q811yescharacteristic
101C42×C61#4111yescharacteristic
102C4⋊Dic61—22no—
103C4⋊Dic61—41no—
104C4×Dic61—81no—
105#11758194—42no—
106C2×#92418—22no—
107C2×#92418—21no—
108C2×#92418—21no—
109C2×#92418—41no—
110C2×#92418—42no—
111C2×#92418—41no—
112C2×#92418C22⋊C411yescharacteristic
113C2×#92418C22⋊C411yescharacteristic
114C2×#92418C22⋊C411yescharacteristic
115C2×#92418C22⋊C411yescharacteristic
116C23×C61—21no—
117C23×C61#12111yescharacteristic
118Q8×C61—41no—
119D4×C61—41no—
120D4×C61—41no—
121D4×C61—41no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—21no—
128C2×C4×C61—21no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61—42no—
133C2×C4×C61—81no—
134C2×C4×C61—81no—
135C2×C4×C61C2×C22⋊C411yescharacteristic
136C2×C4×C61C2×C4⋊C411yescharacteristic
137C2×C4×C61#11711yescharacteristic
138C2×C4×C61#12111yescharacteristic
139C2×C4×C61C22⋊Q811yescharacteristic
140C2×C4×C61C22⋊Q811yescharacteristic
141C2×Dic61—42no—
142C2×Dic61—81no—
143C2×Dic61—81no—
144C8×C61—81no—
145#92418—21no—
146#92418—21no—
147#92418—21no—
148#92418—41no—
149#92418—41no—
150#92418—81no—
151#92418#10412yes—
152#8697—611nomaximal, Sylow
153C22×C61—21no—
154C22×C61—21no—
155C22×C61—41no—
156C22×C61—41no—
157C22×C61—41no—
158C22×C61#38011yescharacteristic
159C22×C61#47211yescharacteristic
160C22×C61#47311yescharacteristic
161C4×C61—21no—
162C4×C61—21no—
163C4×C61—21no—
164C4×C61—21no—
165C4×C61—21no—
166C4×C61—41no—
167C4×C61—41no—
168C4×C61—41no—
169C4×C61—82no—
170C4×C61C2×#10411yescharacteristic
171C4×C61#42111yescharacteristic
172Dic61—82no—
173#2966—611no—
174C4×#107—611no—
175#1568—611no—
176#1509—611no—
177#1509D6111yescharacteristic
178#1285—611no—
179#1285—611no—
180C2×C61—21no—
181C2×C61—21no—
182C2×C61—81no—
183C2×C61#180611yescharacteristic
184C2×C61#181211yescharacteristic
185C2×C61#181611yescharacteristic
186#529C2×D6111yescharacteristic
187#446—611no—
188C8×Q8—611no—
189#436—611no—
190#435—1221no—
191C8×D4—611no—
192C8×D4—1221no—
193#433—1221no—
194#432—611no—
195#432—611no—
196#432C2×D6111yescharacteristic
197C2×#107—1221no—
198#405—1221no—
199C2×C4×C8—611no—
200C4×C4⋊C4—611no—
201C4×C4⋊C4C2×D6111yescharacteristic
202#340—1221no—
203#335—1221no—
204#334—1221no—
205#328—21no—
206#328—1221no—
207#327—21no—
208#327—1221no—
209C61#869711yescharacteristic, complemented, Sylow
210C2×M16—1221no—
211C22×C8—611no—
212C22×C8—611no—
213C22×C8—1221no—
214#126—21no—
215#125Dic12211yescharacteristic
216#124Dic12211yescharacteristic
217#123—21no—
218C22⋊Q8C22⋊D6111yescharacteristic
219#121C22⋊D6111yescharacteristic
220C4×Q8C4×D6111yescharacteristic
221C4×D4—21no—
222C4×D4C4×D6111yescharacteristic
223#117—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—1221no—
226C2×C42—21no—
227C2×C42—1221no—
228C2×C42C22×D6111yescharacteristic
229#107—1224no—
230C4⋊C8—21no—
231C4⋊C8—611no—
232C4⋊C8—611no—
233C4⋊C8—1221no—
234C4⋊C8—1221no—
235C4⋊C8—1221no—
236C22⋊C8—21no—
237C22⋊C8—1221no—
238C22⋊C8—1221no—
239C22⋊C8—1221no—
240C22⋊C8—1221no—
241#97—21no—
242#97—1221no—
243#97—1221no—
244#97—1221no—
245C4×C8—611no—
246C4×C8—611no—
247C4×C8—611no—
248C4×C8—611no—
249C4×C8—1221no—
250#95—21no—
251#95—1221no—
252C2×Q8#1175819611yescharacteristic
253C2×D4—21no—
254C2×D4#1175819611yescharacteristic
255C22×C4—21no—
256C22×C4—41no—
257C22×C4—2441no—
258C22×C4D4×D6111yescharacteristic
259C22×C4#1175822411yescharacteristic
260C22×C4C2×C22⋊D6111yescharacteristic
261M16—2442no—
262C2×C8—41no—
263C2×C8—611no—
264C2×C8—611no—
265C2×C8—611no—
266C2×C8—611no—
267C2×C8—1222no—
268C2×C8—1221no—
269C2×C8—1221no—
270C2×C8—2441no—
271C2×C8—2442no—
272C2×C8—2441no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—22no—
278C4⋊C4—41no—
279C4⋊C4—1222no—
280C4⋊C4—2441no—
281C4⋊C4#1175819611yescharacteristic
282C4⋊C4#1175819611yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C42—21no—
289C42—81no—
290C42—4881no—
291C42C2×Dic12211yescharacteristic
292C42C2×C4×D6111yescharacteristic
293C42D4×D6111yescharacteristic
294C42#1175822411yescharacteristic
295C23—21no—
296C23#1451974911yescharacteristic
297Q8—41no—
298D4—41no—
299D4—41no—
300D4—41no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—4881no—
315C2×C4—4881no—
316C2×C4—2442no—
317C2×C4C22⋊C4×D6111yescharacteristic
318C2×C4C2×#1175819611yescharacteristic
319C2×C4#1451973311yescharacteristic
320C2×C4#1451974911yescharacteristic
321C2×C4#1451975411yescharacteristic
322C2×C4#1451975411yescharacteristic
323C8—81no—
324C8—612no—
325C8—1221no—
326C8—1221no—
327C8—1221no—
328C8—4881no—
329C8—2441no—
330C8—2441no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22#1683962011yescharacteristic
337C22#1684003011yescharacteristic
338C22#1684003511yescharacteristic
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—82no—
348C4—4882no—
349C4#104×D6111yescharacteristic
350C4#1683977711yescharacteristic
351C2—21no—
352C2—21no—
353C2—81no—
354C2#2144583711yescharacteristic
355C2#2144586311yescharacteristic
356C2#2144588111yescharacteristic
subgroupG/H
centerC22#16839620
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1509C2
socleC22×C61#380
prime radical#121×C61D4
cosocleC2×C42×C61C23
hypercenter#1509D61
hypocenterC61#8697
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1509

Complements

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

NH ≅ G/Ntype
2. C4×#14519710C2semidirect
3. #21444580C2semidirect
5. #21443675C2semidirect
6. #21443674C2semidirect
8. #16839893C4semidirect
9. Q8×#92418C4semidirect
10. #16839840C4semidirect
13. D4×#92418C4semidirect
209. C61#8697semidirect

Maximal subgroups

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

subgroupindex
#214547952
C4×#145197102
#214445802
#214443442
#214436752
#214436742
C61×#15092
#869761