Finite Group Databasebeta

← #29596247

Subgroups — #29596247

Order 15616.

416 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454235C211yesmaximal, characteristic
2C2×#16839543C211yesmaximal, characteristic
3C2×#16839543C211yesmaximal, characteristic
4#21443906C211yesmaximal, characteristic, complemented
5#21443905C211yesmaximal, characteristic, complemented
6#21443904C211yesmaximal, characteristic, complemented
7C61×#1378C211yesmaximal, characteristic, complemented
8C22×#11758189C2211yescharacteristic
9C2×M16×D61C411yescharacteristic
10#117×D61C411yescharacteristic
11C2×#14519717C2211yescharacteristic
12C2×#14519717C2211yescharacteristic
13#16839720C411yescharacteristic
14#16839719—21no—
15#16839718C411yescharacteristic
16#16839717C2211yescharacteristic
17#16839716—21no—
18#16839714—21no—
19C4×#11758189—21no—
20#16839543—21no—
21#16839543—21no—
22#16839543—21no—
23#16839543—21no—
24C2×M32×C61C2211yescharacteristic
25C2×M32×C61C2211yescharacteristic
26C61×#405C2211yescharacteristic
27C22×C4×D61C2×C411yescharacteristic
28C2×#11758194C2×C411yescharacteristic
29C2×#11758189—21no—
30C2×#11758189—21no—
31C2×#11758189—21no—
32C2×#11758189—21no—
33C2×#11758189—21no—
34C2×#11758189D411yescharacteristic
35C2×#11758189D411yescharacteristic
36C2×#11758189D411yescharacteristic
37C2×#11758189D411yescharacteristic
38M16×D61—22no—
39M16×D61—22no—
40C22×#92418C2×C411yescharacteristic
41C2×C8×D61—21no—
42#14519736C2×C411yescharacteristic
43#14519735—21no—
44#14519734—21no—
45#14519733C812yes—
46C4⋊C4×D61—21no—
47C22⋊C4×D61C812yes—
48C42×D61—21no—
49#14519717—21no—
50#14519717—21no—
51#14519717—21no—
52#14519717—21no—
53C2×#11758185—21no—
54C2×#11758185—21no—
55#14519688—21no—
56#14519687—21no—
57#14519687—21no—
58C4×#92418—21no—
59C8×Dic61—21no—
60C8×Dic61—21no—
61C2×M16×C61C2×C411yescharacteristic
62C22×C8×C61C2311yescharacteristic
63#117×C61C2×C411yescharacteristic
64M32×C61—21no—
65M32×C61—21no—
66M32×C61—21no—
67M32×C61—21no—
68C2×C16×C61—21no—
69C2×C16×C61—21no—
70#97×C61—21no—
71C4×C8×C61—21no—
72C23×D61C2×C811yescharacteristic
73C22×C4×C61C22×C411yescharacteristic
74C22×Dic61C2×C811yescharacteristic
75C2×C4×D61—21no—
76C2×C4×D61—21no—
77C2×C4×D61—21no—
78C2×C4×D61—21no—
79C2×C4×D61—22no—
80C2×C4×D61C22⋊C411yescharacteristic
81C2×C4×D61C22⋊C411yescharacteristic
82C2×C4×D61M1611yescharacteristic
83C2×C4×D61M1611yescharacteristic
84M16×C61—21no—
85M16×C61M1612yes—
86C2×C8×C61—21no—
87C2×C8×C61C2×D411yescharacteristic
88C2×C8×C61C2×D411yescharacteristic
89C2×C8×C61#4111yescharacteristic
90C2×C8×C61#4111yescharacteristic
91C2×C8×C61#4111yescharacteristic
92C2×C8×C61#4111yescharacteristic
93C4⋊C4×C61—21no—
94C22⋊C4×C61C2×C812yes—
95C42×C61—21no—
96C22⋊Dic61C2×C812yes—
97#11758198—22no—
98C4⋊Dic61—21no—
99#11758196—22no—
100C4×Dic61—21no—
101C4×Dic61—22no—
102#11758194—21no—
103#11758194M1612yes—
104C2×#92418—21no—
105C2×#92418—21no—
106C2×#92418—21no—
107C2×#92418C22⋊C411yescharacteristic
108C2×#92418C22⋊C411yescharacteristic
109#11758189—24no—
110#11758189—21no—
111#11758189—21no—
112#11758189—21no—
113#11758189—21no—
114#11758189—21no—
115#11758189—21no—
116#11758189—21no—
117#11758189—21no—
118C8×D61—24no—
119C16×C61—41no—
120C16×C61—41no—
121#11758185—41no—
122#11758185—41no—
123C23×C61C22×C811yescharacteristic
124C22×D61—21no—
125C22×D61—21no—
126C22×D61—81no—
127C22×D61C22⋊C811yescharacteristic
128C22×D61C22⋊C811yescharacteristic
129C2×C4×C61—22no—
130C2×C4×C61C2×C22⋊C411yescharacteristic
131C2×C4×C61#11711yescharacteristic
132C2×C4×C61#11711yescharacteristic
133C2×C4×C61C2×M1611yescharacteristic
134C2×C4×C61C2×M1611yescharacteristic
135C2×C4×C61C2×M1611yescharacteristic
136C2×Dic61—21no—
137C2×Dic61—21no—
138C2×Dic61—22no—
139C2×Dic61C22⋊C811yescharacteristic
140C2×Dic61C22⋊C811yescharacteristic
141C4×D61—21no—
142C4×D61—21no—
143C4×D61—21no—
144C4×D61—21no—
145C4×D61—21no—
146C4×D61—21no—
147C4×D61—21no—
148C4×D61—21no—
149C4×D61—82no—
150C8×C61—22no—
151C8×C61#12311yescharacteristic
152C8×C61#12311yescharacteristic
153C8×C61#12311yescharacteristic
154C8×C61#12311yescharacteristic
155#92418—21no—
156#92418—21no—
157#92418—22no—
158#12851—611nomaximal, Sylow
159C22×C61—41no—
160C22×C61C2×C22⋊C811yescharacteristic
161C22×C61#43111yescharacteristic
162C22×C61#43111yescharacteristic
163C2×D61—21no—
164C2×D61—21no—
165C2×D61—41no—
166C2×D61—41no—
167C2×D61—81no—
168C2×D61—81no—
169C4×C61—42no—
170C4×C61#38111yescharacteristic
171C4×C61#40711yescharacteristic
172C4×C61#43211yescharacteristic
173C4×C61#43211yescharacteristic
174Dic61—21no—
175Dic61—21no—
176Dic61—42no—
177#2873—611no—
178C2×#349—611no—
179C2×#349—611no—
180#1378—611no—
181#1378—611no—
182#1378—611no—
183#1378D6111yescharacteristic, complemented
184C2×C61—21no—
185C2×C61—21no—
186C2×C61—21no—
187C2×C61#181711yescharacteristic
188D61—41no—
189D61—41no—
190C22×M16—611no—
191C22×M16—611no—
192C2×#117—611no—
193C2×M32—611no—
194C2×M32—611no—
195C2×M32C2×D6111yescharacteristic
196C2×M32C2×D6111yescharacteristic
197#405—611no—
198#405—611no—
199#405—611no—
200#405—1221no—
201#405—1221no—
202#405C2×D6111yescharacteristic
203C4×M16—1221no—
204C4×M16—1221no—
205#349—1221no—
206#349—1221no—
207#349—1221no—
208#349—1221no—
209C61#1285111yescharacteristic, complemented, Sylow
210C23×C4—611no—
211C2×M16—611no—
212C2×M16—611no—
213C2×M16—611no—
214C2×M16—611no—
215C2×M16—611no—
216C2×M16—1221no—
217C2×M16—1222no—
218C2×M16—1222no—
219C2×M16—1221no—
220C2×M16—1221no—
221C2×M16—1221no—
222C2×M16—1221no—
223C2×M16C4×D6111yescharacteristic
224C22×C8—611no—
225C22×C8—1221no—
226C22×C8C22×D6111yescharacteristic
227#117—611no—
228#117—612no—
229#117—1221no—
230#117—1221no—
231#117C4×D6111yescharacteristic
232C2×C4⋊C4—1221no—
233C2×C22⋊C4—612no—
234C2×C42—1221no—
235M32—21no—
236M32—21no—
237M32—21no—
238M32—21no—
239M32—1221no—
240M32—1221no—
241M32—1221no—
242M32—1221no—
243C2×C16—21no—
244C2×C16—21no—
245C2×C16—1221no—
246C2×C16—1221no—
247#97—21no—
248#97—1221no—
249#97—1221no—
250#97—1221no—
251C4×C8—21no—
252C4×C8—1221no—
253C4×C8—1221no—
254C4×C8—1221no—
255C24—611no—
256C22×C4—611no—
257C22×C4—611no—
258C22×C4—611no—
259C22×C4—611no—
260C22×C4—611no—
261C22×C4—1222no—
262C22×C4—1221no—
263C22×C4—1221no—
264C22×C4—1221no—
265C22×C4—1221no—
266C22×C4C2×C4×D6111yescharacteristic
267M16—21no—
268M16—612no—
269M16—1224no—
270M16—1221no—
271M16—1221no—
272M16—1221no—
273M16—1221no—
274M16—1221no—
275M16—1221no—
276M16—1221no—
277M16—1221no—
278M16—1221no—
279M16#1175818912yes—
280C2×C8—21no—
281C2×C8—611no—
282C2×C8—611no—
283C2×C8—1224no—
284C2×C8—1221no—
285C2×C8—1221no—
286C2×C8—1221no—
287C2×C8#1175822211yescharacteristic
288C2×C8#1175822211yescharacteristic
289C2×C8D4×D6111yescharacteristic
290C2×C8D4×D6111yescharacteristic
291C2×C8#1175822411yescharacteristic
292C2×C8#1175822611yescharacteristic
293C4⋊C4—21no—
294C4⋊C4—1222no—
295C4⋊C4—1221no—
296C22⋊C4—612no—
297C22⋊C4—1222no—
298C22⋊C4C8×D6112yes—
299C42—21no—
300C42—1222no—
301C42—1221no—
302C16—41no—
303C16—41no—
304C16—2441no—
305C16—2441no—
306C23—611no—
307C23—611no—
308C23—1221no—
309C23—1221no—
310C23—4881no—
311C23C2×C8×D6111yescharacteristic
312C2×C4—22no—
313C2×C4—611no—
314C2×C4—611no—
315C2×C4—1221no—
316C2×C4—1221no—
317C2×C4—1221no—
318C2×C4—1221no—
319C2×C4—1221no—
320C2×C4—1221no—
321C2×C4—1221no—
322C2×C4—1221no—
323C2×C4—1221no—
324C2×C4—1221no—
325C2×C4—1222no—
326C2×C4—4882no—
327C2×C4C22⋊C4×D6111yescharacteristic
328C2×C4#1451973411yescharacteristic
329C2×C4#1451973511yescharacteristic
330C2×C4M16×D6111yescharacteristic
331C2×C4M16×D6111yescharacteristic
332C2×C4C2×#1175818911yescharacteristic
333C8—22no—
334C8—1222no—
335C8—1221no—
336C8—1221no—
337C8#1451975911yescharacteristic
338C8#1451975911yescharacteristic
339C8#1451976211yescharacteristic
340C8#1451976211yescharacteristic
341C22—41no—
342C22—1221no—
343C22—1221no—
344C22—4881no—
345C22—4881no—
346C22—2441no—
347C22—2441no—
348C22C22⋊C8×D6111yescharacteristic
349C22#1683981311yescharacteristic
350C22#1683981411yescharacteristic
351C4—42no—
352C4—1221no—
353C4—1221no—
354C4—2442no—
355C4#1683962411yescharacteristic
356C4#1683972411yescharacteristic
357C4#1683981711yescharacteristic
358C4#1683981811yescharacteristic
359C2—21no—
360C2—21no—
361C2—21no—
362C2—2441no—
363C2—2441no—
364C2#2144588611yescharacteristic
subgroupG/H
centerC4#16839624
commutatorC23×C61C22×C8
FrattiniC22×C8C22×D61
FittingC61×#1378C2
socleC2×C61#1817
prime radicalC23×D61C2×C8
cosocleC22×C8×C61C23
hypercenter#1378D61
hypocenterC61#12851
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C8 ◁ #1378

Complements

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

NH ≅ G/Ntype
4. #21443906C2semidirect
5. #21443905C2semidirect
6. #21443904C2semidirect
7. C61×#1378C2semidirect
183. #1378D61semidirect
209. C61#12851semidirect

Maximal subgroups

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

subgroupindex
#214542352
C2×#168395432
#214439062
#214439052
#214439042
C61×#13782
#1285161