Finite Group Databasebeta

← #29596940

Subgroups — #29596940

Order 15616.

470 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#21458921C211yesmaximal, characteristic, complemented
2#21454524C211yesmaximal, characteristic
3#21447439C211yesmaximal, characteristic, complemented
4C2×#16839543C211yesmaximal, characteristic, complemented
5C2×#16839543C211yesmaximal, characteristic, complemented
6#21443905C211yesmaximal, characteristic, complemented
7C61×#1378C211yesmaximal, characteristic, complemented
8C2×#14519783C2211yescharacteristic
9C22×#11758189C2211yescharacteristic
10#16840274C411yescharacteristic
11#16840180—22no—
12#16840178—22no—
13C2×#14519717C2211yescharacteristic
14C2×#14519717C2211yescharacteristic
15M32×D61—22no—
16#16839841—22no—
17#16839835—22no—
18#16839789C411yescharacteristic
19#16839730C411yescharacteristic
20#16839725C411yescharacteristic
21#16839543—22no—
22#16839543—22no—
23#16839543—22no—
24C2×M32×C61C2211yescharacteristic
25C2×M32×C61C2211yescharacteristic
26C61×#405C2211yescharacteristic
27C2×#11758222C2×C411yescharacteristic
28C22×C4×D61C2×C411yescharacteristic
29#14519783—21no—
30#14519783—21no—
31#14519783—21no—
32#14519783—21no—
33C2×#11758194C2×C411yescharacteristic
34C2×#11758189—21no—
35C2×#11758189—21no—
36C2×#11758189—21no—
37C2×#11758189—21no—
38C2×#11758189—21no—
39C2×#11758189—21no—
40C2×#11758189D412yes—
41C22×#92418C2×C411yescharacteristic
42C2×C8×D61D412yes—
43#14519761C812yes—
44#14519757—21no—
45#14519752—21no—
46#14519747C812yes—
47C4×D244—22no—
48#14519717—21no—
49#14519717—21no—
50#14519717—21no—
51#14519717D412yescomplemented
52#14519716—24no—
53C2×#11758185—21no—
54C2×#11758185—21no—
55C16×D61—24no—
56C4⋊#92418—22no—
57#14519689—22no—
58#14519689—22no—
59C2×M16×C61C2×C411yescharacteristic
60C22×C8×C61C2311yescharacteristic
61#117×C61C2×C411yescharacteristic
62M32×C61—21no—
63M32×C61—21no—
64M32×C61—21no—
65M32×C61D412yescomplemented
66C2×C16×C61—21no—
67C2×C16×C61—21no—
68#97×C61—21no—
69C4×C8×C61—21no—
70C23×D61C2×C811yescharacteristic
71C22×C4×C61C22×C411yescharacteristic
72C2×C22⋊D61C2×C812yes—
73C22×Dic61C2×C811yescharacteristic
74#11758222—21no—
75#11758222—21no—
76#11758222—21no—
77#11758222—21no—
78C2×D244—22no—
79C2×D244M1611yescharacteristic
80C2×C4×D61—21no—
81C2×C4×D61—21no—
82C2×C4×D61—21no—
83C2×C4×D61—21no—
84C2×C4×D61—21no—
85C2×C4×D61—21no—
86C2×C4×D61C22⋊C412yes—
87C2×Dic122M1611yescharacteristic
88M16×C61—21no—
89M16×C61—21no—
90C2×C8×C61—21no—
91C2×C8×C61C2×D411yescharacteristic
92C2×C8×C61C2×D411yescharacteristic
93C2×C8×C61C2×D411yescharacteristic
94C2×C8×C61C2×D411yescharacteristic
95C2×C8×C61#4111yescharacteristic
96C2×C8×C61#4111yescharacteristic
97C4⋊C4×C61—21no—
98C22⋊C4×C61C2×C812yes—
99C42×C61—21no—
100#11758198—22no—
101#11758198—22no—
102C4⋊Dic61—22no—
103#11758194—21no—
104#11758194—21no—
105C2×#92418—21no—
106C2×#92418—21no—
107C2×#92418—21no—
108C2×#92418C22⋊C412yes—
109#11758189—22no—
110#11758189—22no—
111#11758189—41no—
112#11758189—41no—
113#11758189—41no—
114#11758189—41no—
115C8×D61—22no—
116C8×D61—22no—
117C16×C61—22no—
118C16×C61—41no—
119#11758185—22no—
120#11758185—41no—
121C23×C61C22×C811yescharacteristic
122C22×D61—21no—
123C22×D61—21no—
124C22×D61—21no—
125C22×D61—81no—
126C22×D61C22⋊C812yes—
127C2×C4×C61—22no—
128C2×C4×C61C2×C22⋊C411yescharacteristic
129C2×C4×C61C4×D411yescharacteristic
130C2×C4×C61C4×D411yescharacteristic
131C2×C4×C61C2×M1611yescharacteristic
132C2×C4×C61#13111yescharacteristic
133C2×C4×C61#13111yescharacteristic
134C22⋊D61—22no—
135C22⋊D61—22no—
136C2×Dic61—21no—
137C2×Dic61—21no—
138C2×Dic61—21no—
139C2×Dic61C22⋊C812yes—
140D244—21no—
141D244—21no—
142D244—82no—
143C4×D61—22no—
144C4×D61—22no—
145C4×D61—41no—
146C4×D61—41no—
147C4×D61—41no—
148C4×D61—41no—
149Dic122—21no—
150Dic122—21no—
151C8×C61—41no—
152C8×C61#12011yescharacteristic
153C8×C61#12111yescharacteristic
154C8×C61#12111yescharacteristic
155C8×C61#12311yescharacteristic
156#92418—22no—
157#92418—41no—
158#12989—611nomaximal, Sylow
159C22×C61—41no—
160C22×C61C2×C22⋊C811yescharacteristic
161C22×C61C8×D411yescharacteristic
162C22×C61C8×D411yescharacteristic
163C2×D61—22no—
164C2×D61—42no—
165C2×D61—41no—
166C2×D61—41no—
167C2×D61—41no—
168C2×D61—81no—
169C2×D61—81no—
170C4×C61—42no—
171C4×C61#38611yescharacteristic
172C4×C61#40811yescharacteristic
173C4×C61#43511yescharacteristic
174C4×C61#43511yescharacteristic
175Dic61—22no—
176Dic61—41no—
177#3407—611no—
178#2930—611no—
179#2115—611no—
180C2×#349—611no—
181C2×#349—611no—
182#1378—611no—
183#1378D6111yescharacteristic, complemented
184C2×C61—21no—
185C2×C61—21no—
186C2×C61—21no—
187C2×C61#186911yescharacteristic
188D61—42no—
189D61—81no—
190C2×#131—611no—
191C22×M16—611no—
192#525—611no—
193#504—1222no—
194#504—1222no—
195C2×M32—611no—
196C2×M32—611no—
197C2×M32—1222no—
198C2×M32C2×D6111yescharacteristic
199C2×M32C2×D6111yescharacteristic
200#436—1222no—
201#435—1222no—
202#424—611no—
203#408—611no—
204#407—611no—
205#405C2×D6111yescharacteristic
206#349—1222no—
207#349—1222no—
208#349—1222no—
209C61#1298911yescharacteristic, complemented, Sylow
210C2×#41—611no—
211C23×C4—611no—
212#131—1221no—
213#131—1221no—
214#131—1221no—
215#131—1221no—
216C2×M16—611no—
217C2×M16—612no—
218C2×M16—1221no—
219C2×M16—1221no—
220C2×M16—1221no—
221C2×M16—1221no—
222C2×M16—1221no—
223C2×M16—1221no—
224C2×M16C4×D6111yescharacteristic
225C22×C8—612no—
226C22×C8—611no—
227C22×C8C22×D6111yescharacteristic
228#123—612no—
229C22⋊Q8—1221no—
230#121—1221no—
231#120—612no—
232C4×D4—1222no—
233#117C4×D6111yescharacteristic
234M32—21no—
235M32—21no—
236M32—21no—
237M32—612no—
238M32—1224no—
239M32—1221no—
240M32—1221no—
241M32—1221no—
242M32D24412yescomplemented
243C2×C16—21no—
244C2×C16—21no—
245C2×C16—1224no—
246C2×C16—1221no—
247C2×C16—1221no—
248C4⋊C8—1222no—
249C22⋊C8—1222no—
250C22⋊C8—1222no—
251#97—21no—
252C4×C8—21no—
253C24—611no—
254#41—1221no—
255#41—1221no—
256#41—1221no—
257#41—1221no—
258C2×Q8—611no—
259C2×D4—612no—
260C2×D4—611no—
261C2×D4—1222no—
262C22×C4—612no—
263C22×C4—611no—
264C22×C4—1221no—
265C22×C4—1221no—
266C22×C4—1221no—
267C22×C4—1221no—
268C22×C4—1221no—
269C22×C4—1221no—
270C22×C4C2×C4×D6111yescharacteristic
271M16—21no—
272M16—21no—
273M16—1221no—
274M16—1221no—
275M16—1222no—
276M16—1222no—
277M16—2441no—
278M16—2441no—
279M16—2441no—
280M16—2441no—
281C2×C8—21no—
282C2×C8—612no—
283C2×C8—1222no—
284C2×C8—1222no—
285C2×C8—1221no—
286C2×C8—1221no—
287C2×C8—1221no—
288C2×C8C2×D24411yescharacteristic
289C2×C8#1175822211yescharacteristic
290C2×C8D4×D6111yescharacteristic
291C2×C8D4×D6111yescharacteristic
292C2×C8D4×D6111yescharacteristic
293C2×C8#1175822611yescharacteristic
294C4⋊C4—21no—
295C4⋊C4—1222no—
296C22⋊C4—1222no—
297C22⋊C4—1222no—
298C22⋊C4C8×D6112yes—
299C42—21no—
300C16—22no—
301C16—41no—
302C16—1222no—
303C16—2441no—
304C23—612no—
305C23—1221no—
306C23—1221no—
307C23—1221no—
308C23—4881no—
309C23C2×C8×D6111yescharacteristic
310Q8—1221no—
311Q8—1221no—
312D4—1222no—
313D4—1222no—
314D4—1221no—
315D4—1221no—
316D4—4882no—
317C2×C4—22no—
318C2×C4—612no—
319C2×C4—1222no—
320C2×C4—1222no—
321C2×C4—1221no—
322C2×C4—1221no—
323C2×C4—1221no—
324C2×C4—2441no—
325C2×C4—2441no—
326C2×C4—2441no—
327C2×C4—2441no—
328C2×C4C22⋊C4×D6111yescharacteristic
329C2×C4#1451973911yescharacteristic
330C2×C4C4×D24411yescharacteristic
331C2×C4M16×D6111yescharacteristic
332C2×C4#1451978311yescharacteristic
333C2×C4#1451978511yescharacteristic
334C8—41no—
335C8—1222no—
336C8—2441no—
337C8#1451974711yescharacteristic
338C8#1451975011yescharacteristic
339C8#1451975211yescharacteristic
340C8#1451976211yescharacteristic
341C22—41no—
342C22—1222no—
343C22—4881no—
344C22—4881no—
345C22—2441no—
346C22—2441no—
347C22—2442no—
348C22—2441no—
349C22C22⋊C8×D6111yescharacteristic
350C22#1683982711yescharacteristic
351C22C8×D24411yescharacteristic
352C4—42no—
353C4—1222no—
354C4—2441no—
355C4#1683964011yescharacteristic
356C4#1683972711yescharacteristic
357C4#1683983211yescharacteristic
358C4#1683983511yescharacteristic
359C2—21no—
360C2—21no—
361C2—21no—
362C2—4881no—
363C2—2442no—
364C2#2144611011yescharacteristic
subgroupG/H
centerC4#16839640
commutatorC23×C61C22×C8
FrattiniC22×C8C22×D61
FittingC61×#1378C2
socleC2×C61#1869
prime radical#16840274C4
cosocleC22×C8×C61C23
hypercenter#1378D61
hypocenterC61#12989
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
1. #21458921C2semidirect
3. #21447439C2semidirect
4. C2×#16839543C2semidirect
5. C2×#16839543C2semidirect
6. #21443905C2semidirect
7. C61×#1378C2semidirect
51. #14519717D4semidirect
65. M32×C61D4semidirect
183. #1378D61semidirect
209. C61#12989semidirect
242. M32D244semidirect

Maximal subgroups

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

subgroupindex
#214589212
#214545242
#214474392
C2×#168395432
#214439052
C61×#13782
#1298961