Finite Group Databasebeta

← #30624269

Subgroups — #30624269

Order 15616.

646 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840821C211yesmaximal, characteristic, complemented
2C2×#16840820C211yesmaximal, characteristic, complemented
3#21462210C214yesmaximal, complemented
4#21462207C214yesmaximal, complemented
5#21462162C211yesmaximal, characteristic, complemented
6#21462063C211yesmaximal, characteristic, complemented
7#21462061C211yesmaximal, characteristic, complemented
8#21462028C211yesmaximal, characteristic, complemented
9#21454718C211yesmaximal, characteristic
10#16840821—21no—
11#16840821—21no—
12#16840821C2212yescomplemented
13#16840821C2212yescomplemented
14#16840820—21no—
15#16840820—21no—
16#16840820C2212yescomplemented
17#16840820C2212yescomplemented
18C2×#14519852—21no—
19C2×#14519851—21no—
20C2×#14519851C2211yescharacteristic
21C2×#14519851C2211yescharacteristic
22C2×#14519851C2211yescharacteristic
23C2×D61⋊C16—21no—
24C22×#11758187—21no—
25C22×#11758187C2211yescharacteristic
26#16840685C2214yescomplemented
27#16840661C2214yescomplemented
28#16840660C2214yescomplemented
29#16840650C2212yescomplemented
30C4×#11758187C2212yescomplemented
31#16840560C411yescharacteristic
32C2×#14519785C2211yescharacteristic
33C2×#14519784C2211yescharacteristic
34#16839881C414yes—
35#16839876C2214yes—
36#16839810C411yescharacteristic
37#16839780C411yescharacteristic
38#16839718C2211yescharacteristic
39C61×#421C411yescharacteristic
40#14519852—22no—
41#14519852—22no—
42#14519851—21no—
43#14519851—21no—
44#14519851—21no—
45#14519851—21no—
46#14519851—22no—
47#14519851—22no—
48#14519851C2312yes—
49#14519851C2312yes—
50D61⋊C16—22no—
51D61⋊C16—22no—
52C2×#11758187—21no—
53C2×#11758187—21no—
54C2×#11758187—22no—
55C2×#11758187—22no—
56C2×#11758187D411yescharacteristic
57C2×#11758187D411yescharacteristic
58C2×#11758187D411yescharacteristic
59C2×#11758187D411yescharacteristic
60C2×#11758187C2312yes—
61C2×#11758187C2311yescharacteristic
62C2×#11758187C2311yescharacteristic
63C2×#11758222C2×C411yescharacteristic
64#14519808C812yes—
65#14519807C812yes—
66#14519803C812yes—
67#14519802C812yes—
68#14519797C2×C414yes—
69#14519785—21no—
70#14519785—21no—
71#14519785C2×C412yes—
72#14519785C2312yes—
73#14519784—21no—
74#14519784—21no—
75#14519784C2×C412yes—
76#14519784C2312yes—
77C2×#11758194—21no—
78C2×#11758194C2×C411yescharacteristic
79C2×#11758194C2×C411yescharacteristic
80C2×#11758189—21no—
81C22×#92418—21no—
82C22×#92418C2311yescharacteristic
83C2×C8×D61—21no—
84#14519712C2×C414yes—
85#14519705C2×C414yes—
86#14519688C2×C412yes—
87C4×#92418C2312yes—
88C2×#41×C61C2×C411yescharacteristic
89C2×M16×C61C2×C411yescharacteristic
90#117×C61C2×C411yescharacteristic
91#104×C61C2×C414yes—
92#41×C61—21no—
93#41×C61—21no—
94#41×C61C2×C812yes—
95#41×C61C22×C412yes—
96C2×Q8×C61C2×C811yescharacteristic
97C2×D4×C61—21no—
98C2×D4×C61C2×C811yescharacteristic
99C22×C4×C61—21no—
100C22×C4×C61C22×C411yescharacteristic
101C2×C22⋊D61—21no—
102#11758222—21no—
103#11758222—21no—
104#11758222C2×C812yes—
105#11758222C22×C412yes—
106C2×D244C2×C811yescharacteristic
107C2×C4×D61—21no—
108C2×Dic122C2×C811yescharacteristic
109M16×C61—21no—
110M16×C61C22×C412yes—
111C2×C8×C61—21no—
112C4⋊C4×C61—21no—
113C22⋊C4×C61—21no—
114C42×C61C22×C412yes—
115#11758202C2×C814yes—
116#11758201C2×C814yes—
117#11758200C2×C814yes—
118#11758199C2×C814yes—
119#11758194—21no—
120#11758194—22no—
121#11758194—22no—
122#11758194C2×C812yes—
123#11758194C2×C812yes—
124#11758194C22×C412yes—
125C2×#92418—21no—
126C2×#92418—22no—
127C2×#92418—22no—
128C2×#92418C22×C411yescharacteristic
129C2×#92418C22×C412yes—
130C2×#92418C2×D411yescharacteristic
131C2×#92418C2×D411yescharacteristic
132C2×#92418#4111yescharacteristic
133C2×#92418#4111yescharacteristic
134C2×#92418C2411yescharacteristic
135#11758189—22no—
136#11758189—22no—
137C8×D61—22no—
138C8×D61—22no—
139#11758187—22no—
140#11758187—22no—
141#11758187—22no—
142#11758187C2×D411yescharacteristic
143#11758187C2×D411yescharacteristic
144#11758187C2×D411yescharacteristic
145#11758187C2×D411yescharacteristic
146C23×C61—21no—
147C23×C61#13111yescharacteristic
148C22×D61—21no—
149Q8×C61—21no—
150Q8×C61C22×C812yes—
151D4×C61—22no—
152D4×C61—22no—
153D4×C61—21no—
154D4×C61C22×C812yes—
155C2×C4×C61—21no—
156C2×C4×C61—22no—
157C2×C4×C61—22no—
158C2×C4×C61—22no—
159C2×C4×C61C4×D411yescharacteristic
160C2×C4×C61C4×D411yescharacteristic
161C2×C4×C61C22×C811yescharacteristic
162C2×C4×C61C22×C811yescharacteristic
163C2×C4×C61#13111yescharacteristic
164C2×C4×C61C23×C411yescharacteristic
165C22⋊D61—22no—
166C22⋊D61—22no—
167C2×Dic61—21no—
168D244—21no—
169D244C22×C812yes—
170C4×D61—22no—
171C4×D61—22no—
172Dic122—21no—
173Dic122C22×C812yes—
174C8×C61—22no—
175#92418—22no—
176#92418C4×D411yescharacteristic
177#92418C4×D411yescharacteristic
178#92418C22×C814yes—
179#92418C22×D411yescharacteristic
180#92418C2×#4111yescharacteristic
181#33441—611nomaximal, Sylow
182C22×C61—22no—
183C22×C61—41no—
184C22×C61—42no—
185C22×C61C8×D411yescharacteristic
186C22×C61C8×D411yescharacteristic
187C22×C61C2×#13111yescharacteristic
188C2×D61—22no—
189C2×D61—42no—
190C4×C61—22no—
191C4×C61—42no—
192C4×C61C8×D411yescharacteristic
193C4×C61C8×D411yescharacteristic
194C4×C61C2×C4×D411yescharacteristic
195C4×C61C23×C811yescharacteristic
196Dic61—22no—
197C2×#504—611no—
198C2×#504—611no—
199#2955—611no—
200#2171—614no—
201#2170—614no—
202#2153—611no—
203#2117—611no—
204#2117—611no—
205#2108—611no—
206C2×C61—21no—
207C2×C61—21no—
208C2×C61—21no—
209C2×C61—42no—
210C2×C61C2×C8×D411yescharacteristic
211D61—42no—
212#575—611no—
213C2×#131—611no—
214C2×#131—611no—
215#504—612no—
216#504—612no—
217#504—612no—
218#504—612no—
219#504—1221no—
220#504—1221no—
221#504—1221no—
222#504—1221no—
223C2×M32—611no—
224C2×M32—611no—
225C2×M32—611no—
226C2×M32—1221no—
227C2×M32—1221no—
228C22×C16—611no—
229C22×C16—1221no—
230C22×C16—1221no—
231#444—614no—
232#443—614no—
233#430—611no—
234#421—611no—
235#421#692311yescharacteristic
236#405—611no—
237#364—614no—
238#350—614no—
239#350—614no—
240#346—612no—
241C4×C16—612no—
242C61#3344111yescharacteristic, complemented, Sylow
243C2×#41—611no—
244C2×#41C2×#692311yescharacteristic
245#137—612no—
246#137—612no—
247AGL(1,ℤ/8)—612no—
248AGL(1,ℤ/8)—612no—
249#135—614no—
250#131—612no—
251#131—612no—
252#131—612no—
253#131—612no—
254#131—1221no—
255#131—1221no—
256#131—1221no—
257#131—1221no—
258C2×M16—611no—
259C2×M16—611no—
260C2×M16—1221no—
261C2×M16—1221no—
262C2×M16C2×#692311yescharacteristic
263C22×C8—611no—
264C22×C8—1221no—
265C22×C8—1221no—
266#117C2×#692311yescharacteristic
267M32—612no—
268M32—612no—
269M32—1222no—
270M32—1222no—
271M32—1222no—
272M32—1222no—
273M32—1221no—
274M32—1221no—
275M32—1221no—
276M32—1221no—
277C2×C16—611no—
278C2×C16—611no—
279C2×C16—611no—
280C2×C16—611no—
281C2×C16—611no—
282C2×C16—611no—
283C2×C16—612no—
284C2×C16—1222no—
285C2×C16—1222no—
286C2×C16—1222no—
287C2×C16—1222no—
288C2×C16—1221no—
289C2×C16—1221no—
290#108—614no—
291#104—614no—
292#104C2×#692314yes—
293#97—612no—
294C4×C8—612no—
295#41—21no—
296#41—21no—
297#41—612no—
298#41—612no—
299#41—1221no—
300#41—1221no—
301#41D61⋊C812yes—
302#41C22×#692312yes—
303C2×Q8—611no—
304C2×Q8D61⋊C811yescharacteristic
305C2×D4—21no—
306C2×D4—611no—
307C2×D4—1221no—
308C2×D4D61⋊C811yescharacteristic
309C22×C4—21no—
310C22×C4—1221no—
311C22×C4C22×#692311yescharacteristic
312Q16—614no—
313SD16—614no—
314SD16—614no—
315D8—614no—
316M16—21no—
317M16—612no—
318M16—612no—
319M16—612no—
320M16—1222no—
321M16—1222no—
322M16—1222no—
323M16—1222no—
324M16—1221no—
325M16C22×#692312yes—
326C2×C8—21no—
327C2×C8—611no—
328C2×C8—611no—
329C2×C8—611no—
330C2×C8—611no—
331C2×C8—611no—
332C2×C8—612no—
333C2×C8—611no—
334C2×C8—1222no—
335C2×C8—1222no—
336C2×C8—1222no—
337C2×C8—1222no—
338C2×C8—1221no—
339C4⋊C4—21no—
340C22⋊C4—21no—
341C42C22×#692312yes—
342C16—611no—
343C16—611no—
344C16—611no—
345C16—611no—
346C16—1222no—
347C16—1222no—
348C16—1222no—
349C23—21no—
350C23—1221no—
351C23#1451986411yescharacteristic
352Q8—21no—
353Q8—612no—
354Q8—1221no—
355Q8C2×D61⋊C812yes—
356D4—22no—
357D4—22no—
358D4—21no—
359D4—612no—
360D4—1222no—
361D4—1222no—
362D4—1221no—
363D4C2×D61⋊C812yes—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—1221no—
369C2×C4—1222no—
370C2×C4—1222no—
371C2×C4D4×#692311yescharacteristic
372C2×C4D4×#692311yescharacteristic
373C2×C4C2×D61⋊C811yescharacteristic
374C2×C4C2×D61⋊C811yescharacteristic
375C2×C4#1451986411yescharacteristic
376C2×C4C23×#692311yescharacteristic
377C8—22no—
378C8—611no—
379C8—611no—
380C8—611no—
381C8—611no—
382C8—614no—
383C8—1222no—
384C22—22no—
385C22—41no—
386C22—42no—
387C22—1222no—
388C22—2442no—
389C22#1684078911yescharacteristic
390C22#1684078911yescharacteristic
391C22C2×#1451986411yescharacteristic
392C4—22no—
393C4—42no—
394C4—1222no—
395C4#1684078911yescharacteristic
396C4#1684078911yescharacteristic
397C4C2×D4×#692311yescharacteristic
398C4C22×D61⋊C811yescharacteristic
399C2—21no—
400C2—21no—
401C2—21no—
402C2—42no—
403C2—2442no—
404C2C2×#1684078911yescharacteristic
subgroupG/H
centerC4C2×D4×#6923
commutatorC4×C61C23×C8
FrattiniC2×C4C23×#6923
FittingC61×#421C4
socleC2×C61C2×C8×D4
prime radical#16840560C4
cosocleC2×#92418C24
hypercenter#421#6923
hypocenterC61#33441
LCSG ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
1. C2×#16840821C2semidirect
2. C2×#16840820C2semidirect
3. #21462210C2semidirect
4. #21462207C2semidirect
5. #21462162C2semidirect
6. #21462063C2semidirect
7. #21462061C2semidirect
8. #21462028C2semidirect
12. #16840821C22semidirect
13. #16840821C22semidirect
16. #16840820C22semidirect
17. #16840820C22semidirect
26. #16840685C22semidirect
27. #16840661C22semidirect
28. #16840660C22semidirect
29. #16840650C22semidirect
30. C4×#11758187C22semidirect
242. C61#33441semidirect

Maximal subgroups

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

subgroupindex
C2×#168408212
C2×#168408202
#214622102
#214622072
#214621622
#214620632
#214620612
#214620282
#214547182
#3344161