Finite Group Databasebeta

← #29833462

Subgroups — #29833462

Order 15616.

652 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458955C211yesmaximal, characteristic
2C2×#16839883C211yesmaximal, characteristic, complemented
3#21454619C211yesmaximal, characteristic, complemented
4#21447716C214yesmaximal, complemented
5#21447709C214yesmaximal, complemented
6#21447566C211yesmaximal, characteristic, complemented
7#21447554C211yesmaximal, characteristic, complemented
8C2×#16839539C211yesmaximal, characteristic, complemented
9C61×#2155C211yesmaximal, characteristic, complemented
10C2×#14519799C2211yescharacteristic
11C2×#14519799C2211yescharacteristic
12C2×#14519785—21no—
13#16840230—21no—
14#16840205C412yescomplemented
15#16840205C2212yes—
16#16840200C412yescomplemented
17#16840200C2212yes—
18#16840194C414yescomplemented
19C2×#14519717C2211yescharacteristic
20#16839883—22no—
21#16839883C2214yes—
22#16839873C2212yescomplemented
23#16839868C2212yescomplemented
24#16839861—22no—
25C2×#14519703—21no—
26#16839756—21no—
27C2×#14519687C2211yescharacteristic
28#16839577C2212yescomplemented
29#16839574C2212yescomplemented
30#16839571C2214yes—
31#16839539C2214yescomplemented
32C2×M32×C61C2211yescharacteristic
33C2×#108×C61C2211yescharacteristic
34C61×#427C2211yescharacteristic
35C61×#365C2214yescomplemented
36C2×#11758222—21no—
37C2×#11758222—21no—
38#14519799—21no—
39#14519799—21no—
40#14519799—21no—
41#14519799—21no—
42#14519799C2×C412yes—
43#14519799C2×C412yes—
44#14519799C2×C412yescomplemented
45#14519799C2312yes—
46C2×Dic244C2×C411yescharacteristic, complemented
47C2×Dic244C2311yescharacteristic
48C2×#11758190—21no—
49C2×#11758190—21no—
50C2×D488C2×C411yescharacteristic, complemented
51C2×D488C2311yescharacteristic
52#14519785—24no—
53#14519785—42no—
54C2×#11758189—41no—
55C22×#92418D411yescharacteristic
56C2×C8×D61—41no—
57#14519742—22no—
58#14519739—22no—
59#14519737—42no—
60#14519734—41no—
61C2×C4×Dic61D411yescharacteristic
62C42×D61—41no—
63Dic488C2×C414yescomplemented
64#14519722C2×C414yescomplemented
65#14519722C2×C414yescomplemented
66D976C2×C414yescomplemented
67#14519717—21no—
68#14519717C2312yes—
69C2×#11758185—21no—
70#14519703—24no—
71#14519700—22no—
72#14519697—22no—
73#14519687D412yes—
74#14519687C2312yes—
75C2×M16×C61—21no—
76C22×C8×C61C2311yescharacteristic
77#117×C61—21no—
78M32×C61C2×C412yescomplemented
79M32×C61C2312yes—
80C2×C16×C61C2×C412yescomplemented
81#108×C61—21no—
82#108×C61C2312yes—
83#107×C61—21no—
84#106×C61C2312yes—
85C22×C4×C61C2×D411yescharacteristic
86C2×C22⋊D61—41no—
87C2×C22⋊D61—41no—
88C22×Dic61—21no—
89#11758222—22no—
90#11758222—22no—
91#11758222—22no—
92#11758222—22no—
93#11758222—41no—
94#11758222—41no—
95#11758222—41no—
96#11758222—41no—
97C2×D244—21no—
98C2×D244—21no—
99C2×C4×D61—41no—
100C2×C4×D61—41no—
101C2×C4×D61—42no—
102C2×Dic122—21no—
103C2×Dic122—21no—
104M16×C61—22no—
105M16×C61—41no—
106C2×C8×C61—41no—
107C2×C8×C61C22×C411yescharacteristic
108C2×C8×C61C22×C411yescharacteristic
109C2×C8×C61C2×D411yescharacteristic
110C2×C8×C61#4111yescharacteristic
111C2×C8×C61#4111yescharacteristic
112C2×C8×C61C2411yescharacteristic
113C4⋊C4×C61—22no—
114C22⋊C4×C61—41no—
115C42×C61—41no—
116#11758198—42no—
117#11758196—42no—
118C4×Dic61—21no—
119C4×Dic61—42no—
120C4×Dic61C2×D412yes—
121C2×#92418—21no—
122C2×#92418—21no—
123C2×#92418C2×D412yes—
124Dic244—21no—
125Dic244—21no—
126Dic244C22×C412yes—
127Dic244C22×C412yes—
128D488—21no—
129D488—21no—
130D488C22×C412yes—
131D488C22×C412yes—
132#11758190—22no—
133#11758190—22no—
134#11758190—22no—
135#11758190—22no—
136#11758189—44no—
137C8×D61—44no—
138C16×C61C22×C414yes—
139#11758185—22no—
140C23×C61AGL(1,ℤ/8)11yescharacteristic
141C22×D61—41no—
142C22×D61—41no—
143C2×C4×C61—42no—
144C2×C4×C61C4×D411yescharacteristic
145C2×C4×C61C4×D411yescharacteristic
146C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C61C22×D411yescharacteristic
150C22⋊D61—42no—
151C22⋊D61—42no—
152C22⋊D61—42no—
153C22⋊D61—42no—
154C2×Dic61—22no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61—81no—
158D244—22no—
159D244—22no—
160D244—41no—
161D244—41no—
162C4×D61—42no—
163C4×D61—42no—
164C4×D61—42no—
165C4×D61—42no—
166C4×D61—84no—
167Dic122—22no—
168Dic122—22no—
169Dic122—41no—
170Dic122—41no—
171C8×C61—42no—
172C8×C61C4×D411yescharacteristic
173C8×C61C4×D411yescharacteristic
174C8×C61C23×C411yescharacteristic
175C8×C61C2×#4111yescharacteristic
176#92418—22no—
177#33468—611nomaximal, Sylow
178C22×C61—41no—
179C22×C61#44211yescharacteristic
180C22×C61#44211yescharacteristic
181C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
182C2×D61—42no—
183C2×D61—42no—
184C2×D61—82no—
185C2×D61—82no—
186C4×C61—82no—
187C4×C61#44211yescharacteristic
188C4×C61#44211yescharacteristic
189C4×C61C2×C4×D411yescharacteristic
190C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
191Dic61—42no—
192Dic61—42no—
193Dic61—42no—
194#3414—611no—
195C2×#444—611no—
196#2943—611no—
197AGL(1,ℤ/16)—614no—
198#2180—614no—
199#2155D6111yescharacteristic, complemented
200#2148—611no—
201#2145—611no—
202C2×#347—611no—
203C2×C61—21no—
204C2×C61—21no—
205C2×C61—21no—
206C2×C61C2×#44211yescharacteristic
207D61—82no—
208D61—82no—
209C2×#135—611no—
210C2×#135—611no—
211C2×#131—1221no—
212C4×#41—1221no—
213#510—612no—
214#510—612no—
215#509—612no—
216#509—612no—
217#508—614no—
218C2×M32—611no—
219C2×M32C2×D6111yescharacteristic
220#444—614no—
221#444—1222no—
222#442—612no—
223#441—612no—
224#440—1222no—
225C2×#108C2×D6111yescharacteristic
226#427C2×D6111yescharacteristic
227C2×#104—1221no—
228#416—1221no—
229C2×#97—611no—
230#365C2×D6114yescomplemented
231#362—612no—
232#361—612no—
233#360—614no—
234#347—614no—
235C61#3346811yescharacteristic, complemented, Sylow
236C2×#41—1221no—
237C2×#41—1221no—
238#135—612no—
239#135—612no—
240#135—612no—
241#135—612no—
242#135—1221no—
243#135—1221no—
244#135—1221no—
245#135—1221no—
246C2×Q16—611no—
247C2×Q16—611no—
248C2×SD16—1221no—
249C2×SD16—1221no—
250C2×D8—611no—
251C2×D8—611no—
252#131—1224no—
253#131—2442no—
254C2×M16—21no—
255C2×M16—2441no—
256C22×C8—611no—
257C22×C8—2441no—
258C22×C8C22×D6111yescharacteristic
259C4×Q8—1222no—
260C4×D4—1222no—
261C4×D4—2442no—
262#117—21no—
263#117—2441no—
264C2×C42—611no—
265C2×C42—2441no—
266Q32—614no—
267SD32—614no—
268SD32—614no—
269D16—614no—
270M32—612no—
271M32—1221no—
272M32C4×D6112yescomplemented
273M32C22×D6112yes—
274C2×C16—1221no—
275C2×C16C4×D6112yescomplemented
276#108—21no—
277#108C22×D6112yes—
278#107—21no—
279#106C22×D6112yes—
280#104—1224no—
281#103—1222no—
282#102—1222no—
283#97—612no—
284#97—612no—
285#41—1222no—
286#41—1222no—
287#41—1222no—
288#41—1222no—
289#41—2441no—
290#41—2441no—
291#41—2441no—
292#41—2441no—
293C2×Q8—1221no—
294C2×Q8—1221no—
295C2×D4—1221no—
296C2×D4—1221no—
297C2×D4—2441no—
298C2×D4—2441no—
299C22×C4—1221no—
300C22×C4—2442no—
301C22×C4—2441no—
302C22×C4—2441no—
303C22×C4D4×D6111yescharacteristic
304Q16—612no—
305Q16—612no—
306Q16—1221no—
307Q16—1221no—
308SD16—1222no—
309SD16—1222no—
310SD16—1222no—
311SD16—1222no—
312D8—612no—
313D8—612no—
314D8—1221no—
315D8—1221no—
316M16—22no—
317M16—41no—
318M16—2444no—
319C2×C8—41no—
320C2×C8—612no—
321C2×C8—1221no—
322C2×C8—1221no—
323C2×C8—2444no—
324C2×C8C2×C4×D6111yescharacteristic
325C2×C8C2×C4×D6111yescharacteristic
326C2×C8D4×D6111yescharacteristic
327C2×C8#1175822611yescharacteristic
328C2×C8#1175822611yescharacteristic
329C2×C8C23×D6111yescharacteristic
330C4⋊C4—22no—
331C4⋊C4—2442no—
332C22⋊C4—41no—
333C22⋊C4—2442no—
334C42—41no—
335C42—612no—
336C42—1221no—
337C42—2442no—
338C16—1222no—
339C16C2×C4×D6114yes—
340C23—2441no—
341C23—2441no—
342C23#1451980411yescharacteristic
343Q8—1222no—
344Q8—1222no—
345Q8—2441no—
346Q8—2441no—
347D4—1222no—
348D4—1222no—
349D4—2442no—
350D4—2442no—
351D4—2442no—
352D4—2442no—
353D4—2441no—
354D4—2441no—
355C2×C4—42no—
356C2×C4—1222no—
357C2×C4—4884no—
358C2×C4—4881no—
359C2×C4—2441no—
360C2×C4—2441no—
361C2×C4—2442no—
362C2×C4—2442no—
363C2×C4—2442no—
364C2×C4—2442no—
365C2×C4#1451973911yescharacteristic
366C2×C4#1451973911yescharacteristic
367C2×C4#1451980411yescharacteristic
368C2×C4#1451980411yescharacteristic
369C2×C4#1451980411yescharacteristic
370C2×C4C2×D4×D6111yescharacteristic
371C8—42no—
372C8—1222no—
373C8#1451973911yescharacteristic
374C8#1451973911yescharacteristic
375C8C22×C4×D6111yescharacteristic
376C8C2×#1175822611yescharacteristic
377C22—41no—
378C22—4882no—
379C22—4882no—
380C22—2442no—
381C22—2442no—
382C22#1683987311yescharacteristic
383C22#1683987311yescharacteristic
384C22C2×#1451980411yescharacteristic
385C4—82no—
386C4—2442no—
387C4—2442no—
388C4—2442no—
389C4#1683987311yescharacteristic
390C4#1683987311yescharacteristic
391C4C2×#1451973911yescharacteristic
392C4C2×#1451980411yescharacteristic
393C2—21no—
394C2—21no—
395C2—21no—
396C2—4882no—
397C2—4882no—
398C2C2×#1683987311yescharacteristic
subgroupG/H
centerC4#16839873
commutatorC8×C61C23×C4
FrattiniC2×C8C23×D61
FittingC61×#2155C2
socleC2×C61C2×#442
prime radical#21458955C2
cosocleC2×C8×C61C24
hypercenter#2155D61
hypocenterC61#33468
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ #2155

Complements

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

NH ≅ G/Ntype
2. C2×#16839883C2semidirect
3. #21454619C2semidirect
4. #21447716C2semidirect
5. #21447709C2semidirect
6. #21447566C2semidirect
7. #21447554C2semidirect
8. C2×#16839539C2semidirect
9. C61×#2155C2semidirect
14. #16840205C4semidirect
16. #16840200C4semidirect
18. #16840194C4semidirect
22. #16839873C22semidirect
23. #16839868C22semidirect
28. #16839577C22semidirect
29. #16839574C22semidirect
31. #16839539C22semidirect
35. C61×#365C22semidirect
44. #14519799C2×C4semidirect
46. C2×Dic244C2×C4semidirect
50. C2×D488C2×C4semidirect
63. Dic488C2×C4semidirect
64. #14519722C2×C4semidirect
65. #14519722C2×C4semidirect
66. D976C2×C4semidirect
78. M32×C61C2×C4semidirect
80. C2×C16×C61C2×C4semidirect
199. #2155D61semidirect
230. #365C2×D61semidirect
235. C61#33468semidirect
272. M32C4×D61semidirect
275. C2×C16C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214589552
C2×#168398832
#214546192
#214477162
#214477092
#214475662
#214475542
C2×#168395392
C61×#21552
#3346861