Finite Group Databasebeta

← #29663172

Subgroups — #29663172

Order 15616.

726 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456746C211yesmaximal, characteristic, complemented
2#21456739C211yesmaximal, characteristic
3C2×#16840109C211yesmaximal, characteristic, complemented
4#21454532C211yesmaximal, characteristic, complemented
5#21445381C214yesmaximal, complemented
6#21445257C214yesmaximal, complemented
7#21444503C211yesmaximal, characteristic, complemented
8#21444486C211yesmaximal, characteristic, complemented
9C61×#1564C211yesmaximal, characteristic, complemented
10C2×#14519783—21no—
11#16840312C2211yescharacteristic
12C2×#14519777C2211yescharacteristic
13C2×#14519774C2211yescharacteristic
14#16840139—21no—
15#16840135—21no—
16#16840113—21no—
17#16840109—21no—
18#16840109—21no—
19#16840109—21no—
20#16840109C2212yes—
21#16840109C2212yes—
22#16839968C2214yescomplemented
23#16839958C2214yes—
24#16839903—22no—
25#16839898—22no—
26#16839841—21no—
27#16839841C2212yescomplemented
28C8×D244—21no—
29C8×D244C2212yescomplemented
30#16839786C2211yescharacteristic
31C2×#14519699—21no—
32C2×#14519699—21no—
33#16839728—21no—
34#16839509C2214yescomplemented
35#16839505C2214yes—
36C61×#431C2211yescharacteristic
37C61×#431C2211yescharacteristic
38C2×C4×C8×C61C2211yescharacteristic
39C61×#335C2212yescomplemented
40C61×#334C2212yescomplemented
41C2×#11758222—21no—
42C22×Dic122—21no—
43C22×D244—21no—
44C22×D244—21no—
45C22×D244—21no—
46C2×#11758190—24no—
47C2×D488—24no—
48#14519783—44no—
49C2×#11758194—21no—
50C2×#11758189—22no—
51C2×C8×D61—22no—
52#14519777—21no—
53#14519777—21no—
54#14519777—21no—
55#14519777D411yescharacteristic
56#14519777C2312yes—
57#14519774—21no—
58#14519774—21no—
59#14519774—21no—
60#14519774D411yescharacteristic
61#14519774C2312yes—
62#14519765—21no—
63#14519751—21no—
64#14519751D412yes—
65C4×D244—21no—
66C4×D244C2312yes—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C4⋊#92418—21no—
70C4⋊#92418C2312yes—
71#14519702—24no—
72#14519699—22no—
73#14519699—22no—
74#14519699—22no—
75#14519699—22no—
76#14519699—24no—
77#14519689—22no—
78#14519689—22no—
79C22×C8×C61D411yescharacteristic
80C22×C8×C61D411yescharacteristic
81C2×C42×C61C2311yescharacteristic
82C4⋊C8×C61C2312yes—
83C4⋊C8×C61C2312yes—
84C22⋊C8×C61—21no—
85C22⋊C8×C61—21no—
86C4×C8×C61—21no—
87C4×C8×C61—21no—
88C4×C8×C61D411yescharacteristic
89C4×C8×C61D411yescharacteristic
90C4×C8×C61C2311yescharacteristic
91C4×C8×C61C2311yescharacteristic
92C23×D61—41no—
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61#4111yescharacteristic
96C2×C22⋊D61—22no—
97C22×Dic61—41no—
98#11758222—42no—
99#11758222—42no—
100C2×D244—21no—
101C2×D244—21no—
102C2×D244—22no—
103C2×D244—22no—
104C2×D244—22no—
105C2×D244—42no—
106C2×D244—42no—
107C2×D244—41no—
108C2×D244—41no—
109C2×D244—41no—
110C2×D244—82no—
111C2×D244—82no—
112C2×D244C2×D412yes—
113C2×C4×D61—22no—
114C2×Dic122—21no—
115C2×Dic122—22no—
116C2×Dic122—41no—
117C2×Dic122—82no—
118C2×C8×C61—22no—
119C2×C8×C61—22no—
120C2×C8×C61—22no—
121C2×C8×C61—22no—
122C2×C8×C61C2×D411yescharacteristic
123C2×C8×C61C2×D411yescharacteristic
124C2×C8×C61C2×D411yescharacteristic
125C2×C8×C61C2×D411yescharacteristic
126C42×C61C2×D411yescharacteristic
127C42×C61C2×D411yescharacteristic
128C42×C61#4111yescharacteristic
129C42×C61C2411yescharacteristic
130#11758198—22no—
131#11758198—22no—
132C4⋊Dic61—21no—
133C4⋊Dic61—22no—
134C4⋊Dic61—22no—
135C4⋊Dic61—41no—
136C4⋊Dic61—41no—
137C4⋊Dic61C2×D412yes—
138#11758194—42no—
139C2×#92418—22no—
140D488—84no—
141#11758190—84no—
142#11758189—44no—
143C8×D61—44no—
144C23×C61#12111yescharacteristic
145C22×D61—22no—
146C22×D61—42no—
147C22×D61—82no—
148C22×D61—82no—
149C22×D61—82no—
150C2×C4×C61—22no—
151C2×C4×C61—22no—
152C2×C4×C61#12111yescharacteristic
153C2×C4×C61#12111yescharacteristic
154C2×C4×C61#12111yescharacteristic
155C2×C4×C61#13512yes—
156C2×C4×C61AGL(1,ℤ/8)12yes—
157C2×C4×C61C22×D411yescharacteristic
158C2×C4×C61C22×D411yescharacteristic
159C2×C4×C61C2×#4111yescharacteristic
160C22⋊D61—44no—
161C2×Dic61—22no—
162C2×Dic61—42no—
163C2×Dic61—82no—
164D244—24no—
165D244—42no—
166D244—42no—
167D244—44no—
168D244—42no—
169D244—42no—
170D244—82no—
171D244—82no—
172D244—82no—
173D244—82no—
174C4×D61—42no—
175C4×D61—42no—
176Dic122—42no—
177Dic122—44no—
178Dic122—82no—
179C8×C61—21no—
180C8×C61—21no—
181C8×C61—21no—
182C8×C61—21no—
183C8×C61—42no—
184C8×C61—42no—
185#92418—42no—
186#19198—611nomaximal, Sylow
187C22×C61—21no—
188C22×C61—21no—
189C22×C61#46312yes—
190C22×C61C2×#12111yescharacteristic
191C2×D61—42no—
192C2×D61—42no—
193C2×D61—42no—
194C2×D61—82no—
195C2×D61—82no—
196C2×D61—82no—
197C2×D61—164no—
198C4×C61—21no—
199C4×C61—21no—
200C4×C61#46312yes—
201C4×C61#47112yes—
202C4×C61C2×#13511yescharacteristic
203C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
204C4×C61#57612yes—
205Dic61—42no—
206Dic61—82no—
207#3206—611no—
208#3205—611no—
209C2×#486—611no—
210#2931—611no—
211#1701—614no—
212#1681—614no—
213#1564D6111yescharacteristic, complemented
214#1549—611no—
215#1545—611no—
216C2×C61—22no—
217C2×C61C2×#46311yescharacteristic
218C2×C61#305811yescharacteristic
219C2×C61C2×#47111yescharacteristic
220D61—82no—
221D61—162no—
222C2×#131—1221no—
223#532—611no—
224C2×C4⋊Q8—611no—
225C2×#127—611no—
226#493—1221no—
227#492—1221no—
228#487—1221no—
229#486—612no—
230#486—612no—
231#486—1221no—
232#486—1221no—
233#486—1221no—
234#461—614no—
235#459—614no—
236#448—1222no—
237#447—1222no—
238#436—612no—
239#436—1221no—
240C8×D4—612no—
241C8×D4—1221no—
242#431C2×D6111yescharacteristic
243#431C2×D6111yescharacteristic
244#423—611no—
245C2×#102—1221no—
246C2×#102—1221no—
247#408—1221no—
248C2×C4×C8C2×D6111yescharacteristic
249#335C2×D6112yescomplemented
250#334C2×D6112yescomplemented
251#332—614no—
252#331—614no—
253C61#1919811yescharacteristic, complemented, Sylow
254C2×#41—1221no—
255C22×Q8—1221no—
256C22×D4—1221no—
257C22×D4—1221no—
258C22×D4—1221no—
259C2×SD16—1224no—
260C2×D8—1224no—
261#131—2444no—
262C2×M16—1221no—
263C2×M16—1222no—
264C22×C8—1222no—
265C22×C8D24411yescharacteristic
266C22×C8D24411yescharacteristic
267C4⋊Q8—611no—
268C4⋊Q8—612no—
269C4⋊Q8—1221no—
270C4⋊Q8—1221no—
271C4⋊Q8—1221no—
272#127—611no—
273#127—612no—
274#127—1221no—
275#127—1221no—
276#127—1221no—
277#124—1221no—
278#121—612no—
279#121—1221no—
280C4×D4—612no—
281C4×D4—1221no—
282C2×C4⋊C4—1221no—
283C2×C4⋊C4—1221no—
284C2×C42C22×D6111yescharacteristic
285C4⋊C8—612no—
286C4⋊C8—1221no—
287C4⋊C8C22×D6112yes—
288C4⋊C8C22×D6112yes—
289#103—1224no—
290#102—1224no—
291#102—1222no—
292#102—1222no—
293#102—1222no—
294#102—1222no—
295C22⋊C8—21no—
296C22⋊C8—21no—
297C22⋊C8—1222no—
298C22⋊C8—1222no—
299C4×C8—21no—
300C4×C8—21no—
301C4×C8D24411yescharacteristic
302C4×C8D24411yescharacteristic
303C4×C8C22×D6111yescharacteristic
304C4×C8C22×D6111yescharacteristic
305C24—2441no—
306#41—2442no—
307#41—2442no—
308C2×Q8—1221no—
309C2×Q8—1222no—
310C2×Q8—4882no—
311C2×Q8—2441no—
312C2×D4—612no—
313C2×D4—1222no—
314C2×D4—1222no—
315C2×D4—1222no—
316C2×D4—1222no—
317C2×D4—1221no—
318C2×D4—1221no—
319C2×D4—4882no—
320C2×D4—4882no—
321C2×D4—2441no—
322C2×D4—2441no—
323C2×D4—2441no—
324C2×D4—2442no—
325C2×D4—2442no—
326C22×C4—1222no—
327C22×C4—2441no—
328C22×C4C2×D24411yescharacteristic
329C22×C4D4×D6111yescharacteristic
330C22×C4#1175822611yescharacteristic
331SD16—4884no—
332D8—4884no—
333M16—2444no—
334M16—2442no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—1222no—
340C2×C8—2444no—
341C2×C8C2×D24411yescharacteristic
342C2×C8C2×D24411yescharacteristic
343C2×C8C2×D24411yescharacteristic
344C2×C8C2×D24411yescharacteristic
345C4⋊C4—612no—
346C4⋊C4—1221no—
347C4⋊C4—1222no—
348C4⋊C4—1222no—
349C4⋊C4—2441no—
350C4⋊C4—2441no—
351C22⋊C4—1222no—
352C22⋊C4—1222no—
353C42C2×D24411yescharacteristic
354C42D4×D6111yescharacteristic
355C42#1175822611yescharacteristic
356C42C23×D6111yescharacteristic
357C23—1222no—
358C23—4882no—
359C23—4882no—
360C23—4882no—
361C23—2442no—
362C23#1451975211yescharacteristic
363Q8—4882no—
364Q8—2444no—
365Q8—2442no—
366D4—1224no—
367D4—4882no—
368D4—4882no—
369D4—4882no—
370D4—4882no—
371D4—2444no—
372D4—2442no—
373D4—2442no—
374D4—2442no—
375D4—2442no—
376D4—2444no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—1222no—
380C2×C4—4882no—
381C2×C4—2442no—
382C2×C4—2442no—
383C2×C4—2442no—
384C2×C4#1451975211yescharacteristic
385C2×C4#1451975211yescharacteristic
386C2×C4#1451975211yescharacteristic
387C2×C4#1451979912yes—
388C2×C4#1451980512yes—
389C2×C4C2×D4×D6111yescharacteristic
390C2×C4C22×D24411yescharacteristic
391C2×C4C2×#1175822611yescharacteristic
392C8—21no—
393C8—21no—
394C8—21no—
395C8—21no—
396C8—42no—
397C8—42no—
398C8—2442no—
399C22—21no—
400C22—21no—
401C22—9764no—
402C22—4882no—
403C22—4882no—
404C22—4882no—
405C22—2442no—
406C22—2442no—
407C22—2442no—
408C22#1683997812yes—
409C22C2×#1451975211yescharacteristic
410C4—21no—
411C4—21no—
412C4—4882no—
413C4—2442no—
414C4#1683997812yes—
415C4#1684002612yes—
416C4C2×#1451979911yescharacteristic
417C4C2×#1451980511yescharacteristic
418C4#1684056412yes—
419C2—22no—
420C2—9762no—
421C2—4882no—
422C2C2×#1683997811yescharacteristic
423C2#2145546811yescharacteristic
424C2C2×#1684002611yescharacteristic
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1564C2
socleC22×C61C2×#121
prime radical#21456739C2
hypercenter#1564D61
hypocenterC61#19198
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1564

Complements

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

NH ≅ G/Ntype
1. #21456746C2semidirect
3. C2×#16840109C2semidirect
4. #21454532C2semidirect
5. #21445381C2semidirect
6. #21445257C2semidirect
7. #21444503C2semidirect
8. #21444486C2semidirect
9. C61×#1564C2semidirect
22. #16839968C22semidirect
27. #16839841C22semidirect
29. C8×D244C22semidirect
34. #16839509C22semidirect
39. C61×#335C22semidirect
40. C61×#334C22semidirect
213. #1564D61semidirect
249. #335C2×D61semidirect
250. #334C2×D61semidirect
253. C61#19198semidirect

Maximal subgroups

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

subgroupindex
#214567462
#214567392
C2×#168401092
#214545322
#214453812
#214452572
#214445032
#214444862
C61×#15642
#1919861