Finite Group Databasebeta

← #29665173

Subgroups — #29665173

Order 15616.

738 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456762C211yesmaximal, characteristic, complemented
2#21455616C211yesmaximal, characteristic, complemented
3C2×#16839957C211yesmaximal, characteristic
4#21445228C214yesmaximal, complemented
5#21445160C214yesmaximal, complemented
6#21444590C211yesmaximal, characteristic, complemented
7#21444486C211yesmaximal, characteristic, complemented
8#21444474C211yesmaximal, characteristic, complemented
9C61×#2972C211yesmaximal, characteristic, complemented
10C22×#11758199—21no—
11#16840319C2211yescharacteristic
12C2×#14519774C2211yescharacteristic
13#16840135C2212yescomplemented
14#16840109C2212yescomplemented
15#16840062—22no—
16#16840052C2212yescomplemented
17#16840042C2212yescomplemented
18#16839963—22no—
19#16839957—22no—
20#16839957C2214yes—
21#16839915—22no—
22#16839802—21no—
23#16839786C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25#16839767—21no—
26C2×#14519697—21no—
27#16839513C2214yescomplemented
28#16839510C2212yescomplemented
29#16839509C2212yescomplemented
30#16839505C2214yescomplemented
31C2×C4×D4×C61C2211yescharacteristic
32C8×D4×C61—22no—
33C8×D4×C61C2214yescomplemented
34C61×#431C2211yescharacteristic
35C61×#431C2211yescharacteristic
36C2×C22⋊C8×C61—22no—
37C22×D244—21no—
38C22×D244—21no—
39C22×D244—21no—
40C2×#11758190—24no—
41C2×#11758199—24no—
42C2×#11758199—44no—
43C2×#11758194—21no—
44C22×#92418—21no—
45#14519777C2311yescharacteristic
46#14519777C2311yescharacteristic
47#14519774—21no—
48#14519774—21no—
49#14519774—21no—
50#14519774C2312yes—
51#14519769—21no—
52#14519756—21no—
53#14519756—21no—
54C4×Dic122—21no—
55C4×Dic122—21no—
56#14519736—21no—
57#14519710—22no—
58#14519708—22no—
59C4⋊#92418—21no—
60C4⋊#92418—21no—
61C4⋊#92418C2312yes—
62C4⋊#92418C2312yes—
63#14519699—24no—
64#14519698—22no—
65#14519698—22no—
66#14519697—24no—
67C22×D4×C61D411yescharacteristic
68C23×C4×C61—21no—
69C22×C8×C61—24no—
70C4×D4×C61—21no—
71C4×D4×C61—21no—
72C4×D4×C61D412yes—
73C4×D4×C61C2312yes—
74C2×C4⋊C4×C61D411yescharacteristic
75C2×C22⋊C4×C61—21no—
76C2×C42×C61C2311yescharacteristic
77C4⋊C8×C61D412yes—
78C4⋊C8×C61C2312yes—
79C22⋊C8×C61—21no—
80C22⋊C8×C61—24no—
81C22⋊C8×C61D412yes—
82C4×C8×C61—21no—
83C4×C8×C61C2312yes—
84C24×C61—21no—
85C23×D61—41no—
86C2×D4×C61—21no—
87C2×D4×C61—22no—
88C2×D4×C61SD1614yescomplemented
89C2×D4×C61C2×D412yes—
90C22×C4×C61—21no—
91C22×C4×C61—22no—
92C22×C4×C61—22no—
93C22×C4×C61—82no—
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61#4111yescharacteristic
97C2×D244—22no—
98C2×D244—22no—
99C2×D244—22no—
100C2×D244—42no—
101C2×D244—42no—
102C2×D244—41no—
103C2×D244—41no—
104C2×D244—41no—
105C2×D244—82no—
106C2×D244—82no—
107C2×Dic122—21no—
108C2×Dic122—21no—
109C2×C8×C61—22no—
110C2×C8×C61—84no—
111C2×C8×C61C2×D414yes—
112C4⋊C4×C61—21no—
113C4⋊C4×C61C2×D412yes—
114C22⋊C4×C61—22no—
115C22⋊C4×C61—22no—
116C42×C61C2×D411yescharacteristic
117C42×C61C2×D411yescharacteristic
118C42×C61#4111yescharacteristic
119C42×C61C2411yescharacteristic
120C22⋊Dic61—41no—
121#11758199—48no—
122C4⋊Dic61—21no—
123C4⋊Dic61—21no—
124C4⋊Dic61—21no—
125C4⋊Dic61—21no—
126#11758196—41no—
127#11758196—41no—
128C4×Dic61—41no—
129#11758194—42no—
130C2×#92418—22no—
131C2×#92418—22no—
132C2×#92418—42no—
133#11758190—84no—
134C23×C61—22no—
135C23×C61—24no—
136C23×C61—81no—
137C23×C61—81no—
138C23×C61#12111yescharacteristic
139C22×D61—42no—
140C22×D61—82no—
141C22×D61—82no—
142C22×D61—82no—
143D4×C61—22no—
144D4×C61—24no—
145D4×C61C2×SD1614yes—
146C2×C4×C61—22no—
147C2×C4×C61—22no—
148C2×C4×C61—41no—
149C2×C4×C61—44no—
150C2×C4×C61—82no—
151C2×C4×C61—81no—
152C2×C4×C61#12111yescharacteristic
153C2×C4×C61#12111yescharacteristic
154C2×C4×C61#12111yescharacteristic
155C2×C4×C61C2×SD1612yes—
156C2×C4×C61#13512yes—
157C2×C4×C61C22×D411yescharacteristic
158C2×C4×C61C22×D411yescharacteristic
159C2×C4×C61C2×#4111yescharacteristic
160C2×Dic61—41no—
161C2×Dic61—41no—
162D244—42no—
163D244—42no—
164D244—44no—
165D244—82no—
166D244—82no—
167D244—82no—
168D244—82no—
169Dic122—42no—
170Dic122—81no—
171C8×C61—24no—
172C8×C61—42no—
173#92418—42no—
174#92418—42no—
175#19384—611nomaximal, Sylow
176C22×C61—21no—
177C22×C61—21no—
178C22×C61—24no—
179C22×C61—42no—
180C22×C61—42no—
181C22×C61—84no—
182C22×C61#46512yes—
183C22×C61C2×#12111yescharacteristic
184C2×D61—82no—
185C2×D61—82no—
186C2×D61—82no—
187C2×D61—164no—
188C4×C61—21no—
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—42no—
192C4×C61#46512yes—
193C4×C61C22×SD1611yescharacteristic
194C4×C61C2×#13511yescharacteristic
195C4×C61#57612yes—
196Dic61—81no—
197Dic61—81no—
198#3208—611no—
199#3077—611no—
200C2×#459—611no—
201#2972D6111yescharacteristic, complemented
202#1677—614no—
203#1665—614no—
204#1570—611no—
205#1545—611no—
206#1542—611no—
207C2×C61—22no—
208C2×C61—44no—
209C2×C61#304111yescharacteristic
210C2×C61C2×#46511yescharacteristic
211C2×C61#306811yescharacteristic
212D61—162no—
213C22×D8—1221no—
214#533—611no—
215C2×#127—611no—
216C2×C4×D4C2×D6111yescharacteristic
217#492—612no—
218#486—612no—
219#478—1222no—
220#476—612no—
221#474—612no—
222#460—1222no—
223#459—614no—
224#459—1222no—
225#450—1222no—
226C8×D4—22no—
227C8×D4C2×D6114yescomplemented
228#431C2×D6111yescharacteristic
229#431C2×D6111yescharacteristic
230#428—1221no—
231#423—611no—
232C2×C4⋊C8—611no—
233#418—1221no—
234C2×#102—1221no—
235C2×C22⋊C8—22no—
236#334—614no—
237#333—612no—
238#332—612no—
239#331—614no—
240C61#1938411yescharacteristic, complemented, Sylow
241C22×D4—1221no—
242C22×D4—1221no—
243C22×D4—1221no—
244C22×D4D24411yescharacteristic
245C23×C4—21no—
246C2×SD16—1224no—
247C2×D8—1224no—
248C2×D8—2444no—
249C2×M16—1221no—
250C22×C8—24no—
251C22×C8—1221no—
252C4⋊Q8—611no—
253C4⋊Q8—611no—
254#127—612no—
255#127—1221no—
256#127—1221no—
257#127—1221no—
258#125—1221no—
259C22⋊Q8—1221no—
260C22⋊Q8—1221no—
261C4×Q8—1221no—
262C4×Q8—1221no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4D24412yes—
266C4×D4C22×D6112yes—
267#117—1221no—
268C2×C4⋊C4D24411yescharacteristic
269C2×C22⋊C4—21no—
270C2×C42C22×D6111yescharacteristic
271#107—1222no—
272#106—1222no—
273C4⋊C8—612no—
274C4⋊C8—612no—
275C4⋊C8—1221no—
276C4⋊C8—1221no—
277C4⋊C8C22⋊D6112yes—
278C4⋊C8C22×D6112yes—
279#102—1222no—
280#102—1222no—
281#102—1224no—
282#102—1224no—
283C22⋊C8—21no—
284C22⋊C8—24no—
285C22⋊C8C22⋊D6112yes—
286C4×C8—21no—
287C4×C8C22×D6112yes—
288C24—21no—
289C24—2441no—
290C2×Q8—1221no—
291C2×Q8—1221no—
292C2×D4—21no—
293C2×D4—22no—
294C2×D4—1222no—
295C2×D4—1222no—
296C2×D4—1222no—
297C2×D4—4882no—
298C2×D4—4882no—
299C2×D4—2441no—
300C2×D4—2441no—
301C2×D4—2441no—
302C2×D4—2442no—
303C2×D4—2442no—
304C2×D4#1175819014yescomplemented
305C2×D4C2×D24412yes—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—82no—
310C22×C4C2×D24411yescharacteristic
311C22×C4#1175822211yescharacteristic
312C22×C4C2×C22⋊D6111yescharacteristic
313SD16—4884no—
314D8—2448no—
315M16—2442no—
316C2×C8—22no—
317C2×C8—84no—
318C2×C8—1222no—
319C2×C8—1222no—
320C2×C8—2442no—
321C2×C8C2×C22⋊D6114yes—
322C4⋊C4—21no—
323C4⋊C4—1221no—
324C4⋊C4—1221no—
325C4⋊C4—1221no—
326C4⋊C4—1221no—
327C4⋊C4—2441no—
328C4⋊C4—2441no—
329C4⋊C4C2×D24412yes—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—2441no—
333C42—2441no—
334C42C2×D24411yescharacteristic
335C42#1175822211yescharacteristic
336C42C2×C22⋊D6111yescharacteristic
337C42C23×D6111yescharacteristic
338C23—22no—
339C23—24no—
340C23—81no—
341C23—81no—
342C23—4882no—
343C23—4882no—
344C23—4882no—
345C23—2442no—
346C23#1451975111yescharacteristic
347Q8—4881no—
348Q8—2442no—
349D4—24no—
350D4—22no—
351D4—4882no—
352D4—4882no—
353D4—4882no—
354D4—4882no—
355D4—2442no—
356D4—2442no—
357D4—2444no—
358D4C2×#1175819014yes—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—44no—
363C2×C4—82no—
364C2×C4—81no—
365C2×C4—2441no—
366C2×C4—2441no—
367C2×C4#1451975111yescharacteristic
368C2×C4#1451975111yescharacteristic
369C2×C4#1451975111yescharacteristic
370C2×C4C2×#1175819012yes—
371C2×C4#1451979912yes—
372C2×C4C22×C22⋊D6111yescharacteristic
373C2×C4C22×D24411yescharacteristic
374C2×C4C2×#1175822211yescharacteristic
375C8—24no—
376C8—42no—
377C8—2442no—
378C8—2442no—
379C22—21no—
380C22—21no—
381C22—24no—
382C22—42no—
383C22—42no—
384C22—84no—
385C22—9764no—
386C22—4882no—
387C22—4882no—
388C22—4882no—
389C22#1683999112yes—
390C22C2×#1451975111yescharacteristic
391C4—21no—
392C4—21no—
393C4—21no—
394C4—42no—
395C4—4881no—
396C4—4881no—
397C4#1683999112yes—
398C4C22×#1175819011yescharacteristic
399C4C2×#1451979911yescharacteristic
400C4#1684056512yes—
401C2—22no—
402C2—44no—
403C2—9762no—
404C2#2145532211yescharacteristic
405C2C2×#1683999111yescharacteristic
406C2#2145554711yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#2972C2
socleC22×C61C2×#121
prime radicalC2×#16839957C2
hypercenter#2972D61
hypocenterC61#19384
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
1. #21456762C2semidirect
2. #21455616C2semidirect
4. #21445228C2semidirect
5. #21445160C2semidirect
6. #21444590C2semidirect
7. #21444486C2semidirect
8. #21444474C2semidirect
9. C61×#2972C2semidirect
13. #16840135C22semidirect
14. #16840109C22semidirect
16. #16840052C22semidirect
17. #16840042C22semidirect
27. #16839513C22semidirect
28. #16839510C22semidirect
29. #16839509C22semidirect
30. #16839505C22semidirect
33. C8×D4×C61C22semidirect
88. C2×D4×C61SD16semidirect
201. #2972D61semidirect
227. C8×D4C2×D61semidirect
240. C61#19384semidirect
304. C2×D4#11758190semidirect

Maximal subgroups

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

subgroupindex
#214567622
#214556162
C2×#168399572
#214452282
#214451602
#214445902
#214444862
#214444742
C61×#29722
#1938461