Finite Group Databasebeta

← #29834965

Subgroups — #29834965

Order 15616.

590 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#21456393C211yesmaximal, characteristic
2C2×#16839856C211yesmaximal, characteristic, complemented
3#21448004C211yesmaximal, characteristic, complemented
4#21447960C212yesmaximal, complemented
5#21447858C212yesmaximal, complemented
6#21447738C211yesmaximal, characteristic, complemented
7#21447735C211yesmaximal, characteristic, complemented
8#21447674C212yesmaximal, complemented
9#21447670C212yesmaximal, complemented
10#21447615C211yesmaximal, characteristic, complemented
11C61×#2233C211yesmaximal, characteristic, complemented
12C22×Dic244C2211yescharacteristic
13#16840307—21no—
14C2×#14519742—21no—
15C2×#14519722C2212yes—
16C2×#14519721C2212yes—
17#16840166C2211yescharacteristic
18#16840137C2211yescharacteristic
19#16840107—21no—
20#16839998C2211yescharacteristic
21#16839996—21no—
22#16839856—21no—
23#16839856—21no—
24#16839856C2212yes—
25#16839856C2212yes—
26D8×Dic61C2211yescharacteristic
27#16839844C2211yescharacteristic
28#16839842—21no—
29C2×#14519700—21no—
30C2×C8×Dic61C2211yescharacteristic
31#16839589C2212yes—
32#16839580C2212yes—
33#16839567C2212yes—
34#16839565C2212yes—
35#16839562C2212yes—
36C22⋊#11758185C2211yescharacteristic
37C16×Dic61C2212yes—
38C61×#466C2211yescharacteristic
39C2×#107×C61C2211yescharacteristic
40C61×#367C2212yes—
41C61×#357C2212yes—
42C61×C22⋊C16C2211yescharacteristic
43C2×#11758224—41no—
44C22×Dic122—21no—
45C2×Dic244—21no—
46C2×Dic244—24no—
47C2×Dic244C2312yes—
48C2×D488C2311yescharacteristic
49C2×#11758199—21no—
50C22×#92418D411yescharacteristic
51#14519776—21no—
52#14519773—21no—
53#14519763—41no—
54#14519751—21no—
55#14519749—41no—
56#14519742—22no—
57#14519742—22no—
58#14519742—41no—
59#14519742—41no—
60D4×Dic61—21no—
61#14519739—21no—
62#14519737—41no—
63C2×#11758196—41no—
64C2×C4×Dic61—41no—
65C2×C4×Dic61D411yescharacteristic
66#14519722—24no—
67#14519721—24no—
68C2×#11758185C2312yes—
69#14519711C2311yescharacteristic
70#14519710—21no—
71#14519700—22no—
72#14519700—22no—
73#14519699—21no—
74#14519698—21no—
75#14519697—21no—
76C8×Dic61D411yescharacteristic
77C8×Dic61D411yescharacteristic
78C8×Dic61C2311yescharacteristic
79C8×Dic61C2311yescharacteristic
80C2×D8×C61C2311yescharacteristic
81C22×C8×C61C2311yescharacteristic
82#121×C61—21no—
83C2×C4⋊C4×C61—21no—
84C2×C16×C61C2312yes—
85#107×C61—21no—
86#107×C61C2312yes—
87#107×C61C2311yescharacteristic
88#102×C61—21no—
89C2×D4×C61—21no—
90C2×D4×C61—41no—
91C22×C4×C61—41no—
92C22×C4×C61C2×D411yescharacteristic
93C2×C22⋊D61—41no—
94C2×C22⋊D61—41no—
95C2×C22⋊D61—41no—
96C22×Dic61—41no—
97C22×Dic61—41no—
98C22×Dic61D811yescharacteristic
99C22×Dic61D811yescharacteristic
100#11758224—84no—
101C2×D244—21no—
102C2×C4×D61—41no—
103C2×Dic122—24no—
104C2×Dic122—22no—
105C2×Dic122—42no—
106C2×Dic122—41no—
107C2×Dic122—41no—
108D8×C61—22no—
109C2×C8×C61—22no—
110C2×C8×C61C2×D411yescharacteristic
111C2×C8×C61C2411yescharacteristic
112C4⋊C4×C61—21no—
113C4⋊C4×C61—22no—
114C4⋊C4×C61—41no—
115C22⋊C4×C61—41no—
116C22⋊Dic61—41no—
117#11758199—42no—
118#11758198—41no—
119#11758198—41no—
120C4⋊Dic61—21no—
121#11758196—42no—
122#11758196—42no—
123#11758196—41no—
124C4×Dic61—42no—
125C4×Dic61—42no—
126C4×Dic61D811yescharacteristic
127C4×Dic61D811yescharacteristic
128C4×Dic61C2×D411yescharacteristic
129C4×Dic61C2×D411yescharacteristic
130C2×#92418—41no—
131C2×#92418C2×D411yescharacteristic
132C2×#92418C2×D411yescharacteristic
133Dic244—24no—
134Dic244—22no—
135Dic244#4114yescomplemented
136D488—22no—
137C16×C61—22no—
138#11758185—22no—
139C23×C61—41no—
140C23×C61C2×D811yescharacteristic
141C22×D61—41no—
142D4×C61—41no—
143D4×C61—41no—
144D4×C61—82no—
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—42no—
148C2×C4×C61—41no—
149C2×C4×C61—82no—
150C2×C4×C61C2×D811yescharacteristic
151C2×C4×C61C22×D411yescharacteristic
152C22⋊D61—82no—
153C22⋊D61—82no—
154C22⋊D61—82no—
155C2×Dic61—41no—
156C2×Dic61—44no—
157C2×Dic61—42no—
158C2×Dic61—41no—
159C2×Dic61—82no—
160C2×Dic61SD3214yescomplemented
161C2×Dic61C2×D811yescharacteristic
162C2×Dic61C2×D811yescharacteristic
163C2×Dic61C2×D811yescharacteristic
164C2×Dic61C2×D811yescharacteristic
165D244—41no—
166D244—41no—
167C4×D61—82no—
168Dic122—24no—
169Dic122—44no—
170Dic122—42no—
171Dic122—82no—
172C8×C61—21no—
173C8×C61C2×#4112yes—
174#92418—21no—
175#92418—21no—
176#33663—611nomaximal, Sylow
177C22×C61—21no—
178C22×C61—21no—
179C22×C61—81no—
180C22×C61—81no—
181C22×C61—81no—
182C22×C61C2×SD3212yes—
183C22×C61C22×D811yescharacteristic
184C2×D61—81no—
185C2×D61—81no—
186C2×D61—81no—
187C4×C61—21no—
188C4×C61—81no—
189C4×C61—82no—
190C4×C61#52511yescharacteristic
191C4×C61#57511yescharacteristic
192Dic61—21no—
193Dic61—21no—
194Dic61—44no—
195Dic61—81no—
196Dic61C2×SD3212yes—
197Dic61C2×SD3212yes—
198#3169—611no—
199C2×C4×Q16—611no—
200#2233—611no—
201#2233D6111yescharacteristic, complemented
202#2225—612no—
203#2207—612no—
204#2185—611no—
205#2185—611no—
206C4×SD32—612no—
207C4×SD32—612no—
208#2162—611no—
209C2×C61—22no—
210C2×C61—161no—
211C2×C61#310011yescharacteristic
212C2×C61C22×SD3211yescharacteristic
213C2×C61#341411yescharacteristic
214D61—161no—
215C22×Q16—611no—
216#532—1221no—
217C2×C4×Q8—1221no—
218C2×SD32—612no—
219C2×SD32—612no—
220#500—611no—
221#493—611no—
222#486—1221no—
223#466—611no—
224#466—1221no—
225#466C2×D6111yescharacteristic
226C4×Q16—612no—
227C4×Q16—612no—
228C4×Q16—1221no—
229C4×Q16—1221no—
230C4×D8—611no—
231C4×D8—611no—
232C4×D8—1221no—
233C2×#107C2×D6111yescharacteristic
234C2×#103—1221no—
235C2×C4×C8—611no—
236#367—612no—
237#367C2×D6112yes—
238C4⋊C16—612no—
239#358—612no—
240#358—612no—
241#357—612no—
242#357C2×D6112yes—
243C22⋊C16—611no—
244C22⋊C16C2×D6111yescharacteristic
245C4×C16—612no—
246C61#3366311yescharacteristic, complemented, Sylow
247C2×#41—2441no—
248C22×Q8—1221no—
249C2×Q16—612no—
250C2×Q16—1224no—
251C2×Q16—1221no—
252C2×D8—611no—
253C2×D8—1221no—
254C2×D8C22×D6111yescharacteristic
255C22×C8—611no—
256C22×C8C22×D6111yescharacteristic
257C4⋊Q8—1221no—
258#127—1221no—
259#124—2441no—
260#121—21no—
261#121—1221no—
262#121—2441no—
263C4×Q8—1222no—
264C4×Q8—1222no—
265C4×Q8—2441no—
266C4×Q8—2441no—
267C4×D4—1221no—
268C4×D4—1221no—
269C4×D4—2441no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—2441no—
272C2×C42—611no—
273C2×C42—2441no—
274SD32—1224no—
275SD32—1224no—
276C2×C16—612no—
277C2×C16C22×D6112yes—
278#107—21no—
279#107—611no—
280#107—1221no—
281#107C22×D6112yes—
282#107C22×D6111yescharacteristic
283#103—1222no—
284#103—1222no—
285#102—21no—
286#102—1221no—
287#102—1221no—
288#102—1221no—
289C4×C8—611no—
290C4×C8—611no—
291C4×C8—611no—
292C4×C8—611no—
293#41—4884no—
294C2×Q8—1224no—
295C2×Q8—1222no—
296C2×Q8—2442no—
297C2×Q8—2441no—
298C2×Q8—2441no—
299C2×D4—21no—
300C2×D4—41no—
301C2×D4—1221no—
302C2×D4—2441no—
303C2×D4—2441no—
304C2×D4—2441no—
305C22×C4—41no—
306C22×C4—611no—
307C22×C4—611no—
308C22×C4—2441no—
309C22×C4—2441no—
310C22×C4—2441no—
311C22×C4D4×D6111yescharacteristic
312Q16—614no—
313Q16—1224no—
314Q16—1222no—
315D8—22no—
316D8—1222no—
317D8—2442no—
318C2×C8—22no—
319C2×C8—611no—
320C2×C8—611no—
321C2×C8—2441no—
322C2×C8D4×D6111yescharacteristic
323C2×C8C23×D6111yescharacteristic
324C4⋊C4—21no—
325C4⋊C4—22no—
326C4⋊C4—41no—
327C4⋊C4—1221no—
328C4⋊C4—2442no—
329C4⋊C4—2442no—
330C4⋊C4—2441no—
331C22⋊C4—41no—
332C22⋊C4—2441no—
333C22⋊C4—2441no—
334C22⋊C4—2441no—
335C42—611no—
336C42—611no—
337C42—611no—
338C42—611no—
339C42—2442no—
340C42—2442no—
341C16—22no—
342C16—1222no—
343C23—41no—
344C23—2441no—
345C23D8×D6111yescharacteristic
346Q8—1224no—
347Q8—4882no—
348Q8—2444no—
349Q8—2442no—
350D4—41no—
351D4—41no—
352D4—82no—
353D4—4882no—
354D4—4882no—
355D4—4882no—
356D4—2441no—
357D4—2441no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—82no—
363C2×C4—611no—
364C2×C4—611no—
365C2×C4—614no—
366C2×C4—611no—
367C2×C4—611no—
368C2×C4—4882no—
369C2×C4—4882no—
370C2×C4—2441no—
371C2×C4—2441no—
372C2×C4—2444no—
373C2×C4—2442no—
374C2×C4D8×D6111yescharacteristic
375C2×C4C2×D4×D6111yescharacteristic
376C8—21no—
377C8—1221no—
378C8—1221no—
379C8#41×D6112yes—
380C22—21no—
381C22—21no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—4881no—
386C22—4881no—
387C22—4881no—
388C22SD32×D6112yes—
389C22C2×D8×D6111yescharacteristic
390C4—21no—
391C4—82no—
392C4—81no—
393C4—612no—
394C4—612no—
395C4—1221no—
396C4—1221no—
397C4—4881no—
398C4—2444no—
399C4#1684027011yescharacteristic
400C4#1684056111yescharacteristic
401C2—22no—
402C2—161no—
403C2—9761no—
404C2#2145581611yescharacteristic
405C2C2×SD32×D6111yescharacteristic
406C2#2145895111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2233C2
socleC22×C61C22×D8
prime radical#21456393C2
hypercenter#2233D61
hypocenterC61#33663
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2233

Complements

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

NH ≅ G/Ntype
2. C2×#16839856C2semidirect
3. #21448004C2semidirect
4. #21447960C2semidirect
5. #21447858C2semidirect
6. #21447738C2semidirect
7. #21447735C2semidirect
8. #21447674C2semidirect
9. #21447670C2semidirect
10. #21447615C2semidirect
11. C61×#2233C2semidirect
135. Dic244#41semidirect
160. C2×Dic61SD32semidirect
201. #2233D61semidirect
246. C61#33663semidirect

Maximal subgroups

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

subgroupindex
#214563932
C2×#168398562
#214480042
#214479602
#214478582
#214477382
#214477352
#214476742
#214476702
#214476152
C61×#22332
#3366361