Finite Group Databasebeta

← #29834963

Subgroups — #29834963

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×Q16×Dic61C211yesmaximal, characteristic, complemented
3#21448005C211yesmaximal, characteristic, complemented
4#21448002C211yesmaximal, characteristic, complemented
5#21447959C212yesmaximal, complemented
6#21447857C212yesmaximal, complemented
7#21447736C211yesmaximal, characteristic, complemented
8SD32×Dic61C212yesmaximal, complemented
9#21447668C212yesmaximal, complemented
10#21447615C211yesmaximal, characteristic, complemented
11C61×#2185C211yesmaximal, characteristic, complemented
12#16840307—21no—
13C2×Q8×Dic61—21no—
14C2×#14519720C2212yes—
15#16840166C2211yescharacteristic
16#16840137C2211yescharacteristic
17#16840107—21no—
18#16839998C2211yescharacteristic
19#16839996—21no—
20Q16×Dic61—21no—
21Q16×Dic61—21no—
22Q16×Dic61C2212yes—
23Q16×Dic61C2212yes—
24D8×Dic61C2211yescharacteristic
25#16839844C2211yescharacteristic
26#16839842—21no—
27C2×#14519711C2211yescharacteristic
28C2×#14519701—21no—
29C2×C8×Dic61C2211yescharacteristic
30#16839590C2212yes—
31#16839589C2212yes—
32#16839580C2212yes—
33#16839566C2212yes—
34#16839564C2212yes—
35#16839563C2212yes—
36C22⋊#11758185C2211yescharacteristic
37C16×Dic61C2212yes—
38C22×Q16×C61C2211yescharacteristic
39C2×SD32×C61C2212yes—
40C61×#466C2211yescharacteristic
41C61×#358C2212yes—
42C61×C22⋊C16C2211yescharacteristic
43C2×#11758224—41no—
44C2×D488C2311yescharacteristic
45C2×#11758199—21no—
46C22×#92418D411yescharacteristic
47#14519776—21no—
48#14519773—21no—
49#14519763—41no—
50#14519751—21no—
51#14519749—41no—
52Q8×Dic61—22no—
53Q8×Dic61—22no—
54Q8×Dic61—41no—
55Q8×Dic61—41no—
56D4×Dic61—21no—
57#14519739—21no—
58#14519737—41no—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61—41no—
61C2×C4×Dic61—41no—
62C2×C4×Dic61D411yescharacteristic
63#14519720—24no—
64C2×#11758185C2312yes—
65#14519711—21no—
66#14519711C2311yescharacteristic
67#14519711C2312yes—
68#14519710—21no—
69#14519701—22no—
70#14519701—22no—
71#14519699—21no—
72#14519698—21no—
73#14519697—21no—
74C8×Dic61D411yescharacteristic
75C8×Dic61D411yescharacteristic
76C8×Dic61C2311yescharacteristic
77C8×Dic61C2311yescharacteristic
78C22×Q8×C61—21no—
79C2×Q16×C61—24no—
80C2×Q16×C61—21no—
81C2×Q16×C61C2312yes—
82C2×D8×C61C2311yescharacteristic
83C22×C8×C61C2311yescharacteristic
84#121×C61—21no—
85SD32×C61—24no—
86C2×C16×C61C2312yes—
87#107×C61C2311yescharacteristic
88#102×C61—21no—
89C2×Q8×C61—24no—
90C2×Q8×C61—22no—
91C2×Q8×C61—42no—
92C2×Q8×C61—41no—
93C2×D4×C61—21no—
94C2×D4×C61—41no—
95C22×C4×C61—41no—
96C22×C4×C61C2×D411yescharacteristic
97C2×C22⋊D61—41no—
98C2×C22⋊D61—41no—
99C2×C22⋊D61—41no—
100C22×Dic61—41no—
101C22×Dic61—41no—
102C22×Dic61D811yescharacteristic
103C22×Dic61D811yescharacteristic
104#11758224—84no—
105C2×D244—21no—
106C2×C4×D61—41no—
107C2×Dic122—41no—
108Q16×C61—24no—
109Q16×C61—22no—
110Q16×C61#4114yescomplemented
111D8×C61—22no—
112C2×C8×C61—22no—
113C2×C8×C61C2×D411yescharacteristic
114C2×C8×C61C2411yescharacteristic
115C4⋊C4×C61—21no—
116C22⋊C4×C61—41no—
117C22⋊Dic61—41no—
118#11758199—42no—
119#11758198—41no—
120#11758198—41no—
121C4⋊Dic61—21no—
122C4⋊Dic61—22no—
123C4⋊Dic61—41no—
124C4⋊Dic61—42no—
125C4⋊Dic61—42no—
126#11758196—41no—
127C4×Dic61—42no—
128C4×Dic61—42no—
129C4×Dic61D811yescharacteristic
130C4×Dic61D811yescharacteristic
131C4×Dic61C2×D411yescharacteristic
132C4×Dic61C2×D411yescharacteristic
133C2×#92418—41no—
134C2×#92418C2×D411yescharacteristic
135C2×#92418C2×D411yescharacteristic
136D488—22no—
137C16×C61—22no—
138#11758185—22no—
139C23×C61—41no—
140C23×C61C2×D811yescharacteristic
141C22×D61—41no—
142Q8×C61—24no—
143Q8×C61—44no—
144Q8×C61—42no—
145D4×C61—41no—
146D4×C61—41no—
147D4×C61—82no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—44no—
151C2×C4×C61—42no—
152C2×C4×C61—41no—
153C2×C4×C61C2×D811yescharacteristic
154C2×C4×C61C22×D411yescharacteristic
155C22⋊D61—82no—
156C22⋊D61—82no—
157C22⋊D61—82no—
158C2×Dic61—41no—
159C2×Dic61—42no—
160C2×Dic61—41no—
161C2×Dic61—82no—
162C2×Dic61—82no—
163C2×Dic61SD3214yescomplemented
164C2×Dic61C2×D811yescharacteristic
165C2×Dic61C2×D811yescharacteristic
166C2×Dic61C2×D811yescharacteristic
167C2×Dic61C2×D811yescharacteristic
168D244—41no—
169D244—41no—
170C4×D61—82no—
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—44no—
189C4×C61—81no—
190C4×C61#52511yescharacteristic
191C4×C61#57511yescharacteristic
192Dic61—21no—
193Dic61—21no—
194Dic61—81no—
195Dic61—82no—
196Dic61C2×SD3212yes—
197Dic61C2×SD3212yes—
198#3169—611no—
199C2×C4×Q16—611no—
200#2233—611no—
201#2233—611no—
202#2225—612no—
203#2207—612no—
204#2185—611no—
205#2185D6111yescharacteristic, complemented
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×Q16C2×D6111yescharacteristic
216#532—1221no—
217C2×C4×Q8—1221no—
218C2×SD32—612no—
219C2×SD32C2×D6112yes—
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×#107—611no—
234C2×#103—1221no—
235C2×C4×C8—611no—
236#367—612no—
237#367—612no—
238C4⋊C16—612no—
239#358—612no—
240#358C2×D6112yes—
241#357—612no—
242#357—612no—
243C22⋊C16—611no—
244C22⋊C16C2×D6111yescharacteristic
245C4×C16—612no—
246C61#3366311yescharacteristic, complemented, Sylow
247C2×#41—2441no—
248C22×Q8—21no—
249C2×Q16—24no—
250C2×Q16—21no—
251C2×Q16C22×D6112yes—
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—1221no—
271C2×C4⋊C4—2441no—
272C2×C42—611no—
273C2×C42—2441no—
274SD32—24no—
275SD32—1224no—
276C2×C16—612no—
277C2×C16C22×D6112yes—
278#107—612no—
279#107—611no—
280#107—1221no—
281#107—1221no—
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—24no—
295C2×Q8—22no—
296C2×Q8—42no—
297C2×Q8—41no—
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—24no—
313Q16—22no—
314Q16#1175822414yescomplemented
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—1221no—
326C4⋊C4—1222no—
327C4⋊C4—2442no—
328C4⋊C4—2442no—
329C4⋊C4—2441no—
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—24no—
347Q8—44no—
348Q8—42no—
349Q8—4882no—
350D4—41no—
351D4—41no—
352D4—82no—
353D4—4882no—
354D4—4882no—
355D4—4882no—
356D4—2441no—
357D4—2441no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—611no—
364C2×C4—611no—
365C2×C4—614no—
366C2×C4—611no—
367C2×C4—611no—
368C2×C4—4882no—
369C2×C4—4882no—
370C2×C4—4882no—
371C2×C4—2441no—
372C2×C4—2442no—
373C2×C4—2441no—
374C2×C4D8×D6111yescharacteristic
375C2×C4C2×D4×D6111yescharacteristic
376C8—21no—
377C8—1221no—
378C8—1221no—
379C8C2×#1175822412yes—
380C22—21no—
381C22—21no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—4881no—
386C22—4881no—
387C22—4881no—
388C22SD32×D6112yes—
389C22C2×D8×D6111yescharacteristic
390C4—21no—
391C4—44no—
392C4—81no—
393C4—612no—
394C4—612no—
395C4—1221no—
396C4—1221no—
397C4—4882no—
398C4—4881no—
399C4#1684026811yescharacteristic
400C4#1684055811yescharacteristic
401C2—22no—
402C2—161no—
403C2—9761no—
404C2#2145581411yescharacteristic
405C2C2×SD32×D6111yescharacteristic
406C2#2145895411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2185C2
socleC22×C61C22×D8
prime radical#21456393C2
hypercenter#2185D61
hypocenterC61#33663
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2185

Complements

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

NH ≅ G/Ntype
2. C2×Q16×Dic61C2semidirect
3. #21448005C2semidirect
4. #21448002C2semidirect
5. #21447959C2semidirect
6. #21447857C2semidirect
7. #21447736C2semidirect
8. SD32×Dic61C2semidirect
9. #21447668C2semidirect
10. #21447615C2semidirect
11. C61×#2185C2semidirect
110. Q16×C61#41semidirect
163. C2×Dic61SD32semidirect
205. #2185D61semidirect
246. C61#33663semidirect
314. Q16#11758224semidirect

Maximal subgroups

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

subgroupindex
#214563932
C2×Q16×Dic612
#214480052
#214480022
#214479592
#214478572
#214477362
SD32×Dic612
#214476682
#214476152
C61×#21852
#3366361