Finite Group Databasebeta

← #29835002

Subgroups — #29835002

Order 15616.

552 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
1C4×#14519799C211yesmaximal, characteristic
2#21448024C212yesmaximal, complemented
3#21447958C211yesmaximal, characteristic, complemented
4#21447953C211yesmaximal, characteristic, complemented
5#21447866C211yesmaximal, characteristic, complemented
6#21447861C211yesmaximal, characteristic, complemented
7#21447757C212yesmaximal
8C4×#14519723C212yesmaximal, complemented
9C4×#14519720C212yesmaximal, complemented
10#21447618C211yesmaximal, characteristic, complemented
11C61×#3170C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13C4×#11758222—21no—
14C2×#14519723C2212yes—
15C2×#14519720C2212yes—
16C4×Dic244C2211yescharacteristic
17C4×Dic244C2211yescharacteristic
18C4×#11758190—21no—
19C4×#11758190—21no—
20C4×D488C2211yescharacteristic
21C4×D488C2211yescharacteristic
22#16839800C2211yescharacteristic
23#16839759—21no—
24#16839589C2212yes—
25#16839587C2212yes—
26C4⋊#11758185C2211yescharacteristic
27C4⋊#11758185C2211yescharacteristic
28#16839566C2212yes—
29#16839566C2212yes—
30#16839565C2212yes—
31#16839562C2212yes—
32C22⋊#11758185C2212yes—
33C4×#11758185C2212yes—
34C61×#533—21no—
35C61×#500C2211yescharacteristic
36C61×#494C2211yescharacteristic
37C61×#487—21no—
38C61×#467—21no—
39C61×#467C2212yes—
40C4×Q16×C61—21no—
41C4×Q16×C61C2212yes—
42C2×C4×C8×C61C2211yescharacteristic
43C2×#11758222—21no—
44#14519799—22no—
45#14519799—22no—
46C2×Dic244C2311yescharacteristic
47C2×#11758190—21no—
48C2×D488C2311yescharacteristic
49C4×Dic122—21no—
50C4×Dic122—21no—
51C4×D244—21no—
52C4×D244—21no—
53C4×C22⋊D61—41no—
54C4×C22⋊D61—41no—
55#14519736—21no—
56#14519734—41no—
57C42×D61—41no—
58#14519723—24no—
59#14519720—24no—
60C2×#11758185C2312yes—
61C2×#11758185C2312yes—
62#14519711C2311yescharacteristic
63#14519711C2311yescharacteristic
64#14519709—21no—
65#14519702—21no—
66#14519702—21no—
67#14519699—21no—
68#14519699—21no—
69C2×Q16×C61—21no—
70C2×Q16×C61C2312yes—
71C22×C8×C61D411yescharacteristic
72C22×C8×C61C2311yescharacteristic
73C4⋊Q8×C61—21no—
74C4⋊Q8×C61—21no—
75#125×C61—41no—
76C22⋊Q8×C61—22no—
77C22⋊Q8×C61—41no—
78C4×Q8×C61—22no—
79C4×Q8×C61—41no—
80#117×C61—41no—
81C2×C42×C61D411yescharacteristic
82#107×C61—21no—
83#107×C61C2312yes—
84#103×C61—22no—
85#103×C61—22no—
86C4×C8×C61D411yescharacteristic
87C4×C8×C61D411yescharacteristic
88C4×C8×C61C2311yescharacteristic
89C4×C8×C61C2311yescharacteristic
90C2×Q8×C61—22no—
91C2×Q8×C61—41no—
92C22×C4×C61D811yescharacteristic
93C22×C4×C61D811yescharacteristic
94C22×C4×C61C2×D411yescharacteristic
95C2×C22⋊D61—41no—
96#11758222—24no—
97#11758222—42no—
98C2×D244—21no—
99C2×C4×D61—41no—
100C2×C4×D61—41no—
101C2×C4×D61—41no—
102C2×Dic122—21no—
103Q16×C61—22no—
104Q16×C61—22no—
105Q16×C61—42no—
106C2×C8×C61—21no—
107C2×C8×C61—21no—
108C2×C8×C61—41no—
109C2×C8×C61C2×D411yescharacteristic
110C2×C8×C61C2×D411yescharacteristic
111C2×C8×C61C2×D411yescharacteristic
112C2×C8×C61C2411yescharacteristic
113C4⋊C4×C61—22no—
114C4⋊C4×C61—41no—
115C4⋊C4×C61—41no—
116C4⋊C4×C61—42no—
117C4⋊C4×C61—42no—
118C4⋊C4×C61—41no—
119C22⋊C4×C61—42no—
120C42×C61—42no—
121C42×C61D811yescharacteristic
122C42×C61D811yescharacteristic
123C42×C61C2×D411yescharacteristic
124C42×C61C2×D411yescharacteristic
125C22⋊Dic61—41no—
126#11758198—41no—
127#11758198—41no—
128C4⋊Dic61—21no—
129C4⋊Dic61—21no—
130#11758196—41no—
131#11758196—41no—
132C4×Dic61—41no—
133C4×Dic61—41no—
134C4×Dic61—41no—
135Dic244—21no—
136Dic244#4112yes—
137D488—21no—
138D488#4112yes—
139#11758190—24no—
140#11758185—22no—
141#11758185—22no—
142C23×C61C2×D811yescharacteristic
143C22×D61—41no—
144Q8×C61—42no—
145Q8×C61—42no—
146Q8×C61—82no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—22no—
150C2×C4×C61—22no—
151C2×C4×C61—42no—
152C2×C4×C61—42no—
153C2×C4×C61C2×D811yescharacteristic
154C2×C4×C61C2×D811yescharacteristic
155C2×C4×C61C2×D811yescharacteristic
156C2×C4×C61C2×D811yescharacteristic
157C2×C4×C61C2×D811yescharacteristic
158C2×C4×C61C22×D411yescharacteristic
159C22⋊D61—44no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163D244—22no—
164D244—41no—
165C4×D61—44no—
166C4×D61—82no—
167C4×D61—82no—
168Dic122—22no—
169Dic122—41no—
170C8×C61—21no—
171C8×C61—21no—
172C8×C61—21no—
173C8×C61C2×#4111yescharacteristic
174C8×C61C2×#4111yescharacteristic
175#33667—611nomaximal, Sylow
176C22×C61—21no—
177C22×C61—21no—
178C22×C61—21no—
179C22×C61C22×D811yescharacteristic
180C2×D61—42no—
181C2×D61—81no—
182C2×D61—81no—
183C4×C61—21no—
184C4×C61—21no—
185C4×C61—21no—
186C4×C61—82no—
187C4×C61—82no—
188C4×C61#50812yes—
189C4×C61#51012yes—
190C4×C61#52511yescharacteristic
191C4×C61#57511yescharacteristic
192Dic61—42no—
193Dic61—81no—
194Dic61—81no—
195#3170D6111yescharacteristic, complemented
196C4×#135—611no—
197#2237—612no—
198#2225—611no—
199#2224—611no—
200#2209—611no—
201#2208—611no—
202#2189—612no—
203C4×Q32—612no—
204C4×SD32—612no—
205#2162—611no—
206C2×C61—41no—
207C2×C61#310011yescharacteristic
208C2×C61C2×#50811yescharacteristic
209C2×C61C2×#51011yescharacteristic
210D61—82no—
211C2×#135—611no—
212#533—21no—
213C4×#41—1221no—
214C2×Q32—612no—
215C2×SD32—612no—
216#500C2×D6111yescharacteristic
217#494C2×D6111yescharacteristic
218#487—21no—
219#467—21no—
220#467C2×D6112yes—
221C4×Q16—21no—
222C4×Q16—611no—
223C4×Q16—611no—
224C4×Q16C2×D6112yes—
225C4×SD16—1221no—
226C4×SD16—1221no—
227C4×D8—611no—
228C4×D8—611no—
229#427—611no—
230#416—1221no—
231C2×C4×C8C2×D6111yescharacteristic
232#367—612no—
233#366—612no—
234C4⋊C16—611no—
235C4⋊C16—611no—
236#358—612no—
237#358—612no—
238#358—612no—
239#357—612no—
240C22⋊C16—612no—
241C4×C16—612no—
242C61#3366711yescharacteristic, complemented, Sylow
243C2×#41—1221no—
244#135—1222no—
245#135—1222no—
246C2×Q16—21no—
247C2×Q16—611no—
248C2×Q16C22×D6112yes—
249C2×SD16—1221no—
250C2×D8—611no—
251C22×C8C22⋊D6111yescharacteristic
252C22×C8C22×D6111yescharacteristic
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255#125—41no—
256C22⋊Q8—22no—
257C22⋊Q8—41no—
258C4×Q8—22no—
259C4×Q8—41no—
260C4×Q8—1221no—
261C4×Q8—1221no—
262C4×D4—1221no—
263C4×D4—1221no—
264C4×D4—2441no—
265C4×D4—2441no—
266#117—41no—
267#117—1221no—
268#117—2441no—
269C2×C42—2441no—
270C2×C42C22⋊D6111yescharacteristic
271Q32—1224no—
272SD32—1224no—
273C2×C16—612no—
274C2×C16—612no—
275#107—21no—
276#107—611no—
277#107—611no—
278#107C22×D6112yes—
279#106—1221no—
280#103—22no—
281#103—22no—
282#103—1221no—
283#103—1221no—
284#102—1221no—
285#102—1221no—
286C4×C8C22⋊D6111yescharacteristic
287C4×C8C22⋊D6111yescharacteristic
288C4×C8C22×D6111yescharacteristic
289C4×C8C22×D6111yescharacteristic
290#41—1224no—
291#41—2442no—
292C2×Q8—22no—
293C2×Q8—41no—
294C2×Q8—1221no—
295C2×D4—1221no—
296C2×D4—2441no—
297C22×C4—2441no—
298C22×C4—2441no—
299C22×C4—2441no—
300C22×C4#1175819911yescharacteristic
301C22×C4#1175819911yescharacteristic
302C22×C4C2×C22⋊D6111yescharacteristic
303Q16—22no—
304Q16—22no—
305Q16—42no—
306Q16—612no—
307Q16—1221no—
308SD16—1224no—
309D8—612no—
310D8—1221no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—41no—
314C2×C8C2×C22⋊D6111yescharacteristic
315C2×C8C2×C22⋊D6111yescharacteristic
316C2×C8C2×C22⋊D6111yescharacteristic
317C2×C8C23×D6111yescharacteristic
318C4⋊C4—22no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—41no—
324C4⋊C4—1221no—
325C4⋊C4—1221no—
326C4⋊C4—2441no—
327C4⋊C4—2441no—
328C22⋊C4—42no—
329C22⋊C4—2441no—
330C22⋊C4—2441no—
331C22⋊C4—2441no—
332C42—42no—
333C42—2441no—
334C42—2441no—
335C42—2441no—
336C42#1175819911yescharacteristic
337C42#1175819911yescharacteristic
338C42C2×C22⋊D6111yescharacteristic
339C42C2×C22⋊D6111yescharacteristic
340C16—1222no—
341C16—1222no—
342C23—2441no—
343C23C2×#1175819911yescharacteristic
344Q8—42no—
345Q8—42no—
346Q8—82no—
347Q8—1222no—
348Q8—2441no—
349D4—1222no—
350D4—2444no—
351D4—2441no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—4882no—
359C2×C4—4882no—
360C2×C4—2444no—
361C2×C4—2441no—
362C2×C4—2441no—
363C2×C4—2441no—
364C2×C4C2×#1175819911yescharacteristic
365C2×C4C2×#1175819911yescharacteristic
366C2×C4C2×#1175819911yescharacteristic
367C2×C4C2×#1175819911yescharacteristic
368C2×C4C2×#1175819911yescharacteristic
369C2×C4C22×C22⋊D6111yescharacteristic
370C8—21no—
371C8—21no—
372C8—21no—
373C8#41×D6111yescharacteristic
374C8#41×D6111yescharacteristic
375C22—21no—
376C22—21no—
377C22—21no—
378C22—4881no—
379C22—4881no—
380C22—2442no—
381C22C22×#1175819911yescharacteristic
382C4—21no—
383C4—21no—
384C4—21no—
385C4—82no—
386C4—82no—
387C4—4881no—
388C4—4881no—
389C4—2442no—
390C4#1684019212yes—
391C4#1684020312yes—
392C4#1684026911yescharacteristic
393C4#1684056011yescharacteristic
394C2—41no—
395C2—4882no—
396C2#2145581511yescharacteristic
397C2C2×#1684019211yescharacteristic
398C2C2×#1684020311yescharacteristic
subgroupG/H
centerC2×C4C2×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3170C2
socleC22×C61C22×D8
prime radicalC2×#14519799C22
hypercenter#3170D61
hypocenterC61#33667
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3170

Complements

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

NH ≅ G/Ntype
2. #21448024C2semidirect
3. #21447958C2semidirect
4. #21447953C2semidirect
5. #21447866C2semidirect
6. #21447861C2semidirect
8. C4×#14519723C2semidirect
9. C4×#14519720C2semidirect
10. #21447618C2semidirect
11. C61×#3170C2semidirect
195. #3170D61semidirect
242. C61#33667semidirect

Maximal subgroups

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

subgroupindex
C4×#145197992
#214480242
#214479582
#214479532
#214478662
#214478612
#214477572
C4×#145197232
C4×#145197202
#214476182
C61×#31702
#3366761