Finite Group Databasebeta

← #29834991

Subgroups — #29834991

Order 15616.

596 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14519799C211yesmaximal, characteristic, complemented
2#21448007C212yesmaximal, complemented
3#21447968C211yesmaximal, characteristic, complemented
4#21447963C211yesmaximal, characteristic, complemented
5#21447856C211yesmaximal, characteristic, complemented
6#21447851C211yesmaximal, characteristic, complemented
7#21447740C212yesmaximal, complemented
8C4×#14519721C212yesmaximal, complemented
9C4×#14519718C212yesmaximal, complemented
10#21447618C211yesmaximal, characteristic, complemented
11C61×#3169C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13C4×#11758222—21no—
14C2×#14519721C2212yes—
15C2×#14519718C2212yes—
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#16839565C2212yes—
29#16839563C2212yes—
30#16839563C2212yes—
31#16839562C2212yes—
32C22⋊#11758185C2212yes—
33C4×#11758185C2212yes—
34C61×#532—21no—
35C61×#500C2211yescharacteristic
36C61×#493C2211yescharacteristic
37C61×#486—21no—
38C61×#466—21no—
39C61×#466C2212yes—
40C4×D8×C61—21no—
41C4×D8×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#14519721—24no—
59#14519718—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×#41×C61—41no—
70C2×D8×C61—21no—
71C2×D8×C61C2312yes—
72C22×C8×C61D411yescharacteristic
73C22×C8×C61C2311yescharacteristic
74C4⋊Q8×C61—21no—
75#127×C61—21no—
76#124×C61—41no—
77#121×C61—22no—
78#121×C61—41no—
79C4×D4×C61—22no—
80C4×D4×C61—41no—
81C2×C42×C61D411yescharacteristic
82#107×C61—21no—
83#107×C61C2312yes—
84#102×C61—22no—
85#102×C61—22no—
86C4×C8×C61D411yescharacteristic
87C4×C8×C61D411yescharacteristic
88C4×C8×C61C2311yescharacteristic
89C4×C8×C61C2311yescharacteristic
90#41×C61—84no—
91C2×Q8×C61—41no—
92C2×D4×C61—22no—
93C2×D4×C61—42no—
94C2×D4×C61—41no—
95C2×D4×C61—41no—
96C22×C4×C61—42no—
97C22×C4×C61D811yescharacteristic
98C22×C4×C61D811yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C2×C22⋊D61—41no—
101#11758222—24no—
102#11758222—42no—
103C2×D244—21no—
104C2×C4×D61—41no—
105C2×C4×D61—41no—
106C2×C4×D61—41no—
107C2×Dic122—21no—
108D8×C61—22no—
109D8×C61—22no—
110D8×C61—42no—
111C2×C8×C61—21no—
112C2×C8×C61—21no—
113C2×C8×C61—41no—
114C2×C8×C61C2×D411yescharacteristic
115C2×C8×C61C2×D411yescharacteristic
116C2×C8×C61C2×D411yescharacteristic
117C2×C8×C61C2411yescharacteristic
118C4⋊C4×C61—22no—
119C4⋊C4×C61—41no—
120C22⋊C4×C61—42no—
121C22⋊C4×C61—42no—
122C42×C61D811yescharacteristic
123C42×C61D811yescharacteristic
124C42×C61C2×D411yescharacteristic
125C42×C61C2×D411yescharacteristic
126C22⋊Dic61—41no—
127#11758198—41no—
128#11758198—41no—
129C4⋊Dic61—21no—
130C4⋊Dic61—21no—
131#11758196—41no—
132#11758196—41no—
133C4×Dic61—41no—
134C4×Dic61—41no—
135C4×Dic61—41no—
136Dic244—21no—
137Dic244#4112yescomplemented
138D488—21no—
139D488#4112yescomplemented
140#11758190—24no—
141#11758185—22no—
142#11758185—22no—
143C23×C61—42no—
144C23×C61C2×D811yescharacteristic
145C22×D61—41no—
146Q8×C61—82no—
147D4×C61—42no—
148D4×C61—42no—
149D4×C61—82no—
150D4×C61—84no—
151D4×C61—82no—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—22no—
155C2×C4×C61—22no—
156C2×C4×C61—42no—
157C2×C4×C61—84no—
158C2×C4×C61C2×D811yescharacteristic
159C2×C4×C61C2×D811yescharacteristic
160C2×C4×C61C2×D811yescharacteristic
161C2×C4×C61C2×D811yescharacteristic
162C2×C4×C61C2×D811yescharacteristic
163C2×C4×C61C22×D411yescharacteristic
164C22⋊D61—44no—
165C2×Dic61—41no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168D244—22no—
169D244—41no—
170C4×D61—44no—
171C4×D61—82no—
172C4×D61—82no—
173Dic122—22no—
174Dic122—41no—
175C8×C61—21no—
176C8×C61—21no—
177C8×C61—21no—
178C8×C61C2×#4111yescharacteristic
179C8×C61C2×#4111yescharacteristic
180#33666—611nomaximal, Sylow
181C22×C61—21no—
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—82no—
185C22×C61—82no—
186C22×C61—82no—
187C22×C61C22×D811yescharacteristic
188C2×D61—42no—
189C2×D61—81no—
190C2×D61—81no—
191C4×C61—21no—
192C4×C61—21no—
193C4×C61—21no—
194C4×C61—82no—
195C4×C61#50812yes—
196C4×C61#50912yes—
197C4×C61#52511yescharacteristic
198C4×C61#57511yescharacteristic
199Dic61—42no—
200Dic61—81no—
201Dic61—81no—
202#3169D6111yescharacteristic, complemented
203C4×#135—611no—
204#2234—612no—
205#2227—611no—
206#2226—611no—
207#2207—611no—
208#2206—611no—
209#2186—612no—
210C4×SD32—612no—
211C4×D16—612no—
212#2162—611no—
213C2×C61—41no—
214C2×C61—162no—
215C2×C61#310011yescharacteristic
216C2×C61C2×#50811yescharacteristic
217C2×C61C2×#50911yescharacteristic
218D61—82no—
219C2×#135—611no—
220#532—21no—
221C4×#41—1221no—
222C2×SD32—612no—
223C2×D16—612no—
224#500C2×D6111yescharacteristic
225#493C2×D6111yescharacteristic
226#486—21no—
227#466—21no—
228#466C2×D6112yes—
229C4×Q16—611no—
230C4×Q16—611no—
231C4×SD16—1221no—
232C4×SD16—1221no—
233C4×D8—21no—
234C4×D8—611no—
235C4×D8—611no—
236C4×D8C2×D6112yes—
237#427—611no—
238#416—1221no—
239C2×C4×C8C2×D6111yescharacteristic
240#367—612no—
241#366—612no—
242C4⋊C16—611no—
243C4⋊C16—611no—
244#358—612no—
245#357—612no—
246#357—612no—
247#357—612no—
248C22⋊C16—612no—
249C4×C16—612no—
250C61#3366611yescharacteristic, complemented, Sylow
251C2×#41—41no—
252C2×#41—1221no—
253#135—1222no—
254#135—1222no—
255C2×Q16—611no—
256C2×SD16—1221no—
257C2×D8—21no—
258C2×D8—611no—
259C2×D8C22×D6112yes—
260C22×C8C22⋊D6111yescharacteristic
261C22×C8C22×D6111yescharacteristic
262C4⋊Q8—21no—
263#127—21no—
264#124—41no—
265#121—22no—
266#121—41no—
267C4×Q8—1221no—
268C4×Q8—1221no—
269C4×D4—22no—
270C4×D4—41no—
271C4×D4—1221no—
272C4×D4—1221no—
273C4×D4—2441no—
274C4×D4—2441no—
275#117—1221no—
276#117—2441no—
277C2×C42—2441no—
278C2×C42C22⋊D6111yescharacteristic
279SD32—1224no—
280D16—1224no—
281C2×C16—612no—
282C2×C16—612no—
283#107—21no—
284#107—611no—
285#107—611no—
286#107C22×D6112yes—
287#106—1221no—
288#103—1221no—
289#103—1221no—
290#102—22no—
291#102—22no—
292#102—1221no—
293#102—1221no—
294C4×C8C22⋊D6111yescharacteristic
295C4×C8C22⋊D6111yescharacteristic
296C4×C8C22×D6111yescharacteristic
297C4×C8C22×D6111yescharacteristic
298#41—84no—
299#41—1224no—
300#41—2442no—
301C2×Q8—41no—
302C2×Q8—1221no—
303C2×D4—22no—
304C2×D4—42no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—1221no—
308C2×D4—2441no—
309C22×C4—42no—
310C22×C4—2441no—
311C22×C4—2441no—
312C22×C4—2441no—
313C22×C4#1175819911yescharacteristic
314C22×C4#1175819911yescharacteristic
315C22×C4C2×C22⋊D6111yescharacteristic
316Q16—612no—
317Q16—1221no—
318SD16—1224no—
319D8—22no—
320D8—22no—
321D8—42no—
322D8—612no—
323D8—1221no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—41no—
327C2×C8C2×C22⋊D6111yescharacteristic
328C2×C8C2×C22⋊D6111yescharacteristic
329C2×C8C2×C22⋊D6111yescharacteristic
330C2×C8C23×D6111yescharacteristic
331C4⋊C4—22no—
332C4⋊C4—41no—
333C4⋊C4—1221no—
334C4⋊C4—1221no—
335C4⋊C4—2441no—
336C4⋊C4—2441no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—2441no—
340C22⋊C4—2441no—
341C22⋊C4—2441no—
342C42—2441no—
343C42—2441no—
344C42—2441no—
345C42#1175819911yescharacteristic
346C42#1175819911yescharacteristic
347C42C2×C22⋊D6111yescharacteristic
348C42C2×C22⋊D6111yescharacteristic
349C16—1222no—
350C16—1222no—
351C23—42no—
352C23—2441no—
353C23C2×#1175819911yescharacteristic
354Q8—82no—
355Q8—1222no—
356Q8—2441no—
357D4—42no—
358D4—42no—
359D4—82no—
360D4—84no—
361D4—82no—
362D4—1222no—
363D4—2444no—
364D4—2441no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—42no—
370C2×C4—84no—
371C2×C4—4882no—
372C2×C4—4882no—
373C2×C4—2444no—
374C2×C4—2441no—
375C2×C4—2441no—
376C2×C4—2441no—
377C2×C4C2×#1175819911yescharacteristic
378C2×C4C2×#1175819911yescharacteristic
379C2×C4C2×#1175819911yescharacteristic
380C2×C4C2×#1175819911yescharacteristic
381C2×C4C2×#1175819911yescharacteristic
382C2×C4C22×C22⋊D6111yescharacteristic
383C8—21no—
384C8—21no—
385C8—21no—
386C8#41×D6111yescharacteristic
387C8#41×D6111yescharacteristic
388C22—21no—
389C22—21no—
390C22—21no—
391C22—82no—
392C22—82no—
393C22—82no—
394C22—4881no—
395C22—4881no—
396C22—2442no—
397C22C22×#1175819911yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—82no—
402C4—4881no—
403C4—4881no—
404C4—2442no—
405C4#1684019212yes—
406C4#1684019812yes—
407C4#1684026911yescharacteristic
408C4#1684056011yescharacteristic
409C2—41no—
410C2—162no—
411C2—4882no—
412C2#2145581511yescharacteristic
413C2C2×#1684019211yescharacteristic
414C2C2×#1684019811yescharacteristic
subgroupG/H
centerC2×C4C2×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3169C2
socleC22×C61C22×D8
hypercenter#3169D61
hypocenterC61#33666
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3169

Complements

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

NH ≅ G/Ntype
1. C4×#14519799C2semidirect
2. #21448007C2semidirect
3. #21447968C2semidirect
4. #21447963C2semidirect
5. #21447856C2semidirect
6. #21447851C2semidirect
7. #21447740C2semidirect
8. C4×#14519721C2semidirect
9. C4×#14519718C2semidirect
10. #21447618C2semidirect
11. C61×#3169C2semidirect
137. Dic244#41semidirect
139. D488#41semidirect
202. #3169D61semidirect
250. C61#33666semidirect

Maximal subgroups

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

subgroupindex
C4×#145197992
#214480072
#214479682
#214479632
#214478562
#214478512
#214477402
C4×#145197212
C4×#145197182
#214476182
C61×#31692
#3366661