Finite Group Databasebeta

← #29834653

Subgroups — #29834653

Order 15616.

546 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14519799C211yesmaximal, characteristic
2#21447980C212yesmaximal, complemented
3#21447973C212yesmaximal, complemented
4#21447968C211yesmaximal, characteristic, complemented
5#21447963C211yesmaximal, characteristic, complemented
6#21447958C211yesmaximal, characteristic, complemented
7#21447953C211yesmaximal, characteristic, complemented
8#21447601C212yesmaximal, complemented
9C2×C4⋊#11758185C211yesmaximal, characteristic, complemented
10#21447524C212yesmaximal
11C61×#3160C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13C4×#11758222—21no—
14C22×#11758185C2212yes—
15C4×Dic244C2211yescharacteristic
16C4×Dic244C2211yescharacteristic
17C4×#11758190—21no—
18C4×#11758190—21no—
19C4×D488C2211yescharacteristic
20C4×D488C2211yescharacteristic
21#16839800C2211yescharacteristic
22#16839759—21no—
23#16839589C2212yes—
24#16839589C2212yes—
25#16839587C2212yes—
26#16839587C2212yes—
27C4⋊#11758185C2211yescharacteristic
28C4⋊#11758185C2211yescharacteristic
29C4⋊#11758185C2212yes—
30#16839565C2212yes—
31#16839565C2212yes—
32#16839562C2212yes—
33#16839562C2212yes—
34C61×#533—21no—
35C61×#500C2211yescharacteristic
36C61×#500C2211yescharacteristic
37C61×#499—21no—
38C61×#499C2212yes—
39C61×#498—21no—
40C61×#427—21no—
41C61×#427C2212yes—
42C2×C4×C8×C61C2211yescharacteristic
43C2×#11758222—21no—
44#14519799—22no—
45#14519799Q814yescomplemented
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—
58C2×#11758185—24no—
59C2×#11758185C2312yes—
60C2×#11758185C2312yes—
61#14519711C2311yescharacteristic
62#14519711C2311yescharacteristic
63#14519709—21no—
64#14519702—21no—
65#14519702—21no—
66#14519699—21no—
67#14519699—21no—
68C22×C8×C61D411yescharacteristic
69C22×C8×C61C2311yescharacteristic
70C4⋊Q8×C61—21no—
71C4⋊Q8×C61—21no—
72#125×C61—22no—
73C22⋊Q8×C61—41no—
74C22⋊Q8×C61—41no—
75C4×Q8×C61—41no—
76C4×Q8×C61—41no—
77#117×C61—22no—
78C2×C42×C61D411yescharacteristic
79#107×C61—21no—
80#107×C61—21no—
81#107×C61C2312yes—
82#107×C61C2312yes—
83#106×C61—22no—
84#106×C61—22no—
85C4×C8×C61D411yescharacteristic
86C4×C8×C61D411yescharacteristic
87C4×C8×C61C2311yescharacteristic
88C4×C8×C61C2311yescharacteristic
89C2×Q8×C61—41no—
90C2×Q8×C61—41no—
91C22×C4×C61D811yescharacteristic
92C22×C4×C61D811yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C2×C22⋊D61—41no—
95#11758222—24no—
96#11758222—42no—
97C2×D244—21no—
98C2×C4×D61—41no—
99C2×C4×D61—41no—
100C2×C4×D61—41no—
101C2×Dic122—21no—
102C2×C8×C61—22no—
103C2×C8×C61C2×D411yescharacteristic
104C2×C8×C61C2×D411yescharacteristic
105C2×C8×C61C2×D411yescharacteristic
106C2×C8×C61C2×Q812yes—
107C2×C8×C61#4112yes—
108C2×C8×C61C2411yescharacteristic
109C4⋊C4×C61—22no—
110C4⋊C4×C61—22no—
111C4⋊C4×C61—42no—
112C4⋊C4×C61—42no—
113C4⋊C4×C61—41no—
114C4⋊C4×C61—41no—
115C22⋊C4×C61—42no—
116C42×C61—42no—
117C42×C61D811yescharacteristic
118C42×C61D811yescharacteristic
119C42×C61C2×D411yescharacteristic
120C42×C61C2×D411yescharacteristic
121C22⋊Dic61—41no—
122#11758198—41no—
123#11758198—41no—
124C4⋊Dic61—21no—
125C4⋊Dic61—21no—
126#11758196—41no—
127#11758196—41no—
128C4×Dic61—41no—
129C4×Dic61—41no—
130C4×Dic61—41no—
131Dic244—21no—
132Dic244C2×Q812yes—
133D488—21no—
134D488C2×Q812yes—
135#11758190—24no—
136#11758185—22no—
137#11758185—22no—
138C23×C61C2×D811yescharacteristic
139C22×D61—41no—
140Q8×C61—82no—
141Q8×C61—82no—
142C2×C4×C61—22no—
143C2×C4×C61—41no—
144C2×C4×C61—42no—
145C2×C4×C61—42no—
146C2×C4×C61C22⋊Q812yes—
147C2×C4×C61C2×D811yescharacteristic
148C2×C4×C61C2×D811yescharacteristic
149C2×C4×C61C2×D811yescharacteristic
150C2×C4×C61C2×D811yescharacteristic
151C2×C4×C61C2×D811yescharacteristic
152C2×C4×C61#13712yes—
153C2×C4×C61C22×D411yescharacteristic
154C22⋊D61—44no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158D244—22no—
159D244—41no—
160C4×D61—44no—
161C4×D61—82no—
162C4×D61—82no—
163Dic122—22no—
164Dic122—41no—
165C8×C61—21no—
166C8×C61—21no—
167C8×C61C22⋊Q812yes—
168C8×C61C22×Q811yescharacteristic
169C8×C61C2×#4111yescharacteristic
170#33626—611nomaximal, Sylow
171C22×C61—21no—
172C22×C61—21no—
173C22×C61#47412yes—
174C22×C61C22×D811yescharacteristic
175C2×D61—42no—
176C2×D61—81no—
177C2×D61—81no—
178C4×C61—21no—
179C4×C61—21no—
180C4×C61—21no—
181C4×C61—82no—
182C4×C61—82no—
183C4×C61#47412yes—
184C4×C61#50812yes—
185C4×C61C2×C22⋊Q811yescharacteristic
186C4×C61C2×#13711yescharacteristic
187Dic61—42no—
188Dic61—81no—
189Dic61—81no—
190#3160D6111yescharacteristic, complemented
191C4×#135—611no—
192#2229—612no—
193#2228—612no—
194#2227—611no—
195#2226—611no—
196#2225—611no—
197#2224—611no—
198#2158—612no—
199C2×C4⋊C16—611no—
200#2138—612no—
201C2×C61—22no—
202C2×C61C2×#47411yescharacteristic
203C2×C61C2×#50811yescharacteristic
204C2×C61#341411yescharacteristic
205D61—82no—
206C2×#135—611no—
207#533—21no—
208C4×#41—1221no—
209C22×C16—612no—
210#500C2×D6111yescharacteristic
211#500C2×D6111yescharacteristic
212#499—21no—
213#499C2×D6112yes—
214#498—21no—
215C4×Q16—611no—
216C4×Q16—611no—
217C4×SD16—1221no—
218C4×SD16—1221no—
219C4×D8—611no—
220C4×D8—611no—
221#427—21no—
222#427—611no—
223#427C2×D6112yes—
224#416—1221no—
225C2×C4×C8C2×D6111yescharacteristic
226#367—612no—
227#367—612no—
228#366—612no—
229#366—612no—
230C4⋊C16—611no—
231C4⋊C16—611no—
232C4⋊C16—612no—
233#358—612no—
234#358—612no—
235#357—612no—
236#357—612no—
237C61#3362611yescharacteristic, complemented, Sylow
238C2×#41—1221no—
239#135—614no—
240#135—1222no—
241C2×Q16—611no—
242C2×SD16—1221no—
243C2×D8—611no—
244C22×C8C22⋊D6111yescharacteristic
245C22×C8C22×D6111yescharacteristic
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248#125—22no—
249C22⋊Q8—41no—
250C22⋊Q8—41no—
251C4×Q8—41no—
252C4×Q8—41no—
253C4×Q8—1221no—
254C4×Q8—1221no—
255C4×D4—1221no—
256C4×D4—1221no—
257C4×D4—2441no—
258C4×D4—2441no—
259#117—22no—
260#117—1221no—
261#117—2441no—
262C2×C42—2441no—
263C2×C42C22⋊D6111yescharacteristic
264C2×C16—612no—
265C2×C16—612no—
266C2×C16—1224no—
267#107—21no—
268#107—21no—
269#107—611no—
270#107—611no—
271#107C22×D6112yes—
272#107C22×D6112yes—
273#106—22no—
274#106—22no—
275#106—1221no—
276#103—1221no—
277#103—1221no—
278#102—1221no—
279#102—1221no—
280C4×C8C22⋊D6111yescharacteristic
281C4×C8C22⋊D6111yescharacteristic
282C4×C8C22×D6111yescharacteristic
283C4×C8C22×D6111yescharacteristic
284#41—1224no—
285#41—2442no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—1221no—
289C2×D4—1221no—
290C2×D4—2441no—
291C22×C4—2441no—
292C22×C4—2441no—
293C22×C4—2441no—
294C22×C4#1175819911yescharacteristic
295C22×C4#1175819911yescharacteristic
296C22×C4C2×C22⋊D6111yescharacteristic
297Q16—612no—
298Q16—1221no—
299SD16—1224no—
300D8—612no—
301D8—1221no—
302C2×C8—22no—
303C2×C8Q8×D6112yes—
304C2×C8#1175822612yes—
305C2×C8C2×C22⋊D6111yescharacteristic
306C2×C8C2×C22⋊D6111yescharacteristic
307C2×C8C2×C22⋊D6111yescharacteristic
308C2×C8C23×D6111yescharacteristic
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—1221no—
316C4⋊C4—1221no—
317C4⋊C4—2441no—
318C4⋊C4—2441no—
319C22⋊C4—42no—
320C22⋊C4—2441no—
321C22⋊C4—2441no—
322C22⋊C4—2441no—
323C42—42no—
324C42—2441no—
325C42—2441no—
326C42—2441no—
327C42#1175819911yescharacteristic
328C42#1175819911yescharacteristic
329C42C2×C22⋊D6111yescharacteristic
330C42C2×C22⋊D6111yescharacteristic
331C16—1222no—
332C16—1222no—
333C23—2441no—
334C23C2×#1175819911yescharacteristic
335Q8—82no—
336Q8—82no—
337Q8—1222no—
338Q8—2441no—
339D4—1222no—
340D4—2444no—
341D4—2441no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—4882no—
347C2×C4—4882no—
348C2×C4—2441no—
349C2×C4—2441no—
350C2×C4—2441no—
351C2×C4—2444no—
352C2×C4#1451975312yes—
353C2×C4C2×#1175819911yescharacteristic
354C2×C4C2×#1175819911yescharacteristic
355C2×C4C2×#1175819911yescharacteristic
356C2×C4C2×#1175819911yescharacteristic
357C2×C4C2×#1175819911yescharacteristic
358C2×C4#1451980812yes—
359C2×C4C22×C22⋊D6111yescharacteristic
360C8—21no—
361C8—21no—
362C8#1451975312yes—
363C8C2×Q8×D6111yescharacteristic
364C8C2×#1175822611yescharacteristic
365C22—21no—
366C22—21no—
367C22—4881no—
368C22—4881no—
369C22—2442no—
370C22#1684003912yes—
371C22C22×#1175819911yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—82no—
376C4—82no—
377C4—4881no—
378C4—4881no—
379C4—2442no—
380C4#1684003912yes—
381C4#1684019212yes—
382C4C2×#1451975311yescharacteristic
383C4C2×#1451980811yescharacteristic
384C2—22no—
385C2—4882no—
386C2C2×#1684003911yescharacteristic
387C2C2×#1684019211yescharacteristic
388C2#2145895311yescharacteristic
subgroupG/H
centerC2×C4C2×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3160C2
socleC22×C61C22×D8
prime radicalC2×#14519799C22
hypercenter#3160D61
hypocenterC61#33626
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3160

Complements

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

NH ≅ G/Ntype
2. #21447980C2semidirect
3. #21447973C2semidirect
4. #21447968C2semidirect
5. #21447963C2semidirect
6. #21447958C2semidirect
7. #21447953C2semidirect
8. #21447601C2semidirect
9. C2×C4⋊#11758185C2semidirect
11. C61×#3160C2semidirect
45. #14519799Q8semidirect
190. #3160D61semidirect
237. C61#33626semidirect

Maximal subgroups

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

subgroupindex
C4×#145197992
#214479802
#214479732
#214479682
#214479632
#214479582
#214479532
#214476012
C2×C4⋊#117581852
#214475242
C61×#31602
#3362661