Finite Group Databasebeta

← #29835017

Subgroups — #29835017

Order 15616.

552 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
1#21456389C211yesmaximal, characteristic
2#21448016C211yesmaximal, characteristic, complemented
3#21447999C211yesmaximal, characteristic
4#21447983C212yesmaximal, complemented
5#21447881C212yesmaximal, complemented
6#21447753C211yesmaximal, characteristic, complemented
7#21447736C211yesmaximal, characteristic
8C4×#14519723C212yesmaximal, complemented
9C4×#14519720C212yesmaximal, complemented
10#21447618C211yesmaximal, characteristic, complemented
11C2×C4×Q16×C61C211yesmaximal, characteristic, complemented
12#16840296—21no—
13C2×#14519723C2212yes—
14C2×#14519720C2212yes—
15#16840164C2211yescharacteristic
16#16840146C2211yescharacteristic
17#16840117—21no—
18#16840003C2211yescharacteristic
19#16839998C2211yescharacteristic
20#16839991—21no—
21C4×Dic244C2211yescharacteristic
22C4×#11758190—21no—
23C4×D488C2211yescharacteristic
24#16839589C2212yes—
25#16839587C2212yes—
26C4⋊#11758185C2212yes—
27#16839566C2212yes—
28#16839566C2212yes—
29#16839565C2212yes—
30#16839562C2212yes—
31C22⋊#11758185C2211yescharacteristic
32C22⋊#11758185C2211yescharacteristic
33C4×#11758185C2212yes—
34C22×Q16×C61C2211yescharacteristic
35C2×C4×Q8×C61—21no—
36C4×Q16×C61—21no—
37C4×Q16×C61—21no—
38C4×Q16×C61C2212yes—
39C4×Q16×C61C2212yes—
40C2×#107×C61C2211yescharacteristic
41C2×#103×C61—21no—
42C2×C4×C8×C61C2211yescharacteristic
43C2×Dic244C2311yescharacteristic
44C2×#11758190—21no—
45C2×D488C2311yescharacteristic
46#14519771—41no—
47#14519769—21no—
48#14519765—21no—
49#14519760—41no—
50#14519756—21no—
51#14519751—21no—
52C4×Dic122—21no—
53C4×D244—21no—
54C4×C22⋊D61—41no—
55#14519734—41no—
56#14519723—24no—
57#14519720—24no—
58C2×#11758185C2312yes—
59C2×#11758185C2312yes—
60#14519711C2311yescharacteristic
61#14519711C2311yescharacteristic
62#14519709—21no—
63#14519702—21no—
64#14519702—21no—
65#14519699—21no—
66#14519699—21no—
67C22×Q8×C61—21no—
68C2×Q16×C61—24no—
69C2×Q16×C61—21no—
70C2×Q16×C61C2312yes—
71C22×C8×C61D411yescharacteristic
72C22×C8×C61C2311yescharacteristic
73C4×Q8×C61—22no—
74C4×Q8×C61—22no—
75C4×Q8×C61—41no—
76C4×Q8×C61—41no—
77C2×C4⋊C4×C61—21no—
78C2×C4⋊C4×C61—41no—
79C2×C42×C61—41no—
80C2×C42×C61D411yescharacteristic
81#107×C61—21no—
82#107×C61C2312yes—
83#103×C61—22no—
84#103×C61—22no—
85C4×C8×C61D411yescharacteristic
86C4×C8×C61D411yescharacteristic
87C4×C8×C61C2311yescharacteristic
88C4×C8×C61C2311yescharacteristic
89C2×Q8×C61—24no—
90C2×Q8×C61—22no—
91C2×Q8×C61—42no—
92C2×Q8×C61—41no—
93C22×C4×C61—41no—
94C22×C4×C61—41no—
95C22×C4×C61D811yescharacteristic
96C22×C4×C61D811yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C2×C22⋊D61—41no—
99C2×D244—21no—
100C2×C4×D61—41no—
101C2×Dic122—21no—
102Q16×C61—24no—
103Q16×C61—22no—
104Q16×C61#4114yes—
105C2×C8×C61—22no—
106C2×C8×C61—41no—
107C2×C8×C61C2×D411yescharacteristic
108C2×C8×C61C2×D411yescharacteristic
109C2×C8×C61C2×D411yescharacteristic
110C2×C8×C61C2411yescharacteristic
111C4⋊C4×C61—22no—
112C4⋊C4×C61—41no—
113C4⋊C4×C61—42no—
114C4⋊C4×C61—42no—
115C42×C61—42no—
116C42×C61—42no—
117C42×C61D811yescharacteristic
118C42×C61D811yescharacteristic
119C42×C61C2×D411yescharacteristic
120C42×C61C2×D411yescharacteristic
121C22⋊Dic61—41no—
122#11758198—41no—
123#11758198—41no—
124#11758198—41no—
125#11758198—41no—
126C4⋊Dic61—21no—
127C4⋊Dic61—21no—
128#11758196—41no—
129#11758196—41no—
130#11758196—41no—
131#11758196—41no—
132C4×Dic61—41no—
133Dic244—22no—
134D488—22no—
135#11758190—42no—
136#11758185—22no—
137#11758185—22no—
138C23×C61C2×D811yescharacteristic
139C22×D61—41no—
140Q8×C61—24no—
141Q8×C61—44no—
142Q8×C61—42no—
143C2×C4×C61—22no—
144C2×C4×C61—21no—
145C2×C4×C61—21no—
146C2×C4×C61—22no—
147C2×C4×C61—44no—
148C2×C4×C61—42no—
149C2×C4×C61—42no—
150C2×C4×C61—82no—
151C2×C4×C61C2×D811yescharacteristic
152C2×C4×C61C2×D811yescharacteristic
153C2×C4×C61C2×D811yescharacteristic
154C2×C4×C61C2×D811yescharacteristic
155C2×C4×C61C2×D811yescharacteristic
156C2×C4×C61C22×D411yescharacteristic
157C22⋊D61—82no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61—41no—
161D244—41no—
162D244—41no—
163C4×D61—82no—
164Dic122—41no—
165Dic122—41no—
166C8×C61—21no—
167C8×C61—21no—
168C8×C61—21no—
169C8×C61C2×#4112yes—
170#33668—611nomaximal, Sylow
171C22×C61—21no—
172C22×C61—21no—
173C22×C61#50812yes—
174C22×C61C22×D811yescharacteristic
175C2×D61—81no—
176C2×D61—81no—
177C2×D61—81no—
178C4×C61—21no—
179C4×C61—21no—
180C4×C61—21no—
181C4×C61—21no—
182C4×C61—44no—
183C4×C61—82no—
184C4×C61#51012yes—
185C4×C61#52511yescharacteristic
186C4×C61#57511yescharacteristic
187Dic61—81no—
188Dic61—81no—
189Dic61—81no—
190#3168—611no—
191C2×C4×Q16D6111yescharacteristic, complemented
192#2235—611no—
193#2232—611no—
194#2229—612no—
195#2211—612no—
196#2188—611no—
197#2185—611no—
198C4×Q32—612no—
199C4×SD32—612no—
200#2162—611no—
201C2×C61—22no—
202C2×C61#310011yescharacteristic
203C2×C61C2×#50811yescharacteristic
204C2×C61C2×#51011yescharacteristic
205D61—161no—
206C22×Q16C2×D6111yescharacteristic
207#529—1221no—
208C2×C4×Q8—21no—
209C2×Q32—612no—
210C2×SD32—612no—
211#499—611no—
212#495—611no—
213#488—1221no—
214#467—611no—
215#466—611no—
216#465—1221no—
217C4×Q16—21no—
218C4×Q16—21no—
219C4×Q16—611no—
220C4×Q16C2×D6112yes—
221C4×Q16C2×D6112yes—
222C4×SD16—1221no—
223C4×D8—611no—
224C2×#107C2×D6111yescharacteristic
225C2×#103—21no—
226C2×C4×C8C2×D6111yescharacteristic
227#367—612no—
228#366—612no—
229C4⋊C16—612no—
230#358—612no—
231#358—612no—
232#358—612no—
233#357—612no—
234C22⋊C16—611no—
235C22⋊C16—611no—
236C4×C16—612no—
237C61#3366811yescharacteristic, complemented, Sylow
238C22×Q8—21no—
239C2×Q16—24no—
240C2×Q16—21no—
241C2×Q16—611no—
242C2×Q16C22×D6112yes—
243C2×SD16—1221no—
244C2×D8—611no—
245C22×C8C22⋊D6111yescharacteristic
246C22×C8C22×D6111yescharacteristic
247#126—2441no—
248#125—1221no—
249#124—1221no—
250#123—2441no—
251C22⋊Q8—1221no—
252#121—1221no—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×Q8—41no—
256C4×Q8—41no—
257C4×Q8—1221no—
258C4×D4—1221no—
259C4×D4—2441no—
260#117—2441no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—41no—
263C2×C42—41no—
264C2×C42C22⋊D6111yescharacteristic
265Q32—1224no—
266SD32—1224no—
267C2×C16—612no—
268C2×C16—612no—
269#107—21no—
270#107—611no—
271#107—611no—
272#107C22×D6112yes—
273#106—1221no—
274#103—22no—
275#103—22no—
276#103—1221no—
277#103—1221no—
278#102—1221no—
279#102—1221no—
280C4×C8C22⋊D6111yescharacteristic
281C4×C8C22⋊D6111yescharacteristic
282C4×C8C22×D6111yescharacteristic
283C4×C8C22×D6111yescharacteristic
284C2×Q8—24no—
285C2×Q8—22no—
286C2×Q8—42no—
287C2×Q8—41no—
288C2×Q8—1221no—
289C2×D4—1221no—
290C2×D4—2441no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—2441no—
294C22×C4#1175819911yescharacteristic
295C22×C4#1175819911yescharacteristic
296C22×C4C2×C22⋊D6111yescharacteristic
297Q16—24no—
298Q16—22no—
299Q16—1222no—
300Q16#1175822214yes—
301SD16—2442no—
302D8—1222no—
303C2×C8—22no—
304C2×C8—41no—
305C2×C8C2×C22⋊D6111yescharacteristic
306C2×C8C2×C22⋊D6111yescharacteristic
307C2×C8C2×C22⋊D6111yescharacteristic
308C2×C8C23×D6111yescharacteristic
309C4⋊C4—22no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—1221no—
314C4⋊C4—1221no—
315C4⋊C4—2441no—
316C4⋊C4—2441no—
317C4⋊C4—2441no—
318C4⋊C4—2441no—
319C22⋊C4—2441no—
320C22⋊C4—2441no—
321C22⋊C4—2441no—
322C22⋊C4—2441no—
323C22⋊C4—2441no—
324C42—42no—
325C42—42no—
326C42—2441no—
327C42#1175819911yescharacteristic
328C42#1175819911yescharacteristic
329C42C2×C22⋊D6111yescharacteristic
330C42C2×C22⋊D6111yescharacteristic
331C16—1222no—
332C16—1222no—
333C23—2441no—
334C23C2×#1175819911yescharacteristic
335Q8—24no—
336Q8—44no—
337Q8—42no—
338Q8—2441no—
339Q8—2441no—
340D4—4882no—
341D4—2441no—
342D4—2441no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—82no—
351C2×C4—4882no—
352C2×C4—2441no—
353C2×C4—2441no—
354C2×C4—2441no—
355C2×C4C2×#1175819911yescharacteristic
356C2×C4C2×#1175819911yescharacteristic
357C2×C4C2×#1175819911yescharacteristic
358C2×C4C2×#1175819911yescharacteristic
359C2×C4C2×#1175819911yescharacteristic
360C2×C4C22×C22⋊D6111yescharacteristic
361C8—21no—
362C8—21no—
363C8—21no—
364C8C2×#1175822212yes—
365C22—21no—
366C22—21no—
367C22—4881no—
368C22—4881no—
369C22—4881no—
370C22#1684019212yes—
371C22C22×#1175819911yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—44no—
377C4—82no—
378C4—4881no—
379C4—4881no—
380C4—4881no—
381C4#1684020312yes—
382C4#1684027311yescharacteristic
383C4#1684056011yescharacteristic
384C2—22no—
385C2—9761no—
386C2#2145581911yescharacteristic
387C2C2×#1684019211yescharacteristic
388C2C2×#1684020311yescharacteristic
subgroupG/H
centerC2×C4C2×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C4×Q16×C61C2
socleC22×C61C22×D8
prime radical#16839998C22
hypercenterC2×C4×Q16D61
hypocenterC61#33668
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×Q16

Complements

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

NH ≅ G/Ntype
2. #21448016C2semidirect
4. #21447983C2semidirect
5. #21447881C2semidirect
6. #21447753C2semidirect
8. C4×#14519723C2semidirect
9. C4×#14519720C2semidirect
10. #21447618C2semidirect
11. C2×C4×Q16×C61C2semidirect
191. C2×C4×Q16D61semidirect
237. C61#33668semidirect

Maximal subgroups

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

subgroupindex
#214563892
#214480162
#214479992
#214479832
#214478812
#214477532
#214477362
C4×#145197232
C4×#145197202
#214476182
C2×C4×Q16×C612
#3366861