Finite Group Databasebeta

← #29886114

Subgroups — #29886114

Order 15616.

4086 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840309C216yesmaximal, complemented
2#21454216C211yesmaximal, characteristic, complemented
3#21454213C211yesmaximal, characteristic, complemented
4#21454173C216yesmaximal, complemented
5#21454165C211yesmaximal, characteristic, complemented
6#21454163C216yesmaximal, complemented
7#21454161C211yesmaximal, characteristic, complemented
8C4×#14519774C211yesmaximal, characteristic, complemented
9C4×#14519773C216yesmaximal, complemented
10C43×D61C211yesmaximal, characteristic, complemented
11C4×#127×C61C211yesmaximal, characteristic, complemented
12C22×#11758222C2214yes—
13#16840309—248no—
14C2×#14519776C2216yes—
15C2×#14519774C2211yescharacteristic
16C2×#14519773C2216yes—
17C2×#14519773C2216yes—
18C2×#14519763C22112yes—
19C2×#14519752C2216yes—
20C2×#14519750C22112yes—
21C2×#14519748C2216yes—
22C2×C4×D244C2216yes—
23C2×C4×C22⋊D61C22112yes—
24C2×C42×D61C2211yescharacteristic
25C2×C42×D61C2213yes—
26C2×C42×D61C2213yes—
27#16839665C2216yes—
28#16839664C2216yes—
29#16839663C2216yes—
30#16839662C22124yes—
31#16839632C2211yescharacteristic
32#16839631C2216yes—
33C42×Dic61C2213yes—
34C42×Dic61C2211yescharacteristic
35C42×Dic61C2213yes—
36C2×#127×C61C2211yescharacteristic
37C2×C4×D4×C61C2216yes—
38C61×#390C2216yes—
39C61×#384C2211yescharacteristic
40C43×C61C2211yescharacteristic
41C2×#11758222—248no—
42C22×Dic122C2314yes—
43C22×D244C2316yes—
44C22×D244C2314yes—
45C22×C22⋊D61C23112yes—
46C22×C22⋊D61C2314yes—
47C22×C22⋊D61C2314yes—
48C22×C4×D61C2314yes—
49C22×C4×D61C2313yes—
50C22×C4×D61C2313yes—
51C22×C4×D61C2311yescharacteristic
52#14519776—224no—
53#14519774—24no—
54#14519773—224no—
55#14519773—224no—
56#14519763—248no—
57#14519752—224no—
58#14519750—248no—
59#14519748—224no—
60C4×D244—224no—
61C4×C22⋊D61—248no—
62C2×C4⋊Dic61C2316yes—
63C2×#11758196C23112yes—
64C2×C22⋊Dic61C23112yes—
65C2×#11758198C23112yes—
66C2×#11758198C23112yes—
67C2×C4×Dic61C2316yes—
68C2×C4×Dic61C2313yes—
69C2×C4×Dic61C2313yes—
70C2×C4×Dic61C2313yes—
71C2×C4×Dic61C2313yes—
72C2×C4×Dic61C2313yes—
73C42×D61—24no—
74C42×D61—212no—
75C42×D61D4124yescomplemented
76C22×D4×C61C2316yes—
77C23×C4×C61C2314yes—
78#127×C61—24no—
79C4×D4×C61—224no—
80C2×C4⋊C4×C61C2316yes—
81C2×C22⋊C4×C61C23112yes—
82C2×C42×C61C2313yes—
83C2×C42×C61C2313yes—
84C2×C42×C61C2311yescharacteristic
85C24×C61C2414yes—
86C23×D61C2414yes—
87C23×D61C2411yescharacteristic
88C2×D4×C61—212no—
89C2×D4×C61—212no—
90C2×D4×C61—212no—
91C22×C4×C61—224no—
92C22×C4×C61C2414yes—
93C22×C4×C61C2413yes—
94C22×C4×C61C2411yescharacteristic
95C22×C4×C61C2413yes—
96C2×C22⋊D61—224no—
97C2×C22⋊D61—224no—
98C2×C22⋊D61—224no—
99C2×C22⋊D61—248no—
100C22×Dic61C2414yes—
101C22×Dic61C2413yes—
102C22×Dic61C2411yescharacteristic
103C22×Dic61C2413yes—
104#11758222—464no—
105C2×D244—224no—
106C2×D244—212no—
107C2×D244—212no—
108C2×D244—212no—
109C2×C4×D61—26no—
110C2×C4×D61—26no—
111C2×C4×D61—212no—
112C2×C4×D61—224no—
113C2×C4×D61C2×D4112yes—
114C2×C4×D61C2×D4112yes—
115C2×C4×D61C2×D4112yes—
116C2×Dic122—224no—
117C4⋊C4×C61—212no—
118C22⋊C4×C61—224no—
119C42×C61—26no—
120C42×C61C2×D4112yes—
121C42×C61#4114yescomplemented
122C22⋊Dic61—224no—
123#11758198—224no—
124#11758198—224no—
125C4⋊Dic61—212no—
126#11758196—224no—
127C4×Dic61—26no—
128C4×Dic61—212no—
129C4×Dic61C2×D4112yes—
130C4×Dic61C2×D4112yes—
131C4×Dic61C2×D4112yes—
132C4×Dic61#41112yescomplemented
133C23×C61—212no—
134C23×C61—212no—
135C23×C61—24no—
136C23×C61C2511yescharacteristic
137C22×D61—23no—
138C22×D61—21no—
139C22×D61—212no—
140C22×D61—24no—
141C22×D61—212no—
142C22×D61C22×D416yes—
143D4×C61—424no—
144C2×C4×C61—26no—
145C2×C4×C61—212no—
146C2×C4×C61—416no—
147C2×C4×C61C22×D416yes—
148C2×C4×C61C22×D416yes—
149C2×C4×C61C22×D416yes—
150C2×C4×C61C2×#4116yes—
151C2×C4×C61C2×#4116yes—
152C22⋊D61—416no—
153C22⋊D61—416no—
154C22⋊D61—448no—
155C2×Dic61—23no—
156C2×Dic61—23no—
157C2×Dic61—212no—
158C2×Dic61C22×D416yes—
159C2×Dic61C22×D416yes—
160C2×Dic61C22×D416yes—
161C2×Dic61C2×#41112yes—
162D244—416no—
163D244—424no—
164C4×D61—224no—
165C4×D61—224no—
166C4×D61—416no—
167C4×D61#127116yescomplemented
168Dic122—416no—
169#36644—611nomaximal, Sylow
170C22×C61—44no—
171C22×C61—412no—
172C22×C61—412no—
173C22×C61C23×D413yes—
174C22×C61C22×#4113yes—
175C22×C61C22×#4111yescharacteristic
176C2×D61—26no—
177C2×D61—26no—
178C2×D61—44no—
179C2×D61—412no—
180C2×D61—412no—
181C2×D61C2×#12714yes—
182C4×C61—26no—
183C4×C61—44no—
184C4×C61C2×#12714yes—
185C4×C61#532112yescomplemented
186Dic61—26no—
187Dic61—44no—
188Dic61C2×#12714yes—
189Dic61#532112yescomplemented
190C2×#532—616no—
191#2868—611no—
192#2867—611no—
193#2858—616no—
194#2856—616no—
195#2856—611no—
196#2856—611no—
197C4×#127—611no—
198C4×#127—616no—
199C4×#127D6111yescharacteristic, complemented
200C2×C43—611no—
201C2×C61—84no—
202C2×C61C22×#12711yescharacteristic
203C2×C61C2×#53213yes—
204C2×C61C2×#53213yes—
205D61—24no—
206D61—84no—
207C22×#41—614no—
208#532—12248no—
209C2×C4⋊Q8—616no—
210C2×#127—616no—
211C2×#127—616no—
212C2×#127—611no—
213C2×#127C2×D6111yescharacteristic
214C2×#124—6112no—
215C2×#121—616no—
216C2×#121—616no—
217C2×#121—6112no—
218C2×C4×D4—6112no—
219C2×C4×D4—616no—
220C2×C4×D4C2×D6116yes—
221C22×C42—613no—
222C22×C42—613no—
223C22×C42—611no—
224#390—616no—
225#390—616no—
226#390—6124no—
227#390—616no—
228#390C2×D6116yes—
229#384—616no—
230#384—611no—
231#384C2×D6111yescharacteristic
232C43—613no—
233C43—613no—
234C43—611no—
235C43C2×D6111yescharacteristic
236C61#3664411yescharacteristic, complemented, Sylow
237C2×#41—12248no—
238C22×Q8—614no—
239C22×D4—6112no—
240C22×D4—614no—
241C22×D4—614no—
242C22×D4—616no—
243C22×D4—614no—
244C22×D4C22×D6116yes—
245C23×C4—611no—
246C23×C4—613no—
247C23×C4—613no—
248C23×C4—614no—
249C23×C4C22×D6114yes—
250C4⋊Q8—12224no—
251#127—24no—
252#127—12224no—
253#127—12224no—
254#127—1224no—
255#124—12248no—
256#121—12248no—
257#121—12224no—
258#121—12224no—
259C4×D4—224no—
260C4×D4—12248no—
261C4×D4—12224no—
262C2×C4⋊C4—6112no—
263C2×C4⋊C4—616no—
264C2×C4⋊C4C22×D6116yes—
265C2×C22⋊C4—6112no—
266C2×C22⋊C4—6112no—
267C2×C22⋊C4—6112no—
268C2×C22⋊C4C22×D61112yes—
269C2×C42—613no—
270C2×C42—613no—
271C2×C42—613no—
272C2×C42—613no—
273C2×C42—6124no—
274C2×C42—616no—
275C2×C42—613no—
276C2×C42—12212no—
277C2×C42—1224no—
278C2×C42C22×D6113yes—
279C2×C42C22×D6111yescharacteristic
280C2×C42C22×D6113yes—
281C24—611no—
282C24—614no—
283C24C23×D6114yes—
284#41—24464no—
285C2×Q8—12224no—
286C2×D4—212no—
287C2×D4—212no—
288C2×D4—212no—
289C2×D4—12224no—
290C2×D4—12224no—
291C2×D4—12224no—
292C2×D4—12248no—
293C2×D4—12224no—
294C2×D4—12212no—
295C2×D4—12212no—
296C2×D4—12212no—
297C22×C4—224no—
298C22×C4—613no—
299C22×C4—611no—
300C22×C4—6112no—
301C22×C4—613no—
302C22×C4—6112no—
303C22×C4—6112no—
304C22×C4—614no—
305C22×C4—12212no—
306C22×C4—1226no—
307C22×C4—1226no—
308C22×C4—12224no—
309C22×C4C23×D6114yes—
310C22×C4C23×D6113yes—
311C22×C4C23×D6113yes—
312C22×C4C23×D6111yescharacteristic
313C4⋊C4—212no—
314C4⋊C4—12224no—
315C4⋊C4—12212no—
316C22⋊C4—224no—
317C22⋊C4—12224no—
318C22⋊C4—12224no—
319C22⋊C4—12224no—
320C42—26no—
321C42—6112no—
322C42—6112no—
323C42—6112no—
324C42—6112no—
325C42—12212no—
326C42—1226no—
327C42#1175822214yescomplemented
328C42D4×D61112yes—
329C23—212no—
330C23—212no—
331C23—24no—
332C23—616no—
333C23—1223no—
334C23—1221no—
335C23—12212no—
336C23—12212no—
337C23—1224no—
338C23C24×D6111yescharacteristic
339Q8—24416no—
340D4—424no—
341D4—24416no—
342D4—24448no—
343D4—24416no—
344D4—24416no—
345D4—24424no—
346C2×C4—26no—
347C2×C4—212no—
348C2×C4—416no—
349C2×C4—616no—
350C2×C4—616no—
351C2×C4—6112no—
352C2×C4—6116no—
353C2×C4—616no—
354C2×C4—12224no—
355C2×C4—1223no—
356C2×C4—1223no—
357C2×C4—12224no—
358C2×C4—12212no—
359C2×C4—24416no—
360C2×C4C2×D4×D6116yes—
361C2×C4C2×D4×D6116yes—
362C2×C4C2×D4×D6116yes—
363C2×C4#41×D6116yes—
364C2×C4C2×#1175822216yes—
365C22—44no—
366C22—412no—
367C22—412no—
368C22—614no—
369C22—1226no—
370C22—1226no—
371C22—2444no—
372C22—24412no—
373C22—24412no—
374C22C22×D4×D6113yes—
375C22C2×#41×D6113yes—
376C22C22×#1175822211yescharacteristic
377C4—26no—
378C4—44no—
379C4—614no—
380C4—6112no—
381C4—1226no—
382C4—2444no—
383C4#127×D6114yes—
384C4#16840309112yescomplemented
385C2—84no—
386C2—1224no—
387C2—4884no—
388C2C2×#127×D6111yescharacteristic
389C2#532×D6113yes—
390C2C2×#1684030913yes—
subgroupG/H
centerC22×C4C23×D61
commutator, socle, cosocleC23×C61C25
FrattiniC23C24×D61
FittingC4×#127×C61C2
hypercenterC4×#127D61
hypocenterC61#36644
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C4×#127

Complements

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

NH ≅ G/Ntype
1. C2×#16840309C2semidirect
2. #21454216C2semidirect
3. #21454213C2semidirect
4. #21454173C2semidirect
5. #21454165C2semidirect
6. #21454163C2semidirect
7. #21454161C2semidirect
8. C4×#14519774C2semidirect
9. C4×#14519773C2semidirect
10. C43×D61C2semidirect
11. C4×#127×C61C2semidirect
75. C42×D61D4semidirect
121. C42×C61#41semidirect
132. C4×Dic61#41semidirect
167. C4×D61#127semidirect
185. C4×C61#532semidirect
189. Dic61#532semidirect
199. C4×#127D61semidirect
236. C61#36644semidirect
327. C42#11758222semidirect
384. C4#16840309semidirect

Maximal subgroups

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

subgroupindex
C2×#168403092
#214542162
#214542132
#214541732
#214541652
#214541632
#214541612
C4×#145197742
C4×#145197732
C43×D612
C4×#127×C612
#3664461