Finite Group Databasebeta

← #30600151

Subgroups — #30600151

Order 15616.

726 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×#92420C211yesmaximal, characteristic
2#21461673C211yesmaximal, characteristic
3C2×#16840612C212yesmaximal, complemented
4#21460655C212yesmaximal, complemented
5#21448347C211yesmaximal, characteristic
6C23×#92420—21no—
7C2×C4⋊#92420C2212yes—
8C2×C22⋊#92420—21no—
9C2×C4×#92420—21no—
10C2×C4×#92420—24no—
11C2×C4×#92420C2212yes—
12#16840628—21no—
13#16840612—24no—
14C2×#14519756C411yescharacteristic
15C2×#14519742C2211yescharacteristic
16C22×C4×Dic61C2211yescharacteristic
17#16839667C411yescharacteristic
18#16839658C411yescharacteristic
19#16839644—21no—
20#16839634—21no—
21C4⋊C4×Dic61C2211yescharacteristic
22#16839613—21no—
23C22⋊C4×Dic61—21no—
24C61×#380C411yescharacteristic
25C22×#92420—24no—
26C22×#92420—24no—
27C22×#92420—24no—
28C22×#92420—22no—
29C22×#92420—22no—
30C4⋊#92420—22no—
31C4⋊#92420—22no—
32C22⋊#92420—42no—
33C4×#92420—24no—
34C4×#92420—22no—
35C4×#92420D414yes—
36C22×Dic122C2×C411yescharacteristic
37C23×Dic61D411yescharacteristic
38C23×Dic61D411yescharacteristic
39#14519756—22no—
40#14519756C814yescomplemented
41#14519742—22no—
42#14519742—22no—
43C2×C4⋊Dic61C2×C411yescharacteristic
44C2×#11758196—21no—
45C2×#11758196—21no—
46C2×#11758196—21no—
47C2×C22⋊Dic61—21no—
48C2×C4×Dic61—22no—
49C2×C4×Dic61—22no—
50C2×C4×Dic61—21no—
51C2×C4×Dic61—21no—
52C2×C4×Dic61C2×C411yescharacteristic
53C2×C4×Dic61D411yescharacteristic
54C2×C4×Dic61D411yescharacteristic
55C2×C4×Dic61C2311yescharacteristic
56#14519684—21no—
57#14519684—21no—
58#14519684—21no—
59#14519684—21no—
60C23×C4×C61C2×C411yescharacteristic
61C2×C4⋊C4×C61C2×C411yescharacteristic
62C2×C4⋊C4×C61C2×C411yescharacteristic
63C2×C22⋊C4×C61—21no—
64#95×C61—21no—
65C24×C61C22⋊C411yescharacteristic
66C22×C4×C61—22no—
67C22×C4×C61—22no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61C22⋊C411yescharacteristic
71C22×C4×C61C2×C812yes—
72C22×C4×C61M1612yes—
73C22×C4×C61C22×C411yescharacteristic
74C22×Dic61—22no—
75C22×Dic61—22no—
76C22×Dic61—21no—
77C22×Dic61—22no—
78C22×Dic61—21no—
79C22×Dic61—22no—
80C22×Dic61—21no—
81C22×Dic61—21no—
82C22×Dic61C22⋊C411yescharacteristic
83C22×Dic61C22⋊C411yescharacteristic
84C22×Dic61SD1612yes—
85C22×Dic61Q1612yes—
86C22×Dic61C2×D411yescharacteristic
87C22×Dic61C2×D411yescharacteristic
88C2×Dic122—22no—
89C2×Dic122—22no—
90C2×Dic122—21no—
91C2×Dic122C2×C812yescomplemented
92C2×#92420—216no—
93C2×#92420—24no—
94C2×#92420—24no—
95C2×#92420—24no—
96C2×#92420—42no—
97C2×#92420—42no—
98C2×#92420—42no—
99C4⋊C4×C61—22no—
100C4⋊C4×C61—22no—
101C22⋊C4×C61—42no—
102C22⋊Dic61—24no—
103C4⋊Dic61—21no—
104C4⋊Dic61C2×C812yescomplemented
105#11758196—24no—
106#11758196—42no—
107#11758196—41no—
108#11758196—41no—
109C4×Dic61—22no—
110C4×Dic61—21no—
111C4×Dic61—22no—
112C4×Dic61—21no—
113C4×Dic61—42no—
114C4×Dic61—42no—
115C4×Dic61C2×D412yes—
116C4×Dic61#4112yes—
117C23×C61—21no—
118C23×C61—21no—
119C23×C61—21no—
120C23×C61—21no—
121C23×C61C22⋊C812yes—
122C23×C61#10312yes—
123C23×C61#10412yes—
124C23×C61C2×C22⋊C411yescharacteristic
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—24no—
128C2×C4×C61—24no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61C22⋊C812yes—
136C2×C4×C61C4×D412yes—
137C2×C4×C61C22×C811yescharacteristic
138C2×C4×C61C2×M1611yescharacteristic
139C2×C4×C61#13112yes—
140C2×Dic61—24no—
141C2×Dic61—24no—
142C2×Dic61—21no—
143C2×Dic61—21no—
144C2×Dic61—21no—
145C2×Dic61—21no—
146C2×Dic61—21no—
147C2×Dic61—21no—
148C2×Dic61—24no—
149C2×Dic61—24no—
150C2×Dic61—22no—
151C2×Dic61—22no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61#10312yes—
157C2×Dic61#10412yes—
158C2×Dic61#12011yescharacteristic
159C2×Dic61#12112yes—
160C2×Dic61#12311yescharacteristic
161C2×Dic61C2×SD1611yescharacteristic
162C2×Dic61C2×Q1611yescharacteristic
163C2×Dic61#13512yes—
164Dic122—24no—
165Dic122—42no—
166#92420—28no—
167#92420—82no—
168#9071—611nomaximal, Sylow
169C22×C61—22no—
170C22×C61—22no—
171C22×C61—22no—
172C22×C61—22no—
173C22×C61—22no—
174C22×C61—22no—
175C22×C61Q8⋊C814yescomplemented
176C22×C61#38611yescharacteristic
177C22×C61C2×C22⋊C811yescharacteristic
178C22×C61#40811yescharacteristic
179C22×C61C2×#10311yescharacteristic
180C22×C61#41611yescharacteristic
181C22×C61C2×#10411yescharacteristic
182C22×C61C2×#10411yescharacteristic
183C4×C61—22no—
184C4×C61—81no—
185C4×C61—81no—
186C4×C61C8×D412yes—
187C4×C61#43512yes—
188Dic61—22no—
189Dic61—21no—
190Dic61—22no—
191Dic61—21no—
192Dic61—42no—
193Dic61—42no—
194Dic61#46512yes—
195Dic61#46712yes—
196C22×C4×C8—611no—
197C4×C22⋊Q8—611no—
198#1797—611no—
199C2×Q8⋊C8—612no—
200#1279—612no—
201C2×C61—24no—
202C2×C61C2×Q8⋊C811yescharacteristic
203C2×C61#147611yescharacteristic
204C2×C61#149311yescharacteristic
205C2×C61#149411yescharacteristic
206C2×C61#186911yescharacteristic
207C2×C61#189211yescharacteristic
208C2×C61#189611yescharacteristic
209C23×C8—1221no—
210C2×C22⋊Q8—611no—
211C2×C4×Q8—611no—
212C22×C42—611no—
213C2×C4⋊C8—612no—
214C2×C22⋊C8—1221no—
215C2×C4×C8—612no—
216C2×C4×C8—1224no—
217C2×C4×C8—1221no—
218#391—611no—
219#389—611no—
220#387—1221no—
221#385—1221no—
222#380#692311yescharacteristic
223C4×C4⋊C4—611no—
224C4×C4⋊C4—1221no—
225C4×C22⋊C4—1221no—
226#336—1221no—
227Q8⋊C8—1224no—
228C61#907111yescharacteristic, complemented, Sylow
229C22×Q8—611no—
230C23×C4—611no—
231C23×C4—611no—
232C23×C4C2×#692311yescharacteristic
233C22×C8—1222no—
234C22×C8—1224no—
235C22×C8—1224no—
236C22×C8—1224no—
237C22×C8—1222no—
238C22⋊Q8—614no—
239C22⋊Q8—1222no—
240C4×Q8—1222no—
241C4×Q8—1222no—
242C2×C4⋊C4—611no—
243C2×C4⋊C4—1221no—
244C2×C4⋊C4—1221no—
245C2×C4⋊C4—1221no—
246C2×C4⋊C4C2×#692311yescharacteristic
247C2×C4⋊C4C2×#692311yescharacteristic
248C2×C22⋊C4—21no—
249C2×C22⋊C4—1221no—
250C2×C42—611no—
251C2×C42—611no—
252C2×C42—611no—
253C2×C42—611no—
254C2×C42—1222no—
255C2×C42—1222no—
256C2×C42—1221no—
257C2×C42—1221no—
258C4⋊C8—1222no—
259C4⋊C8—1222no—
260C22⋊C8—2442no—
261C4×C8—614no—
262C4×C8—1222no—
263C4×C8—1224no—
264#95—21no—
265#95—1221no—
266#95—1221no—
267#95—1221no—
268#95—1221no—
269C24C22⋊#692311yescharacteristic
270C2×Q8—612no—
271C2×Q8—1221no—
272C2×Q8—1222no—
273C2×Q8—1222no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—611no—
279C22×C4—612no—
280C22×C4—612no—
281C22×C4—611no—
282C22×C4—611no—
283C22×C4—611no—
284C22×C4—1221no—
285C22×C4—1222no—
286C22×C4—1221no—
287C22×C4—1222no—
288C22×C4—1222no—
289C22×C4—1222no—
290C22×C4—1221no—
291C22×C4—1221no—
292C22×C4D61⋊C812yes—
293C22×C4#1175821312yes—
294C22×C4C22⋊#692311yescharacteristic
295C22×C4C22×#692311yescharacteristic
296C2×C8—1224no—
297C2×C8—12216no—
298C2×C8—1224no—
299C2×C8—1224no—
300C2×C8—2442no—
301C2×C8—2442no—
302C2×C8—2442no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—612no—
306C4⋊C4—1224no—
307C4⋊C4—1221no—
308C4⋊C4—2442no—
309C4⋊C4—2441no—
310C4⋊C4—2441no—
311C22⋊C4—42no—
312C22⋊C4—1224no—
313C42—612no—
314C42—612no—
315C42—1221no—
316C42—1222no—
317C42—1221no—
318C42—1222no—
319C42—2442no—
320C42—2442no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23#1451983412yes—
326C23#1451984012yes—
327C23#1451984112yes—
328C23C2×C22⋊#692311yescharacteristic
329Q8—1224no—
330Q8—2442no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—612no—
342C2×C4—611no—
343C2×C4—611no—
344C2×C4—612no—
345C2×C4—611no—
346C2×C4—611no—
347C2×C4—612no—
348C2×C4—612no—
349C2×C4—1224no—
350C2×C4—1224no—
351C2×C4—1224no—
352C2×C4—1224no—
353C2×C4—1221no—
354C2×C4—1221no—
355C2×C4—1221no—
356C2×C4—1221no—
357C2×C4—1221no—
358C2×C4—1221no—
359C2×C4—1222no—
360C2×C4—1222no—
361C2×C4—2441no—
362C2×C4—2441no—
363C2×C4—2441no—
364C2×C4—2441no—
365C2×C4#1451983412yes—
366C2×C4D4×#692312yes—
367C2×C4C2×D61⋊C811yescharacteristic
368C2×C4C2×#1175821311yescharacteristic
369C2×C4#1451986412yes—
370C8—1228no—
371C8—4882no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22#1684061214yescomplemented
379C22#1684071011yescharacteristic
380C22C2×#1451983411yescharacteristic
381C22#1684073511yescharacteristic
382C22C2×#1451984011yescharacteristic
383C22#1684075211yescharacteristic
384C22C2×#1451984111yescharacteristic
385C22D61⋊#10411yescharacteristic
386C4—22no—
387C4—81no—
388C4—81no—
389C4—612no—
390C4—612no—
391C4—1222no—
392C4—1221no—
393C4—1222no—
394C4—1221no—
395C4—2442no—
396C4—2442no—
397C4#1684078912yes—
398C4#1684079112yes—
399C2—24no—
400C2C2×#1684061211yescharacteristic
401C2#2146111211yescharacteristic
402C2#2146116711yescharacteristic
403C2#2146117011yescharacteristic
404C2#2146179411yescharacteristic
405C2#2146183211yescharacteristic
406C2#2146183711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
FittingC61×#380C4
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenter#380#6923
hypocenterC61#9071
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. C2×#16840612C2semidirect
4. #21460655C2semidirect
40. #14519756C8semidirect
91. C2×Dic122C2×C8semidirect
104. C4⋊Dic61C2×C8semidirect
175. C22×C61Q8⋊C8semidirect
228. C61#9071semidirect
378. C22#16840612semidirect

Maximal subgroups

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

subgroupindex
C22×C4×#924202
#214616732
C2×#168406122
#214606552
#214483472
#907161