Finite Group Databasebeta

← #30600330

Subgroups — #30600330

Order 15616.

612 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840785C211yesmaximal, characteristic
2C2×#16840616C212yesmaximal
3#21460995C211yesmaximal, characteristic
4#21460692C212yesmaximal
5#21454160C211yesmaximal, characteristic
6#16840785—24no—
7C2×#14519844—21no—
8C2×#14519834C2212yes—
9C2×#14519832—21no—
10C2×(C2×#6923)⋊C4—21no—
11C2×(C2×#6923)⋊C4—21no—
12C2×(C2×#6923)⋊C4C2212yes—
13#16840616—24no—
14C2×#14519753C2211yescharacteristic
15C2×C42×D61C2211yescharacteristic
16#16839669C411yescharacteristic, complemented
17#16839661C411yescharacteristic, complemented
18#16839660C411yescharacteristic, complemented
19#16839645—21no—
20#16839644—21no—
21#16839636—21no—
22#16839624—21no—
23#16839619C2211yescharacteristic
24C61×#391C411yescharacteristic, complemented
25C23×#6923—22no—
26C22×#92420—22no—
27#14519844—24no—
28#14519834—22no—
29#14519834D414yes—
30#14519832—24no—
31(C2×#6923)⋊C4—24no—
32(C2×#6923)⋊C4—22no—
33(C2×#6923)⋊C4—22no—
34(C2×#6923)⋊C4—42no—
35C22×C4×D61D411yescharacteristic
36C22×C4×D61D411yescharacteristic
37C22×C4×D61C2311yescharacteristic
38#14519753—22no—
39#14519753—22no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61C2×C411yescharacteristic
42C2×C4⋊Dic61C2×C411yescharacteristic
43C2×#11758196—21no—
44C2×#11758196—21no—
45C2×#11758198—21no—
46C2×#11758198—21no—
47C2×C4×Dic61C2×C411yescharacteristic
48C2×C4×Dic61D411yescharacteristic
49C2×C4×Dic61D411yescharacteristic
50C42×D61—22no—
51C42×D61—22no—
52#14519684—21no—
53#14519684—21no—
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57C22×Q8×C61C2×C411yescharacteristic
58C2×C4⋊C4×C61C2×C411yescharacteristic
59C2×C42×C61C2×C411yescharacteristic
60#95×C61—21no—
61#95×C61—21no—
62C23×D61C2×D411yescharacteristic
63C22×#6923—24no—
64C22×#6923—42no—
65C22×#6923—42no—
66C22×#6923—42no—
67C22×#6923—42no—
68C22×#6923—42no—
69C22×#6923—42no—
70C2×Q8×C61—22no—
71C2×Q8×C61—41no—
72C2×Q8×C61—41no—
73C22×C4×C61—21no—
74C22×C4×C61—21no—
75C22×C4×C61C22⋊C411yescharacteristic
76C22×C4×C61C22⋊C411yescharacteristic
77C22×C4×C61C22×C411yescharacteristic
78C22×Dic61—21no—
79C22×Dic61—21no—
80C22×Dic61C22⋊C411yescharacteristic
81C22×Dic61C22⋊C411yescharacteristic
82C22×Dic61C2×D411yescharacteristic
83C2×C4×D61—22no—
84C2×C4×D61—22no—
85C2×C4×D61—21no—
86C2×C4×D61—22no—
87C2×C4×D61—21no—
88C2×C4×D61—22no—
89C2×C4×D61—41no—
90C2×C4×D61—41no—
91C2×C4×D61—41no—
92C2×C4×D61C2×D412yes—
93C2×C4×D61#4112yes—
94C2×#92420—24no—
95C2×#92420—42no—
96C2×#92420—42no—
97C4⋊C4×C61—41no—
98C42×C61—22no—
99#11758198—42no—
100#11758198—41no—
101#11758198—41no—
102C4⋊Dic61—22no—
103C4⋊Dic61—42no—
104C4⋊Dic61—41no—
105#11758196—42no—
106#11758196—41no—
107#11758196—41no—
108C4×Dic61—22no—
109C4×Dic61—21no—
110C4×Dic61—21no—
111C4×Dic61—21no—
112C4×Dic61—21no—
113C23×C61C2×C22⋊C411yescharacteristic
114C22×D61—21no—
115C22×D61—21no—
116C22×D61—21no—
117C22×D61—21no—
118C22×D61#12112yes—
119C22×D61AGL(1,ℤ/8)11yescharacteristic
120C22×D61#13712yes—
121C22×D61#13711yescharacteristic
122C2×#6923—44no—
123C2×#6923—44no—
124C2×#6923—44no—
125C2×#6923—82no—
126C2×#6923—82no—
127C2×#6923—82no—
128C2×#6923—82no—
129Q8×C61—82no—
130C2×C4×C61—21no—
131C2×C4×C61—21no—
132C2×C4×C61—21no—
133C2×C4×C61—21no—
134C2×C4×C61—21no—
135C2×C4×C61—21no—
136C2×C4×C61—21no—
137C2×C4×C61—21no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61C4×D412yes—
145C2×Dic61—21no—
146C2×Dic61—21no—
147C2×Dic61—21no—
148C2×Dic61—21no—
149C2×Dic61—21no—
150C2×Dic61—21no—
151C2×Dic61—21no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158C2×Dic61#10412yes—
159C2×Dic61#12011yescharacteristic
160C2×Dic61#12311yescharacteristic
161C4×D61—42no—
162C4×D61—42no—
163C4×D61—44no—
164C4×D61—42no—
165C4×D61—42no—
166#92420—44no—
167#9116—611nomaximal, Sylow
168C22×C61#38611yescharacteristic
169C22×C61#41011yescharacteristic
170C22×C61#41011yescharacteristic
171C22×C61#41711yescharacteristic
172C22×C61#41911yescharacteristic
173C22×C61C2×#10411yescharacteristic
174C22×C61#42111yescharacteristic
175C2×D61—22no—
176C2×D61—22no—
177C2×D61—22no—
178C2×D61—21no—
179C2×D61—21no—
180C2×D61—21no—
181C2×D61—21no—
182C2×D61—22no—
183C2×D61#46812yes—
184C2×D61#47012yes—
185#6923—84no—
186C4×C61—22no—
187C4×C61—41no—
188C4×C61—41no—
189C4×C61—81no—
190C4×C61—81no—
191Dic61—22no—
192Dic61—21no—
193Dic61—21no—
194Dic61—41no—
195Dic61—81no—
196Dic61—81no—
197C2×#432—611no—
198#2855—611no—
199C2×#328—612no—
200#1434—611no—
201#1294—612no—
202C2×C61#150611yescharacteristic
203C2×C61#150911yescharacteristic
204C2×C61#151211yescharacteristic
205C2×C61#151511yescharacteristic
206C2×C61#187111yescharacteristic
207C2×C61#189411yescharacteristic
208C2×C61#189911yescharacteristic
209D61—22no—
210D61—22no—
211C2×C22⋊Q8—611no—
212C22×C42—611no—
213#432—1224no—
214C2×C4⋊C8—1221no—
215C2×C22⋊C8—612no—
216C2×#97—1221no—
217#391—611no—
218#391#692311yescharacteristic, complemented
219#389—611no—
220#389—611no—
221#387—1221no—
222#387—1221no—
223#385—1221no—
224#381—1221no—
225#380—611no—
226C2×#95—612no—
227C2×#95—1221no—
228C2×#95—1221no—
229#328—1224no—
230C61#911611yescharacteristic, complemented, Sylow
231C22×Q8C2×#692311yescharacteristic
232C23×C4—611no—
233C23×C4—611no—
234C23×C4—611no—
235C23×C4—1222no—
236C22×C8—1222no—
237C22⋊Q8—1222no—
238C22⋊Q8—1222no—
239C2×C4⋊C4—611no—
240C2×C4⋊C4—611no—
241C2×C4⋊C4—1221no—
242C2×C4⋊C4—1221no—
243C2×C4⋊C4—1221no—
244C2×C4⋊C4C2×#692311yescharacteristic
245C2×C22⋊C4—1221no—
246C2×C22⋊C4—1221no—
247C2×C42—611no—
248C2×C42—611no—
249C2×C42—611no—
250C2×C42—1222no—
251C2×C42—1222no—
252C2×C42C2×#692311yescharacteristic
253C4⋊C8—1224no—
254C22⋊C8—614no—
255C22⋊C8—1222no—
256#97—1224no—
257#95—21no—
258#95—21no—
259#95—1221no—
260#95—1221no—
261#95—1221no—
262#95—1221no—
263#95—1222no—
264#95—1222no—
265#95—1224no—
266#95—1221no—
267#95—2442no—
268C24—611no—
269C2×Q8—22no—
270C2×Q8—41no—
271C2×Q8—41no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—611no—
275C22×C4—611no—
276C22×C4—612no—
277C22×C4—612no—
278C22×C4—611no—
279C22×C4—1222no—
280C22×C4—1221no—
281C22×C4—1222no—
282C22×C4—1222no—
283C22×C4—1221no—
284C22×C4—1222no—
285C22×C4—1224no—
286C22×C4—1221no—
287C22×C4—1221no—
288C22×C4—2441no—
289C22×C4—2441no—
290C22×C4—2441no—
291C22×C4—2442no—
292C22×C4—2442no—
293C22×C4—2442no—
294C22×C4—2442no—
295C22×C4—2442no—
296C22×C4—2442no—
297C22×C4C22⋊#692311yescharacteristic
298C22×C4C22⋊#692311yescharacteristic
299C22×C4C22×#692311yescharacteristic
300C2×C8—1224no—
301C2×C8—2442no—
302C2×C8—2442no—
303C4⋊C4—41no—
304C4⋊C4—1222no—
305C4⋊C4—2441no—
306C4⋊C4—2442no—
307C4⋊C4—2441no—
308C4⋊C4—2441no—
309C4⋊C4—2442no—
310C22⋊C4—2441no—
311C22⋊C4—2441no—
312C22⋊C4—2442no—
313C42—22no—
314C42—1221no—
315C42—1221no—
316C42—1221no—
317C42—1221no—
318C42—1222no—
319C23—612no—
320C23—611no—
321C23—611no—
322C23—612no—
323C23—1221no—
324C23—1221no—
325C23—1221no—
326C23—1221no—
327C23C2×C22⋊#692311yescharacteristic
328Q8—82no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—611no—
344C2×C4—611no—
345C2×C4—612no—
346C2×C4—1221no—
347C2×C4—1221no—
348C2×C4—1221no—
349C2×C4—1221no—
350C2×C4—1221no—
351C2×C4—1221no—
352C2×C4—1221no—
353C2×C4—4882no—
354C2×C4—4882no—
355C2×C4—4882no—
356C2×C4—4882no—
357C2×C4—2442no—
358C2×C4—2442no—
359C2×C4—2444no—
360C2×C4—2442no—
361C2×C4—2442no—
362C2×C4—2444no—
363C2×C4—2444no—
364C2×C4—2444no—
365C2×C4—2441no—
366C2×C4—2441no—
367C2×C4—2441no—
368C2×C4—2441no—
369C2×C4—2441no—
370C2×C4—2441no—
371C2×C4D4×#692312yes—
372C8—2444no—
373C22—612no—
374C22—612no—
375C22—1222no—
376C22—1222no—
377C22—1222no—
378C22—1221no—
379C22—1221no—
380C22—1221no—
381C22—1221no—
382C22—1222no—
383C22#1684071011yescharacteristic
384C22#1684074111yescharacteristic
385C22#1684074211yescharacteristic
386C22#1684075311yescharacteristic
387C22#1684075511yescharacteristic
388C22D61⋊#10411yescharacteristic
389C22#1684076111yescharacteristic
390C4—22no—
391C4—41no—
392C4—41no—
393C4—81no—
394C4—81no—
395C4—1222no—
396C4—1221no—
397C4—1221no—
398C4—4884no—
399C4—4881no—
400C4—4881no—
401C4—2441no—
402C2—1222no—
403C2—1222no—
404C2#2146121211yescharacteristic
405C2#2146122011yescharacteristic
406C2#2146123211yescharacteristic
407C2#2146124311yescharacteristic
408C2#2146180111yescharacteristic
409C2#2146183411yescharacteristic
410C2#2146184211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#391C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenter#391#6923
hypocenterC61#9116
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
16. #16839669C4semidirect
17. #16839661C4semidirect
18. #16839660C4semidirect
24. C61×#391C4semidirect
218. #391#6923semidirect
230. C61#9116semidirect

Maximal subgroups

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

subgroupindex
C2×#168407852
C2×#168406162
#214609952
#214606922
#214541602
#911661