Finite Group Databasebeta

← #30601761

Subgroups — #30601761

Order 15616.

630 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 104 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
1#21461546C211yesmaximal, characteristic
2C2×#16840616C212yesmaximal
3C2×#16840615C212yesmaximal
4#21460995C211yesmaximal, characteristic
5#21454168C211yesmaximal, characteristic
6C2×#14519834C2212yes—
7C2×#14519832—21no—
8C2×(C2×#6923)⋊C4—21no—
9C2×(C2×#6923)⋊C4—21no—
10C2×(C2×#6923)⋊C4C2212yes—
11#16840628—21no—
12#16840616—24no—
13#16840615—24no—
14C2×#14519775C411yescharacteristic, complemented
15C2×#14519757C2211yescharacteristic
16C2×#14519753C2211yescharacteristic
17C2×C42×D61C2211yescharacteristic
18#16839668C411yescharacteristic, complemented
19#16839659—21no—
20#16839659—21no—
21#16839631C411yescharacteristic, complemented
22#16839621—21no—
23#16839621—21no—
24C61×#391C411yescharacteristic, complemented
25C23×#6923—22no—
26C22×#92420—22no—
27#14519834—22no—
28#14519834D414yes—
29#14519832—42no—
30(C2×#6923)⋊C4—24no—
31(C2×#6923)⋊C4—22no—
32(C2×#6923)⋊C4—22no—
33(C2×#6923)⋊C4—42no—
34C22×Dic122C2×C411yescharacteristic
35C22×C4×D61D411yescharacteristic
36C22×C4×D61D411yescharacteristic
37C22×C4×D61C2311yescharacteristic
38#14519775—42no—
39#14519757—22no—
40#14519757—22no—
41#14519753—22no—
42#14519753—22no—
43C2×C4⋊Dic61—21no—
44C2×C4⋊Dic61C2×C411yescharacteristic
45C2×#11758196—21no—
46C2×#11758196—21no—
47C2×#11758196—21no—
48C2×#11758196—21no—
49C2×#11758196—21no—
50C2×#11758198—21no—
51C2×#11758198—21no—
52C2×C4×Dic61C2×C411yescharacteristic
53C2×C4×Dic61D411yescharacteristic
54C2×C4×Dic61D411yescharacteristic
55C42×D61—42no—
56#14519684—21no—
57#14519684—21no—
58C22×Q8×C61C2×C411yescharacteristic
59C2×C4⋊C4×C61C2×C411yescharacteristic
60C2×C42×C61C2×C411yescharacteristic
61#95×C61—21no—
62#95×C61—21no—
63C23×D61C2×D411yescharacteristic
64C22×#6923—24no—
65C22×#6923—42no—
66C22×#6923—42no—
67C22×#6923—42no—
68C22×#6923—42no—
69C22×#6923—42no—
70C22×#6923—42no—
71C2×Q8×C61—22no—
72C2×Q8×C61—41no—
73C2×Q8×C61—41no—
74C22×C4×C61—21no—
75C22×C4×C61—21no—
76C22×C4×C61C22⋊C411yescharacteristic
77C22×C4×C61C22⋊C411yescharacteristic
78C22×C4×C61C22×C411yescharacteristic
79C22×Dic61—21no—
80C22×Dic61—21no—
81C22×Dic61C22⋊C411yescharacteristic
82C22×Dic61C22⋊C411yescharacteristic
83C22×Dic61C2×D411yescharacteristic
84C2×C4×D61—22no—
85C2×C4×D61—22no—
86C2×C4×D61—22no—
87C2×C4×D61—22no—
88C2×C4×D61—41no—
89C2×C4×D61—41no—
90C2×C4×D61—41no—
91C2×C4×D61—41no—
92C2×C4×D61C2×D412yes—
93C2×C4×D61#4112yes—
94C2×Dic122—22no—
95C2×Dic122—41no—
96C2×Dic122—41no—
97C2×#92420—24no—
98C2×#92420—42no—
99C2×#92420—42no—
100C4⋊C4×C61—22no—
101C42×C61—41no—
102#11758198—41no—
103#11758198—41no—
104#11758198—41no—
105#11758198—41no—
106C4⋊Dic61—22no—
107C4⋊Dic61—42no—
108#11758196—42no—
109#11758196—42no—
110#11758196—42no—
111#11758196—42no—
112#11758196—42no—
113C4×Dic61—22no—
114C4×Dic61—22no—
115C4×Dic61—41no—
116C23×C61C2×C22⋊C411yescharacteristic
117C22×D61—21no—
118C22×D61—21no—
119C22×D61—21no—
120C22×D61—21no—
121C22×D61#12411yescharacteristic
122C22×D61#12711yescharacteristic
123C22×D61#13712yes—
124C22×D61#13712yes—
125C2×#6923—44no—
126C2×#6923—44no—
127C2×#6923—44no—
128C2×#6923—82no—
129C2×#6923—82no—
130C2×#6923—82no—
131C2×#6923—82no—
132Q8×C61—82no—
133C2×C4×C61—21no—
134C2×C4×C61—21no—
135C2×C4×C61—21no—
136C2×C4×C61—21no—
137C2×C4×C61—21no—
138C2×C4×C61—21no—
139C2×C4×C61—21no—
140C2×C4×C61—21no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61C4×D412yes—
148C2×Dic61—21no—
149C2×Dic61—21no—
150C2×Dic61—21no—
151C2×Dic61—21no—
152C2×Dic61—21no—
153C2×Dic61—21no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61#9912yes—
161C2×Dic61#10412yes—
162C2×Dic61#12112yes—
163C4×D61—42no—
164C4×D61—42no—
165C4×D61—44no—
166C4×D61—44no—
167Dic122—82no—
168#92420—44no—
169#9683—611nomaximal, Sylow
170C22×C61#39011yescharacteristic
171C22×C61C2×#9911yescharacteristic
172C22×C61#41011yescharacteristic
173C22×C61#41911yescharacteristic
174C22×C61#41911yescharacteristic
175C22×C61C2×#10411yescharacteristic
176C22×C61#42111yescharacteristic
177C2×D61—21no—
178C2×D61—21no—
179C2×D61—21no—
180C2×D61—21no—
181C2×D61—22no—
182C2×D61—22no—
183C2×D61—22no—
184C2×D61—22no—
185C2×D61#49112yes—
186C2×D61#49712yes—
187#6923—84no—
188C4×C61—22no—
189C4×C61—41no—
190C4×C61—41no—
191C4×C61—81no—
192C4×C61—81no—
193Dic61—22no—
194Dic61—22no—
195Dic61—22no—
196Dic61—81no—
197Dic61—81no—
198#2857—611no—
199#1749—611no—
200C2×#328—612no—
201C2×#328—612no—
202#1434—611no—
203C2×C61#150611yescharacteristic
204C2×C61#150811yescharacteristic
205C2×C61#151111yescharacteristic
206C2×C61#151511yescharacteristic
207C2×C61#196011yescharacteristic
208C2×C61#196411yescharacteristic
209C2×C61#198011yescharacteristic
210D61—22no—
211D61—22no—
212C2×C4⋊Q8—611no—
213C2×C22⋊Q8—611no—
214C2×C22⋊Q8—611no—
215C22×C42—611no—
216C2×C22⋊C8—612no—
217C2×#97—1221no—
218#391—611no—
219#391#692311yescharacteristic, complemented
220#389—1221no—
221#389—1221no—
222#384—611no—
223#380—1221no—
224#380—1221no—
225C2×#95—612no—
226C2×#95—1221no—
227C2×#95—1221no—
228#336—1221no—
229#328—1224no—
230#328—1224no—
231C61#968311yescharacteristic, complemented, Sylow
232C22×Q8—611no—
233C22×Q8C2×#692311yescharacteristic
234C23×C4—611no—
235C23×C4—611no—
236C23×C4—611no—
237C23×C4—1222no—
238C22×C8—1222no—
239C4⋊Q8—2442no—
240C22⋊Q8—1222no—
241C22⋊Q8—1222no—
242C22⋊Q8—1222no—
243C22⋊Q8—1222no—
244C2×C4⋊C4—611no—
245C2×C4⋊C4—1221no—
246C2×C4⋊C4—1221no—
247C2×C4⋊C4—1221no—
248C2×C4⋊C4—1221no—
249C2×C4⋊C4—1221no—
250C2×C4⋊C4—1221no—
251C2×C4⋊C4C2×#692311yescharacteristic
252C2×C22⋊C4—1221no—
253C2×C22⋊C4—1221no—
254C2×C42—611no—
255C2×C42—611no—
256C2×C42—611no—
257C2×C42—2442no—
258C2×C42C2×#692311yescharacteristic
259C22⋊C8—614no—
260C22⋊C8—1222no—
261#97—2442no—
262#95—21no—
263#95—21no—
264#95—1222no—
265#95—1222no—
266#95—1224no—
267#95—1221no—
268#95—1221no—
269#95—2442no—
270C24—611no—
271C2×Q8—22no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—1222no—
275C2×Q8—2441no—
276C2×Q8—2441no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—611no—
280C22×C4—611no—
281C22×C4—611no—
282C22×C4—612no—
283C22×C4—612no—
284C22×C4—1222no—
285C22×C4—1222no—
286C22×C4—1222no—
287C22×C4—1222no—
288C22×C4—1224no—
289C22×C4—1221no—
290C22×C4—1221no—
291C22×C4—2441no—
292C22×C4—2441no—
293C22×C4—2441no—
294C22×C4—2441no—
295C22×C4—2442no—
296C22×C4—2442no—
297C22×C4—2442no—
298C22×C4—2442no—
299C22×C4—2442no—
300C22×C4—2442no—
301C22×C4C22⋊#692311yescharacteristic
302C22×C4C22⋊#692311yescharacteristic
303C22×C4C22×#692311yescharacteristic
304C2×C8—1224no—
305C2×C8—2442no—
306C2×C8—2442no—
307C4⋊C4—22no—
308C4⋊C4—1222no—
309C4⋊C4—2442no—
310C4⋊C4—2442no—
311C4⋊C4—2442no—
312C4⋊C4—2442no—
313C4⋊C4—2442no—
314C4⋊C4—2442no—
315C22⋊C4—2441no—
316C22⋊C4—2441no—
317C22⋊C4—2441no—
318C22⋊C4—2441no—
319C42—41no—
320C42—1222no—
321C42—1222no—
322C42—2441no—
323C23—611no—
324C23—611no—
325C23—612no—
326C23—612no—
327C23—1221no—
328C23—1221no—
329C23—1221no—
330C23—1221no—
331C23C2×C22⋊#692311yescharacteristic
332Q8—82no—
333Q8—4882no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—612no—
349C2×C4—612no—
350C2×C4—612no—
351C2×C4—1221no—
352C2×C4—1221no—
353C2×C4—1221no—
354C2×C4—1221no—
355C2×C4—1221no—
356C2×C4—1221no—
357C2×C4—4882no—
358C2×C4—4882no—
359C2×C4—4882no—
360C2×C4—4882no—
361C2×C4—2442no—
362C2×C4—2442no—
363C2×C4—2444no—
364C2×C4—2444no—
365C2×C4—2444no—
366C2×C4—2444no—
367C2×C4—2444no—
368C2×C4—2441no—
369C2×C4—2441no—
370C2×C4—2441no—
371C2×C4—2441no—
372C2×C4—2441no—
373C2×C4—2441no—
374C2×C4D4×#692312yes—
375C8—2444no—
376C22—612no—
377C22—612no—
378C22—1221no—
379C22—1221no—
380C22—1222no—
381C22—1221no—
382C22—1221no—
383C22—1222no—
384C22—1222no—
385C22—1222no—
386C22#1684071411yescharacteristic
387C22D61⋊#9911yescharacteristic
388C22#1684074111yescharacteristic
389C22#1684075511yescharacteristic
390C22#1684075511yescharacteristic
391C22D61⋊#10411yescharacteristic
392C22#1684075911yescharacteristic
393C4—22no—
394C4—41no—
395C4—41no—
396C4—81no—
397C4—81no—
398C4—1222no—
399C4—1222no—
400C4—1222no—
401C4—4884no—
402C4—4881no—
403C4—4881no—
404C2—1222no—
405C2—1222no—
406C2#2146121211yescharacteristic
407C2#2146121711yescharacteristic
408C2#2146122811yescharacteristic
409C2#2146124111yescharacteristic
410C2#2146195011yescharacteristic
411C2#2146196011yescharacteristic
412C2#2146198411yescharacteristic
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#9683
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
14. C2×#14519775C4semidirect
18. #16839668C4semidirect
21. #16839631C4semidirect
24. C61×#391C4semidirect
219. #391#6923semidirect
231. C61#9683semidirect

Maximal subgroups

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

subgroupindex
#214615462
C2×#168406162
C2×#168406152
#214609952
#214541682
#968361