Finite Group Databasebeta

← #30601816

Subgroups — #30601816

Order 15616.

624 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 112 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#21461674C211yesmaximal, characteristic
2C2×#16840616C212yesmaximal
3C2×#16840615C212yesmaximal
4#21460995C211yesmaximal, characteristic
5#21454168C211yesmaximal, characteristic
6C2×C4⋊#92420—21no—
7C2×#14519834C2212yes—
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—
27C4⋊#92420—42no—
28#14519834—22no—
29#14519834—22no—
30(C2×#6923)⋊C4—22no—
31(C2×#6923)⋊C4—42no—
32(C2×#6923)⋊C4—42no—
33(C2×#6923)⋊C4D414yes—
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—41no—
87C2×C4×D61—41no—
88C2×C4×D61—41no—
89C2×C4×D61—41no—
90C2×C4×D61—41no—
91C2×C4×D61SD1612yes—
92C2×C4×D61Q1612yes—
93C2×C4×D61C2×D412yes—
94C2×C4×D61#4112yes—
95C2×Dic122—22no—
96C2×Dic122—41no—
97C2×Dic122—41no—
98C2×#92420—42no—
99C2×#92420—42no—
100C2×#92420—42no—
101C4⋊C4×C61—22no—
102C42×C61—41no—
103#11758198—42no—
104#11758198—42no—
105C4⋊Dic61—22no—
106C4⋊Dic61—41no—
107C4⋊Dic61—41no—
108#11758196—42no—
109#11758196—42no—
110#11758196—42no—
111#11758196—42no—
112#11758196—41no—
113#11758196—41no—
114C4×Dic61—22no—
115C4×Dic61—22no—
116C4×Dic61—41no—
117C23×C61C2×C22⋊C411yescharacteristic
118C22×D61—21no—
119C22×D61—21no—
120C22×D61—21no—
121C22×D61—21no—
122C22×D61#12112yes—
123C22×D61C2×SD1611yescharacteristic
124C22×D61C2×Q1611yescharacteristic
125C22×D61#13712yes—
126C2×#6923—44no—
127C2×#6923—44no—
128C2×#6923—44no—
129C2×#6923—82no—
130C2×#6923—82no—
131C2×#6923—82no—
132C2×#6923—82no—
133Q8×C61—82no—
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×C61#10312yes—
148C2×C4×C61C4×D412yes—
149C2×Dic61—21no—
150C2×Dic61—21no—
151C2×Dic61—21no—
152C2×Dic61—21no—
153C2×Dic61—21no—
154C2×Dic61—21no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61—41no—
161C2×Dic61#9912yes—
162C2×Dic61#10312yes—
163C2×Dic61#12411yescharacteristic
164C2×Dic61#12711yescharacteristic
165C4×D61—42no—
166C4×D61—42no—
167C4×D61—42no—
168C4×D61—42no—
169C4×D61—44no—
170Dic122—82no—
171#92420—82no—
172#9714—611nomaximal, Sylow
173C22×C61#39011yescharacteristic
174C22×C61C2×#9911yescharacteristic
175C22×C61#41011yescharacteristic
176C22×C61C2×#10311yescharacteristic
177C22×C61#41911yescharacteristic
178C22×C61#42111yescharacteristic
179C22×C61#42111yescharacteristic
180C2×D61—22no—
181C2×D61—22no—
182C2×D61—22no—
183C2×D61—22no—
184C2×D61—21no—
185C2×D61—21no—
186C2×D61—21no—
187C2×D61—21no—
188C2×D61#46112yes—
189C2×D61#46212yes—
190#6923—84no—
191C4×C61—22no—
192C4×C61—22no—
193C4×C61—41no—
194C4×C61—81no—
195C4×C61—81no—
196Dic61—21no—
197Dic61—21no—
198Dic61—22no—
199Dic61—22no—
200Dic61—81no—
201Dic61—81no—
202#2857—611no—
203#1797—611no—
204C2×#328—612no—
205C2×#328—612no—
206#1434—611no—
207C2×C61#151111yescharacteristic
208C2×C61#151511yescharacteristic
209C2×C61#151811yescharacteristic
210C2×C61#152011yescharacteristic
211C2×C61#196311yescharacteristic
212C2×C61#196411yescharacteristic
213C2×C61#198111yescharacteristic
214D61—22no—
215D61—22no—
216C2×C4⋊Q8—611no—
217C2×C22⋊Q8—611no—
218C2×C22⋊Q8—611no—
219C22×C42—611no—
220C2×C4⋊C8—1221no—
221C2×C22⋊C8—612no—
222#391—611no—
223#391#692311yescharacteristic, complemented
224#389—1221no—
225#389—1221no—
226#384—611no—
227#380—1221no—
228#380—1221no—
229C2×#95—612no—
230C2×#95—1221no—
231C2×#95—1221no—
232#336—1221no—
233#328—1224no—
234#328—1224no—
235C61#971411yescharacteristic, complemented, Sylow
236C22×Q8—611no—
237C22×Q8C2×#692311yescharacteristic
238C23×C4—611no—
239C23×C4—611no—
240C23×C4—611no—
241C23×C4—1222no—
242C22×C8—1222no—
243C4⋊Q8—2442no—
244C22⋊Q8—1222no—
245C22⋊Q8—1222no—
246C22⋊Q8—1222no—
247C22⋊Q8—1222no—
248C2×C4⋊C4—611no—
249C2×C4⋊C4—1221no—
250C2×C4⋊C4—1221no—
251C2×C4⋊C4—1221no—
252C2×C4⋊C4—1221no—
253C2×C4⋊C4—1221no—
254C2×C4⋊C4—1221no—
255C2×C4⋊C4C2×#692311yescharacteristic
256C2×C22⋊C4—1221no—
257C2×C22⋊C4—1221no—
258C2×C42—611no—
259C2×C42—611no—
260C2×C42—611no—
261C2×C42—2442no—
262C2×C42C2×#692311yescharacteristic
263C4⋊C8—2442no—
264C22⋊C8—1222no—
265C22⋊C8—1222no—
266#95—21no—
267#95—21no—
268#95—614no—
269#95—1222no—
270#95—1221no—
271#95—1221no—
272#95—2442no—
273#95—2442no—
274C24—611no—
275C2×Q8—22no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—1222no—
279C2×Q8—2441no—
280C2×Q8—2441no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—611no—
284C22×C4—611no—
285C22×C4—611no—
286C22×C4—612no—
287C22×C4—612no—
288C22×C4—612no—
289C22×C4—612no—
290C22×C4—1222no—
291C22×C4—1222no—
292C22×C4—1224no—
293C22×C4—1221no—
294C22×C4—1221no—
295C22×C4—2441no—
296C22×C4—2441no—
297C22×C4—2441no—
298C22×C4—2441no—
299C22×C4—2441no—
300C22×C4—2442no—
301C22×C4—2442no—
302C22×C4—2442no—
303C22×C4—2442no—
304C22×C4—2442no—
305C22×C4—2442no—
306C22×C4C22⋊#692311yescharacteristic
307C22×C4C22⋊#692311yescharacteristic
308C22×C4C22×#692311yescharacteristic
309C2×C8—2442no—
310C2×C8—2442no—
311C2×C8—2442no—
312C4⋊C4—22no—
313C4⋊C4—1222no—
314C4⋊C4—2441no—
315C4⋊C4—2441no—
316C4⋊C4—2442no—
317C4⋊C4—2441no—
318C4⋊C4—2441no—
319C4⋊C4—2442no—
320C4⋊C4—2442no—
321C4⋊C4—2442no—
322C22⋊C4—2442no—
323C22⋊C4—2442no—
324C42—41no—
325C42—1222no—
326C42—1222no—
327C42—2441no—
328C23—612no—
329C23—611no—
330C23—611no—
331C23—612no—
332C23—1221no—
333C23—1221no—
334C23—1221no—
335C23—1221no—
336C23C2×C22⋊#692311yescharacteristic
337Q8—82no—
338Q8—4882no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—611no—
353C2×C4—611no—
354C2×C4—612no—
355C2×C4—612no—
356C2×C4—1221no—
357C2×C4—1221no—
358C2×C4—1221no—
359C2×C4—1221no—
360C2×C4—1221no—
361C2×C4—1221no—
362C2×C4—4882no—
363C2×C4—4882no—
364C2×C4—4882no—
365C2×C4—4882no—
366C2×C4—2442no—
367C2×C4—2442no—
368C2×C4—2442no—
369C2×C4—2442no—
370C2×C4—2444no—
371C2×C4—2444no—
372C2×C4—2444no—
373C2×C4—2444no—
374C2×C4—2441no—
375C2×C4—2441no—
376C2×C4—2441no—
377C2×C4—2441no—
378C2×C4—2441no—
379C2×C4—2441no—
380C2×C4#1451984012yes—
381C2×C4D4×#692312yes—
382C8—4882no—
383C22—612no—
384C22—612no—
385C22—1221no—
386C22—1221no—
387C22—1221no—
388C22—1221no—
389C22—1222no—
390C22—1222no—
391C22—1222no—
392C22—1222no—
393C22#1684071411yescharacteristic
394C22D61⋊#9911yescharacteristic
395C22#1684074111yescharacteristic
396C22C2×#1451984011yescharacteristic
397C22#1684075511yescharacteristic
398C22#1684075911yescharacteristic
399C22#1684075911yescharacteristic
400C4—22no—
401C4—22no—
402C4—41no—
403C4—81no—
404C4—81no—
405C4—1222no—
406C4—1221no—
407C4—1221no—
408C4—1222no—
409C4—4884no—
410C4—4881no—
411C4—4881no—
412C2—1222no—
413C2—1222no—
414C2#2146122811yescharacteristic
415C2#2146124111yescharacteristic
416C2#2146125211yescharacteristic
417C2#2146125711yescharacteristic
418C2#2146195611yescharacteristic
419C2#2146196011yescharacteristic
420C2#2146198511yescharacteristic
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#9714
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
223. #391#6923semidirect
235. C61#9714semidirect

Maximal subgroups

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

subgroupindex
#214616742
C2×#168406162
C2×#168406152
#214609952
#214541682
#971461