Finite Group Databasebeta

← #30603369

Subgroups — #30603369

Order 15616.

826 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 136 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×D61⋊#101C211yesmaximal, characteristic
2#21462293C211yesmaximal, characteristic
3#21461498C212yesmaximal
4C2×#16840629C212yesmaximal
5#21448402C211yesmaximal, characteristic
6C2×D61⋊M16C2212yes—
7C2×#14519856C2212yes—
8D61⋊#101—24no—
9C2×#14519838—22no—
10(C2×#6923)⋊Q8—24no—
11#16840705—22no—
12#16840632—24no—
13#16840629—24no—
14C22×Q8×D61C2211yescharacteristic
15C2×#14519735C2211yescharacteristic
16C2×#14519734C2211yescharacteristic
17#16839669C411yescharacteristic, complemented
18#16839668C411yescharacteristic, complemented
19#16839667C411yescharacteristic, complemented
20#16839666—22no—
21#16839624—22no—
22C61×#391C411yescharacteristic, complemented
23C2×#11758213—22no—
24D61⋊M16—22no—
25D61⋊M16—22no—
26D61⋊M16—22no—
27D61⋊M16D414yes—
28C2×#11758217—22no—
29C2×D61⋊C8—22no—
30#14519856—22no—
31#14519856—22no—
32#14519856—22no—
33#14519856—22no—
34C2×C4⋊#6923—22no—
35C2×C22⋊#6923—22no—
36C2×C4×#6923D414yes—
37#14519838—28no—
38(C2×#6923)⋊C4—28no—
39C22×Dic122C2×C411yescharacteristic
40C22×Dic122D412yes—
41C2×Q8×D61—22no—
42C2×Q8×D61—22no—
43C2×Q8×D61—22no—
44C2×Q8×D61—22no—
45C2×Q8×D61—22no—
46C2×Q8×D61—22no—
47C22×C4×D61D412yes—
48C22×C4×D61C2311yescharacteristic
49#14519735—22no—
50#14519735—22no—
51#14519734—22no—
52#14519734—22no—
53C2×C4⋊Dic61C2×C411yescharacteristic
54C2×#11758196—21no—
55C2×#11758198—21no—
56C2×#11758198—21no—
57C2×C4×Dic61—21no—
58C2×C4×Dic61C2×C411yescharacteristic
59#14519684—22no—
60#14519684—22no—
61#14519684—22no—
62C22×Q8×C61C2×C411yescharacteristic
63C2×C4⋊C4×C61C2×C411yescharacteristic
64C2×C42×C61C2×C411yescharacteristic
65#95×C61—22no—
66C23×D61C2×D411yescharacteristic
67C22×#6923—24no—
68C2×Q8×C61—21no—
69C2×Q8×C61—21no—
70C2×Q8×C61—22no—
71C2×Q8×C61—22no—
72C22×C4×C61—21no—
73C22×C4×C61—21no—
74C22×C4×C61C22⋊C412yes—
75C22×C4×C61C22×C411yescharacteristic
76C22×Dic61—21no—
77C22×Dic61—21no—
78C22×Dic61C22⋊C412yes—
79C22×Dic61C2×D411yescharacteristic
80Q8×D61—42no—
81Q8×D61—42no—
82Q8×D61—42no—
83Q8×D61—42no—
84Q8×D61—42no—
85Q8×D61—42no—
86Q8×D61—42no—
87Q8×D61—42no—
88C2×C4×D61—22no—
89C2×C4×D61—24no—
90C2×C4×D61—24no—
91C2×C4×D61—22no—
92C2×C4×D61C2×D412yes—
93C2×C4×D61C2×D412yes—
94C2×C4×D61#4112yes—
95C2×C4×D61#4112yes—
96C2×C4×D61#4112yes—
97C2×C4×D61#4112yes—
98C2×Dic122—21no—
99C2×Dic122—21no—
100C2×Dic122—22no—
101C2×Dic122—22no—
102C2×Dic122—22no—
103C2×Dic122—24no—
104C2×Dic122—24no—
105C2×Dic122—22no—
106C22⋊#6923—42no—
107C22⋊#6923—42no—
108#11758217—24no—
109#11758217—42no—
110C2×#92420—42no—
111C4⋊#6923—42no—
112C4⋊#6923—42no—
113C4×#6923—24no—
114C4×#6923—24no—
115#11758213—24no—
116#11758213—42no—
117D61⋊C8—24no—
118D61⋊C8—42no—
119C4⋊C4×C61—22no—
120C42×C61—22no—
121#11758198—41no—
122#11758198—42no—
123#11758198—41no—
124C4⋊Dic61—22no—
125#11758196—41no—
126#11758196—41no—
127C4×Dic61—22no—
128C4×Dic61—42no—
129C23×C61C2×C22⋊C411yescharacteristic
130C22×D61—41no—
131C22×D61—41no—
132C22×D61#12011yescharacteristic
133C22×D61#12112yes—
134C22×D61#12311yescharacteristic
135C22×D61#12312yes—
136C2×#6923—44no—
137C2×#6923—44no—
138C2×#6923—44no—
139Q8×C61—42no—
140Q8×C61—42no—
141C2×C4×C61—22no—
142C2×C4×C61—22no—
143C2×C4×C61—22no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—42no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61#11711yescharacteristic
150C2×C4×C61#11711yescharacteristic
151C2×C4×C61C4×D412yes—
152C2×C4×C61C4×D412yes—
153C2×Dic61—22no—
154C2×Dic61—22no—
155C2×Dic61—22no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158C2×Dic61—42no—
159C2×Dic61—41no—
160C2×Dic61—41no—
161C2×Dic61#12011yescharacteristic
162C2×Dic61#12112yes—
163C2×Dic61#12312yes—
164C2×Dic61#12311yescharacteristic
165C4×D61—44no—
166C4×D61—44no—
167C4×D61#12112yes—
168C4×D61#12112yes—
169C4×D61#12312yes—
170C4×D61#12312yes—
171C4×D61#12412yes—
172C4×D61#12412yes—
173C4×D61#12612yes—
174C4×D61#12612yes—
175Dic122—42no—
176Dic122—42no—
177Dic122—44no—
178Dic122—44no—
179#92420—44no—
180#11156—611nomaximal, Sylow
181C22×C61—21no—
182C22×C61—21no—
183C22×C61#38111yescharacteristic
184C22×C61#38611yescharacteristic
185C22×C61#38611yescharacteristic
186C2×D61—21no—
187C2×D61—21no—
188C2×D61—21no—
189C2×D61—21no—
190C2×D61—42no—
191C2×D61—42no—
192C2×D61#39412yes—
193C2×D61#39712yes—
194#6923—84no—
195C4×C61—42no—
196C4×C61—81no—
197C4×C61—81no—
198C4×C61#38812yes—
199C4×C61#38812yes—
200Dic61—42no—
201Dic61—81no—
202Dic61—81no—
203Dic61#39412yes—
204Dic61#39712yes—
205C22×#101—611no—
206#2310—611no—
207#2310—611no—
208#1733—612no—
209C2×H(3,ℤ/4)—612no—
210C2×C61—21no—
211C2×C61—21no—
212C2×C61#144011yescharacteristic
213C2×C61#191211yescharacteristic
214C2×C61#191211yescharacteristic
215D61—22no—
216D61—22no—
217C23×Q8—611no—
218C22×M16—612no—
219C2×#117—612no—
220C2×#117—611no—
221C2×#117—611no—
222C2×#101—1224no—
223C2×#101—1222no—
224#391—611no—
225#391—611no—
226#391—611no—
227#391—1222no—
228#391—1224no—
229#391#692311yescharacteristic, complemented
230#381—1222no—
231#381—1222no—
232#339—1224no—
233H(3,ℤ/4)—1224no—
234C61#1115611yescharacteristic, complemented, Sylow
235C22×Q8—612no—
236C22×Q8—611no—
237C22×Q8—1222no—
238C22×Q8—1222no—
239C22×Q8—1222no—
240C22×Q8—1222no—
241C22×Q8—1222no—
242C22×Q8—1222no—
243C22×Q8C2×#692311yescharacteristic
244C23×C4—611no—
245C23×C4—612no—
246C2×M16—614no—
247C2×M16—1222no—
248C2×M16—1222no—
249C2×M16—1222no—
250C2×M16—1222no—
251C2×M16—1222no—
252C22×C8—1222no—
253#117—1222no—
254#117—1222no—
255#117—1222no—
256#117—1222no—
257#117—1222no—
258#117—1222no—
259#117—1222no—
260#117—1222no—
261C2×C4⋊C4—611no—
262C2×C4⋊C4—1221no—
263C2×C4⋊C4—1222no—
264C2×C4⋊C4C2×#692311yescharacteristic
265C2×C22⋊C4—1221no—
266C2×C22⋊C4—1221no—
267C2×C22⋊C4—1222no—
268C2×C42—614no—
269C2×C42—611no—
270C2×C42—1221no—
271C2×C42C2×#692311yescharacteristic
272#101—1228no—
273#95—22no—
274#95—1222no—
275#95—1222no—
276#95—1228no—
277#95—1222no—
278C24—611no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—1222no—
284C2×Q8—1224no—
285C2×Q8—1222no—
286C2×Q8—1224no—
287C2×Q8—1221no—
288C2×Q8—1221no—
289C2×Q8—1222no—
290C2×Q8—1222no—
291C2×Q8—2442no—
292C2×Q8—2442no—
293C2×Q8—2442no—
294C2×Q8—2442no—
295C2×Q8—2442no—
296C2×Q8—2442no—
297C2×Q8—2442no—
298C2×Q8—2442no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—611no—
302C22×C4—612no—
303C22×C4—612no—
304C22×C4—612no—
305C22×C4—612no—
306C22×C4—612no—
307C22×C4—612no—
308C22×C4—612no—
309C22×C4—1222no—
310C22×C4—1224no—
311C22×C4—1224no—
312C22×C4—1222no—
313C22×C4—1224no—
314C22×C4—1221no—
315C22×C4—1221no—
316C22×C4C22⋊#692312yes—
317C22×C4C22×#692311yescharacteristic
318M16—1224no—
319M16—1224no—
320M16—2442no—
321M16—2442no—
322C2×C8—1224no—
323C2×C8—2442no—
324C2×C8—2442no—
325C4⋊C4—22no—
326C4⋊C4—1222no—
327C4⋊C4—2441no—
328C4⋊C4—2441no—
329C4⋊C4—2442no—
330C4⋊C4—2442no—
331C22⋊C4—2442no—
332C22⋊C4—2441no—
333C22⋊C4—2441no—
334C22⋊C4—2442no—
335C22⋊C4—2442no—
336C42—22no—
337C42—1224no—
338C42—1224no—
339C42—1222no—
340C42—2442no—
341C23—611no—
342C23—611no—
343C23—612no—
344C23—612no—
345C23—2441no—
346C23—2441no—
347C23C2×C22⋊#692311yescharacteristic
348Q8—42no—
349Q8—42no—
350Q8—2444no—
351Q8—2444no—
352Q8—2442no—
353Q8—2442no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—612no—
363C2×C4—611no—
364C2×C4—611no—
365C2×C4—612no—
366C2×C4—612no—
367C2×C4—612no—
368C2×C4—612no—
369C2×C4—612no—
370C2×C4—612no—
371C2×C4—612no—
372C2×C4—612no—
373C2×C4—612no—
374C2×C4—1222no—
375C2×C4—1222no—
376C2×C4—1222no—
377C2×C4—2444no—
378C2×C4—2444no—
379C2×C4—2441no—
380C2×C4—2441no—
381C2×C4—2442no—
382C2×C4—2444no—
383C2×C4—2444no—
384C2×C4—2444no—
385C2×C4—2441no—
386C2×C4—2441no—
387C2×C4#1451985611yescharacteristic
388C2×C4#1451985611yescharacteristic
389C2×C4D4×#692312yes—
390C2×C4D4×#692312yes—
391C8—2444no—
392C22—21no—
393C22—21no—
394C22—612no—
395C22—612no—
396C22—1221no—
397C22—1221no—
398C22—1221no—
399C22—1221no—
400C22—2442no—
401C22—2442no—
402C22#1684070511yescharacteristic
403C22#1684071011yescharacteristic
404C22#1684071011yescharacteristic
405C4—42no—
406C4—81no—
407C4—81no—
408C4—612no—
409C4—612no—
410C4—4881no—
411C4—4884no—
412C4—4881no—
413C4—2442no—
414C4#1684071212yes—
415C4#1684071212yes—
416C2—21no—
417C2—21no—
418C2—1222no—
419C2—1222no—
420C2#2146100311yescharacteristic
421C2#2146186611yescharacteristic
422C2#2146186711yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#391C4
socleC22×C61#386
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenter#391#6923
hypocenterC61#11156
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
17. #16839669C4semidirect
18. #16839668C4semidirect
19. #16839667C4semidirect
22. C61×#391C4semidirect
229. #391#6923semidirect
234. C61#11156semidirect

Maximal subgroups

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

subgroupindex
C2×D61⋊#1012
#214622932
#214614982
C2×#168406292
#214484022
#1115661