Finite Group Databasebeta

← #30603377

Subgroups — #30603377

Order 15616.

898 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 898 conjugacy classes of subgroups — too many to draw. Only its 144 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#21461506C212yesmaximal
4C2×#16840633C212yesmaximal
5#391×D61C211yesmaximal, characteristic
6C2×D61⋊M16C2212yes—
7C2×#14519856C2212yes—
8D61⋊#101—24no—
9C2×#14519838—22no—
10(C2×#6923)⋊Q8—24no—
11#16840705—22no—
12#16840635—24no—
13#16840633—24no—
14C22×Q8×D61C2211yescharacteristic
15C2×C4⋊C4×D61C2211yescharacteristic
16C2×C42×D61C2211yescharacteristic
17#16839669C411yescharacteristic, complemented
18#16839668C411yescharacteristic, complemented
19#16839667C411yescharacteristic, complemented
20#16839666—22no—
21#95×D61—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⋊#6923D414yes—
35C2×C22⋊#6923—22no—
36C2×C4×#6923—22no—
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×D61—21no—
48C22×C4×D61—21no—
49C22×C4×D61D412yes—
50C22×C4×D61C2311yescharacteristic
51C2×C4⋊Dic61C2×C411yescharacteristic
52C2×#11758196—21no—
53C4⋊C4×D61—22no—
54C4⋊C4×D61—22no—
55C2×C4×Dic61—21no—
56C2×C4×Dic61C2×C411yescharacteristic
57C42×D61—22no—
58C42×D61—22no—
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—24no—
89C2×C4×D61—22no—
90C2×C4×D61—24no—
91C2×C4×D61—24no—
92C2×C4×D61—22no—
93C2×C4×D61—42no—
94C2×C4×D61—42no—
95C2×C4×D61—44no—
96C2×C4×D61—41no—
97C2×C4×D61—41no—
98C2×C4×D61C2×D412yes—
99C2×C4×D61C2×D412yes—
100C2×C4×D61#4112yes—
101C2×C4×D61#4112yes—
102C2×C4×D61#4112yes—
103C2×C4×D61#4112yes—
104C2×Dic122—21no—
105C2×Dic122—21no—
106C2×Dic122—22no—
107C2×Dic122—22no—
108C2×Dic122—22no—
109C2×Dic122—24no—
110C2×Dic122—24no—
111C2×Dic122—22no—
112C22⋊#6923—42no—
113C22⋊#6923—42no—
114#11758217—24no—
115#11758217—42no—
116C2×#92420—42no—
117C4⋊#6923—24no—
118C4⋊#6923—24no—
119C4×#6923—42no—
120C4×#6923—42no—
121#11758213—24no—
122#11758213—42no—
123D61⋊C8—24no—
124D61⋊C8—42no—
125C4⋊C4×C61—22no—
126C42×C61—22no—
127C4⋊Dic61—22no—
128#11758196—42no—
129C4×Dic61—22no—
130C4×Dic61—41no—
131C4×Dic61—41no—
132C23×C61C2×C22⋊C411yescharacteristic
133C22×D61—22no—
134C22×D61—22no—
135C22×D61#12011yescharacteristic
136C22×D61#12112yes—
137C22×D61#12312yes—
138C22×D61#12311yescharacteristic
139C2×#6923—44no—
140C2×#6923—44no—
141C2×#6923—44no—
142Q8×C61—42no—
143Q8×C61—42no—
144C2×C4×C61—22no—
145C2×C4×C61—22no—
146C2×C4×C61—22no—
147C2×C4×C61—22no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—42no—
151C2×C4×C61—41no—
152C2×C4×C61#11711yescharacteristic
153C2×C4×C61#11711yescharacteristic
154C2×C4×C61C4×D412yes—
155C2×C4×C61C4×D412yes—
156C2×Dic61—22no—
157C2×Dic61—22no—
158C2×Dic61—22no—
159C2×Dic61—22no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162C2×Dic61—42no—
163C2×Dic61—41no—
164C2×Dic61#12011yescharacteristic
165C2×Dic61#12112yes—
166C2×Dic61#12312yes—
167C2×Dic61#12311yescharacteristic
168C4×D61—48no—
169C4×D61—44no—
170C4×D61—44no—
171C4×D61—82no—
172C4×D61—82no—
173C4×D61#12112yes—
174C4×D61#12112yes—
175C4×D61#12312yes—
176C4×D61#12312yes—
177C4×D61#12412yes—
178C4×D61#12412yes—
179C4×D61#12612yes—
180C4×D61#12612yes—
181Dic122—42no—
182Dic122—42no—
183Dic122—44no—
184Dic122—44no—
185#92420—44no—
186#11162—611nomaximal, Sylow
187C22×C61—21no—
188C22×C61—21no—
189C22×C61#38111yescharacteristic
190C22×C61#38611yescharacteristic
191C22×C61#38611yescharacteristic
192C2×D61—21no—
193C2×D61—21no—
194C2×D61—24no—
195C2×D61—24no—
196C2×D61#39412yes—
197C2×D61#39712yes—
198C2×D61#45412yes—
199C2×D61#45612yes—
200#6923—84no—
201C4×C61—42no—
202C4×C61—42no—
203C4×C61—81no—
204C4×C61#38812yes—
205C4×C61#38812yes—
206Dic61—42no—
207Dic61—42no—
208Dic61—81no—
209Dic61#39412yes—
210Dic61#39712yes—
211C22×#101—611no—
212#2310—611no—
213C2×#391—611no—
214#1736—612no—
215C2×#339—612no—
216C2×C61—21no—
217C2×C61—21no—
218C2×C61#144011yescharacteristic
219C2×C61#188611yescharacteristic
220C2×C61#188811yescharacteristic
221D61—22no—
222D61#203814yes—
223C23×Q8—611no—
224C22×M16—612no—
225C2×#117—612no—
226C22×C4⋊C4—611no—
227C22×C42—611no—
228C2×#101—1224no—
229C2×#101—1222no—
230#391—611no—
231#391—611no—
232#391—611no—
233#391—1222no—
234#391—1224no—
235#391#692311yescharacteristic, complemented
236#381—1222no—
237C2×#95—1222no—
238#340—1224no—
239#339—1224no—
240C61#1116211yescharacteristic, complemented, Sylow
241C22×Q8—612no—
242C22×Q8—611no—
243C22×Q8—1222no—
244C22×Q8—1222no—
245C22×Q8—1222no—
246C22×Q8—1222no—
247C22×Q8—1222no—
248C22×Q8—1222no—
249C22×Q8C2×#692311yescharacteristic
250C23×C4—611no—
251C23×C4—612no—
252C23×C4—1221no—
253C23×C4—1221no—
254C2×M16—614no—
255C2×M16—1222no—
256C2×M16—1222no—
257C2×M16—1222no—
258C2×M16—1222no—
259C2×M16—1222no—
260C22×C8—1222no—
261#117—1222no—
262#117—1222no—
263#117—1222no—
264#117—1222no—
265C2×C4⋊C4—614no—
266C2×C4⋊C4—611no—
267C2×C4⋊C4—1222no—
268C2×C4⋊C4—1222no—
269C2×C4⋊C4—1221no—
270C2×C4⋊C4C2×#692311yescharacteristic
271C2×C22⋊C4—1222no—
272C2×C42—611no—
273C2×C42—1222no—
274C2×C42—1222no—
275C2×C42—1221no—
276C2×C42—1222no—
277C2×C42C2×#692311yescharacteristic
278#101—1228no—
279#95—22no—
280#95—1222no—
281#95—1222no—
282#95—1228no—
283#95—1222no—
284C24—611no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—1222no—
290C2×Q8—1224no—
291C2×Q8—1222no—
292C2×Q8—1224no—
293C2×Q8—1221no—
294C2×Q8—1221no—
295C2×Q8—1222no—
296C2×Q8—1222no—
297C2×Q8—2442no—
298C2×Q8—2442no—
299C2×Q8—2442no—
300C2×Q8—2442no—
301C2×Q8—2442no—
302C2×Q8—2442no—
303C2×Q8—2442no—
304C2×Q8—2442no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—611no—
308C22×C4—612no—
309C22×C4—612no—
310C22×C4—612no—
311C22×C4—612no—
312C22×C4—612no—
313C22×C4—612no—
314C22×C4—612no—
315C22×C4—1222no—
316C22×C4—1224no—
317C22×C4—1224no—
318C22×C4—1222no—
319C22×C4—1224no—
320C22×C4—1221no—
321C22×C4—1224no—
322C22×C4—1221no—
323C22×C4—2441no—
324C22×C4—2441no—
325C22×C4—2444no—
326C22×C4—2442no—
327C22×C4—2442no—
328C22×C4C22⋊#692312yes—
329C22×C4C22×#692311yescharacteristic
330M16—1224no—
331M16—1224no—
332M16—2442no—
333M16—2442no—
334C2×C8—1224no—
335C2×C8—2442no—
336C2×C8—2442no—
337C4⋊C4—22no—
338C4⋊C4—1224no—
339C4⋊C4—1224no—
340C4⋊C4—1222no—
341C4⋊C4—2442no—
342C22⋊C4—2442no—
343C22⋊C4—2442no—
344C42—22no—
345C42—1222no—
346C42—2441no—
347C42—2441no—
348C42—2442no—
349C42—2442no—
350C23—611no—
351C23—612no—
352C23—612no—
353C23—611no—
354C23—1222no—
355C23—1222no—
356C23C2×C22⋊#692311yescharacteristic
357Q8—42no—
358Q8—42no—
359Q8—2444no—
360Q8—2444no—
361Q8—2442no—
362Q8—2442no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—612no—
372C2×C4—611no—
373C2×C4—611no—
374C2×C4—612no—
375C2×C4—612no—
376C2×C4—612no—
377C2×C4—612no—
378C2×C4—612no—
379C2×C4—612no—
380C2×C4—612no—
381C2×C4—612no—
382C2×C4—612no—
383C2×C4—1222no—
384C2×C4—1222no—
385C2×C4—1222no—
386C2×C4—1222no—
387C2×C4—4882no—
388C2×C4—4882no—
389C2×C4—2444no—
390C2×C4—2444no—
391C2×C4—2448no—
392C2×C4—2442no—
393C2×C4—2444no—
394C2×C4—2444no—
395C2×C4—2444no—
396C2×C4—2441no—
397C2×C4—2441no—
398C2×C4—2441no—
399C2×C4#1451985611yescharacteristic
400C2×C4#1451985611yescharacteristic
401C2×C4D4×#692312yes—
402C2×C4D4×#692312yes—
403C8—2444no—
404C22—21no—
405C22—21no—
406C22—612no—
407C22—612no—
408C22—612no—
409C22—612no—
410C22—1224no—
411C22—1224no—
412C22—1221no—
413C22—1221no—
414C22#1684070511yescharacteristic
415C22#1684071011yescharacteristic
416C22#1684071011yescharacteristic
417C4—42no—
418C4—42no—
419C4—81no—
420C4—612no—
421C4—612no—
422C4—4884no—
423C4—4881no—
424C4—2442no—
425C4—2442no—
426C4#1684071212yes—
427C4#1684071212yes—
428C2—21no—
429C2—21no—
430C2—614no—
431C2—1222no—
432C2#2146100311yescharacteristic
433C2#2146182511yescharacteristic
434C2#2146182711yescharacteristic
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#11162
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
235. #391#6923semidirect
240. C61#11162semidirect

Maximal subgroups

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

subgroupindex
C2×D61⋊#1012
#214622932
#214615062
C2×#168406332
#391×D612
#1116261