Finite Group Databasebeta

← #30600344

Subgroups — #30600344

Order 15616.

696 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 106 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×#16840721C211yesmaximal, characteristic, complemented
2#21461546C211yesmaximal, characteristic, complemented
3C2×#16840617C212yesmaximal, complemented
4#21460695C212yesmaximal, complemented
5#21454119C211yesmaximal, characteristic
6C22×D61⋊C8—21no—
7C2×#14519834—21no—
8#16840721—24no—
9C2×#14519832—21no—
10C2×#14519832C2212yes—
11C2×#14519832C2212yes—
12#16840628—21no—
13#16840617—24no—
14C2×#14519764C2211yescharacteristic
15C2×#14519757C411yescharacteristic
16C2×#14519748C411yescharacteristic
17C2×C42×D61C2211yescharacteristic
18#16839668C411yescharacteristic
19#16839655—21no—
20#16839655—21no—
21#16839638—21no—
22#16839638—21no—
23#16839627C2211yescharacteristic
24C61×#390C411yescharacteristic
25C2×D61⋊C8—24no—
26C2×D61⋊C8—24no—
27C2×D61⋊C8—24no—
28C22×#92420—22no—
29C22×#92420—22no—
30#14519834—24no—
31#14519832—24no—
32#14519832—22no—
33#14519832—22no—
34#14519832—22no—
35#14519832D414yescomplemented
36C22×Dic122C2×C411yescharacteristic
37C22×C22⋊D61—21no—
38C22×C4×D61C2×C411yescharacteristic
39C22×C4×D61D411yescharacteristic
40C22×C4×D61D411yescharacteristic
41#14519764—22no—
42#14519764—22no—
43#14519757—42no—
44#14519748—42no—
45C2×C4⋊Dic61—21no—
46C2×C4⋊Dic61C2×C411yescharacteristic
47C2×C22⋊Dic61—21no—
48C2×C22⋊Dic61—21no—
49C2×C22⋊Dic61—21no—
50C2×#11758198—21no—
51C2×C4×Dic61D411yescharacteristic
52C2×C4×Dic61D411yescharacteristic
53C2×C4×Dic61C2311yescharacteristic
54C42×D61—22no—
55C42×D61—22no—
56#14519684—21no—
57#14519684—21no—
58#14519684—21no—
59#14519684—21no—
60C22×D4×C61C2×C411yescharacteristic
61C2×C4⋊C4×C61C2×C411yescharacteristic
62C2×C22⋊C4×C61—21no—
63C2×C22⋊C4×C61—21no—
64C2×C42×C61C2×C411yescharacteristic
65C24×C61—21no—
66C23×D61C22⋊C411yescharacteristic
67C2×D4×C61—22no—
68C2×D4×C61—41no—
69C2×D4×C61—41no—
70C22×C4×C61—21no—
71C22×C4×C61C22⋊C411yescharacteristic
72C22×C4×C61C22⋊C411yescharacteristic
73C22×C4×C61C22×C411yescharacteristic
74C2×C22⋊D61—42no—
75C2×C22⋊D61—42no—
76C2×C22⋊D61—42no—
77C22×Dic61—21no—
78C22×Dic61—21no—
79C22×Dic61C22⋊C411yescharacteristic
80C22×Dic61C2×D411yescharacteristic
81C22×Dic61C2×D411yescharacteristic
82C2×C4×D61—22no—
83C2×C4×D61—22no—
84C2×C4×D61—21no—
85C2×C4×D61—21no—
86C2×C4×D61—22no—
87C2×C4×D61—22no—
88C2×C4×D61—22no—
89C2×C4×D61—22no—
90C2×C4×D61—41no—
91C2×C4×D61—41no—
92C2×Dic122—22no—
93C2×Dic122—41no—
94C2×Dic122—41no—
95C2×#92420—24no—
96C2×#92420—24no—
97C2×#92420—24no—
98C2×#92420—24no—
99C2×#92420—42no—
100C2×#92420—42no—
101D61⋊C8—216no—
102C4⋊C4×C61—41no—
103C22⋊C4×C61—42no—
104C22⋊C4×C61—42no—
105C42×C61—22no—
106C22⋊Dic61—42no—
107C22⋊Dic61—41no—
108C22⋊Dic61—42no—
109C22⋊Dic61—41no—
110#11758198—42no—
111C4⋊Dic61—42no—
112C4⋊Dic61—41no—
113C4×Dic61—21no—
114C4×Dic61—22no—
115C4×Dic61—21no—
116C4×Dic61C2×D412yes—
117C4×Dic61#4112yes—
118C23×C61—41no—
119C23×C61—41no—
120C23×C61—41no—
121C23×C61—41no—
122C23×C61—41no—
123C23×C61—41no—
124C23×C61—41no—
125C23×C61C2×C22⋊C411yescharacteristic
126C22×D61—21no—
127C22×D61—21no—
128C22×D61—21no—
129C22×D61—21no—
130C22×D61#10012yes—
131C22×D61#10412yes—
132C22×D61#10412yes—
133D4×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#10412yes—
145C2×C4×C61C4×D412yes—
146C22⋊D61—84no—
147C2×Dic61—21no—
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#12011yescharacteristic
161C2×Dic61#12112yes—
162C2×Dic61#12311yescharacteristic
163C4×D61—24no—
164C4×D61—24no—
165C4×D61—44no—
166C4×D61—44no—
167Dic122—82no—
168#92420—28no—
169#92420—44no—
170#9119—611nomaximal, Sylow
171C22×C61—81no—
172C22×C61—81no—
173C22×C61—81no—
174C22×C61—81no—
175C22×C61—81no—
176C22×C61—81no—
177C22×C61—81no—
178C22×C61#38611yescharacteristic
179C22×C61C2×#10011yescharacteristic
180C22×C61#41311yescharacteristic
181C22×C61C2×#10411yescharacteristic
182C22×C61C2×#10411yescharacteristic
183C22×C61C2×#10411yescharacteristic
184C22×C61#42111yescharacteristic
185C2×D61—22no—
186C2×D61—22no—
187C2×D61—22no—
188C2×D61—22no—
189C2×D61—22no—
190C2×D61—22no—
191C2×D61#32914yescomplemented
192C4×C61—22no—
193C4×C61—22no—
194C4×C61—41no—
195C4×C61—81no—
196Dic61—22no—
197Dic61—21no—
198Dic61—21no—
199Dic61—41no—
200Dic61—81no—
201Dic61—81no—
202C22×#97—611no—
203#2850—611no—
204#1749—611no—
205C2×#329—612no—
206#1295—612no—
207C2×C61—161no—
208C2×C61C2×#32911yescharacteristic
209C2×C61#152411yescharacteristic
210C2×C61#153011yescharacteristic
211C2×C61#153211yescharacteristic
212C2×C61#187111yescharacteristic
213C2×C61#189611yescharacteristic
214C2×C61#190011yescharacteristic
215D61—24no—
216C23×C8—1221no—
217C2×#124—611no—
218C2×C22⋊Q8—611no—
219C2×#121—611no—
220C22×C42—611no—
221C2×C22⋊C8—1221no—
222C2×#97—612no—
223C2×#97—612no—
224C2×#97—1224no—
225C2×#97—1221no—
226#391—611no—
227#390#692311yescharacteristic
228#388—1221no—
229#388—1221no—
230#386—1221no—
231#386—1221no—
232#382—611no—
233#336—1221no—
234#329—1224no—
235C61#911911yescharacteristic, complemented, Sylow
236C22×Q8—611no—
237C22×D4—1221no—
238C22×D4C2×#692311yescharacteristic
239C23×C4—611no—
240C23×C4—611no—
241C23×C4—611no—
242C22×C8—1222no—
243C22×C8—1224no—
244C22×C8—1224no—
245C22×C8—1224no—
246C22×C8—1222no—
247#124—1222no—
248#124—1222no—
249C22⋊Q8—2442no—
250#121—2442no—
251C2×C4⋊C4—611no—
252C2×C4⋊C4—1221no—
253C2×C4⋊C4C2×#692311yescharacteristic
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—1221no—
257C2×C22⋊C4—1221no—
258C2×C22⋊C4—1221no—
259C2×C22⋊C4—1221no—
260C2×C42—611no—
261C2×C42—611no—
262C2×C42—611no—
263C2×C42—1222no—
264C2×C42—1222no—
265C2×C42C2×#692311yescharacteristic
266C22⋊C8—1224no—
267#97—614no—
268#97—1222no—
269#97—1222no—
270#97—1222no—
271#97—1224no—
272#95—1221no—
273#95—1221no—
274#95—1221no—
275#95—1221no—
276C24—21no—
277C24—611no—
278C2×Q8—1222no—
279C2×Q8—2441no—
280C2×Q8—2441no—
281C2×D4—22no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—2442no—
285C2×D4—2442no—
286C2×D4—2442no—
287C22×C4—21no—
288C22×C4—611no—
289C22×C4—611no—
290C22×C4—611no—
291C22×C4—1221no—
292C22×C4—1221no—
293C22×C4—1222no—
294C22×C4—1222no—
295C22×C4—1222no—
296C22×C4—1222no—
297C22×C4—1222no—
298C22×C4—1222no—
299C22×C4—1221no—
300C22×C4—1221no—
301C22×C4—2441no—
302C22×C4—2441no—
303C22×C4C22⋊#692311yescharacteristic
304C22×C4C22⋊#692311yescharacteristic
305C22×C4C22×#692311yescharacteristic
306C2×C8—1224no—
307C2×C8—12216no—
308C2×C8—1224no—
309C2×C8—1224no—
310C2×C8—1224no—
311C2×C8—2442no—
312C2×C8—2442no—
313C4⋊C4—41no—
314C4⋊C4—2441no—
315C4⋊C4—2442no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—2442no—
319C22⋊C4—2441no—
320C22⋊C4—2442no—
321C22⋊C4—2441no—
322C22⋊C4—2442no—
323C42—22no—
324C42—612no—
325C42—612no—
326C42—1221no—
327C42—1222no—
328C42—1221no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—612no—
337C23—612no—
338C23—612no—
339C23—1221no—
340C23—1221no—
341C23—1221no—
342C23—1221no—
343C23C2×C22⋊#692311yescharacteristic
344Q8—4882no—
345D4—82no—
346D4—4884no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—611no—
358C2×C4—612no—
359C2×C4—611no—
360C2×C4—1224no—
361C2×C4—1224no—
362C2×C4—1221no—
363C2×C4—1221no—
364C2×C4—1221no—
365C2×C4—1221no—
366C2×C4—1221no—
367C2×C4—1221no—
368C2×C4—1221no—
369C2×C4—2444no—
370C2×C4—2444no—
371C2×C4—2441no—
372C2×C4—2441no—
373C2×C4—2441no—
374C2×C4—2441no—
375C2×C4—2441no—
376C2×C4—2441no—
377C2×C4#1451984112yes—
378C2×C4D4×#692312yes—
379C8—1228no—
380C8—2444no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—614no—
389C22—1222no—
390C22—1222no—
391C22—1222no—
392C22—1222no—
393C22—1222no—
394C22—1222no—
395C22#1684071011yescharacteristic
396C22D61⋊#10011yescharacteristic
397C22#1684074811yescharacteristic
398C22C2×#1451984111yescharacteristic
399C22D61⋊#10411yescharacteristic
400C22D61⋊#10411yescharacteristic
401C22#1684076011yescharacteristic
402C4—22no—
403C4—22no—
404C4—41no—
405C4—81no—
406C4—1222no—
407C4—1221no—
408C4—1221no—
409C4—4881no—
410C4—4881no—
411C4—2441no—
412C2—161no—
413C2—1224no—
414C2D61⋊#32911yescharacteristic
415C2#2146126911yescharacteristic
416C2#2146128911yescharacteristic
417C2#2146129411yescharacteristic
418C2#2146180311yescharacteristic
419C2#2146183711yescharacteristic
420C2#2146184511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC23×C61C2×C22⋊C4
prime radicalC2×#14519748C4
cosocleC2×C4×Dic61C23
hypercenter#390#6923
hypocenterC61#9119
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#16840721C2semidirect
2. #21461546C2semidirect
3. C2×#16840617C2semidirect
4. #21460695C2semidirect
35. #14519832D4semidirect
191. C2×D61#329semidirect
235. C61#9119semidirect

Maximal subgroups

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

subgroupindex
C2×#168407212
#214615462
C2×#168406172
#214606952
#214541192
#911961