Finite Group Databasebeta

← #30605389

Subgroups — #30605389

Order 15616.

536 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 76 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#21461680C211yesmaximal, characteristic, complemented
2#21461056C211yesmaximal, characteristic, complemented
3#21460799C212yesmaximal, complemented
4#21460797C212yesmaximal, complemented
5#21450720C211yesmaximal, characteristic
6C2×#14519836—21no—
7D61⋊(C22⋊C8)—21no—
8#16840704—21no—
9#16840640C2212yes—
10#16840607—21no—
11#16840606C2212yes—
12C2×#14519751—21no—
13C2×C4×D244C2211yescharacteristic
14#16839688C411yescharacteristic, complemented
15#16839688C2211yescharacteristic
16#16839670C411yescharacteristic, complemented
17#16839670C2211yescharacteristic
18#16839638—21no—
19#16839638—21no—
20#16839628C411yescharacteristic, complemented
21#16839622—21no—
22C2×C4×D4×C61C411yescharacteristic, complemented
23C2×D61⋊C8—41no—
24C2×C4⋊#6923—41no—
25C2×C22⋊#6923—22no—
26#14519836—42no—
27#14519834—41no—
28C22⋊#92420—22no—
29(C2×#6923)⋊C4—41no—
30C22×D244D411yescharacteristic
31C22×C22⋊D61—21no—
32C22×C22⋊D61—21no—
33C22×C4×D61D411yescharacteristic
34C22×C4×D61D411yescharacteristic
35#14519751—42no—
36#14519751—42no—
37C4×D244—41no—
38C4×D244—41no—
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61D411yescharacteristic
41C2×C22⋊Dic61—21no—
42C2×C22⋊Dic61—21no—
43C2×#11758198—21no—
44C2×#11758198—21no—
45C2×#11758198C2×C411yescharacteristic
46C2×#11758198C2311yescharacteristic
47#14519684—21no—
48#14519684—21no—
49#14519684—21no—
50#14519684C2×C411yescharacteristic
51#14519684C2×C411yescharacteristic
52C22×D4×C61C2×C411yescharacteristic
53C23×C4×C61—21no—
54C4×D4×C61—41no—
55C4×D4×C61—41no—
56C2×C4⋊C4×C61C2×C411yescharacteristic
57C2×C22⋊C4×C61—21no—
58C2×C42×C61C2×C411yescharacteristic
59C24×C61—21no—
60C23×D61C22⋊C411yescharacteristic
61C23×D61C2×D411yescharacteristic
62C22×#6923—42no—
63C2×D4×C61—41no—
64C2×D4×C61—42no—
65C22×C4×C61—21no—
66C22×C4×C61—42no—
67C22×C4×C61—42no—
68C22×C4×C61—42no—
69C22×C4×C61C22⋊C411yescharacteristic
70C22×C4×C61C22⋊C411yescharacteristic
71C22×C4×C61C22×C411yescharacteristic
72C2×C22⋊D61—42no—
73C2×C22⋊D61—42no—
74C2×C22⋊D61—42no—
75C2×C22⋊D61—41no—
76C2×C22⋊D61—41no—
77C2×C22⋊D61—81no—
78C2×C22⋊D61—81no—
79C22×Dic61—21no—
80C22×Dic61—21no—
81C22×Dic61C22⋊C411yescharacteristic
82C22×Dic61C2×D411yescharacteristic
83C2×D244—41no—
84C2×D244—41no—
85C2×D244D812yes—
86C2×D244SD1612yes—
87C2×C4×D61—21no—
88C2×C4×D61—21no—
89C2×C4×D61—21no—
90C2×C4×D61—21no—
91C2×C4×D61—81no—
92C2×C4×D61—81no—
93C22⋊#6923—42no—
94C22⋊#6923—42no—
95C2×#92420—42no—
96C4⋊#6923—82no—
97D61⋊C8—82no—
98C4⋊C4×C61—41no—
99C22⋊C4×C61—41no—
100C22⋊C4×C61—41no—
101C42×C61—21no—
102C42×C61—21no—
103C22⋊Dic61—81no—
104C22⋊Dic61—81no—
105#11758198—41no—
106#11758198—41no—
107#11758198—42no—
108#11758198—41no—
109#11758198—41no—
110C4⋊Dic61—41no—
111C4⋊Dic61—42no—
112C23×C61—22no—
113C23×C61—41no—
114C23×C61—41no—
115C23×C61—41no—
116C23×C61—41no—
117C23×C61—41no—
118C23×C61—41no—
119C23×C61C2×C22⋊C411yescharacteristic
120C22×D61—21no—
121C22×D61—21no—
122C22×D61—21no—
123C22×D61—21no—
124C22×D61—41no—
125C22×D61—41no—
126C22×D61—41no—
127C22×D61—41no—
128C22×D61#10212yes—
129C22×D61C2×D811yescharacteristic
130C22×D61C2×SD1611yescharacteristic
131C2×#6923—82no—
132C2×#6923—82no—
133C2×#6923—82no—
134D4×C61—42no—
135D4×C61—42no—
136C2×C4×C61—21no—
137C2×C4×C61—21no—
138C2×C4×C61—21no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—84no—
145C2×C4×C61#9911yescharacteristic
146C2×C4×C61#9911yescharacteristic
147C2×C4×C61#10212yes—
148C22⋊D61—84no—
149C22⋊D61—82no—
150C22⋊D61—82no—
151C2×Dic61—21no—
152C2×Dic61—21no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61—41no—
161D244—42no—
162D244—42no—
163C4×D61—42no—
164C4×D61—42no—
165C4×D61—81no—
166C4×D61—81no—
167#92420—82no—
168#11875—611nomaximal, Sylow
169C22×C61—21no—
170C22×C61—21no—
171C22×C61—42no—
172C22×C61—42no—
173C22×C61—42no—
174C22×C61—81no—
175C22×C61—81no—
176C22×C61—81no—
177C22×C61—81no—
178C22×C61C2×#9911yescharacteristic
179C22×C61C2×#10211yescharacteristic
180C22×C61#42111yescharacteristic
181C2×D61—21no—
182C2×D61—22no—
183C2×D61—21no—
184C2×D61—41no—
185C2×D61—41no—
186C2×D61—41no—
187C2×D61—41no—
188C2×D61—42no—
189C2×D61—41no—
190C2×D61—41no—
191C2×D61—81no—
192C2×D61—81no—
193#6923—162no—
194C4×C61—21no—
195C4×C61—21no—
196C4×C61—22no—
197C4×C61—41no—
198C4×C61—81no—
199Dic61—41no—
200Dic61—41no—
201Dic61—81no—
202Dic61—81no—
203#2535—611no—
204#1799—611no—
205#1459—611no—
206#1344—612no—
207#1343—612no—
208C2×C61—21no—
209C2×C61—21no—
210C2×C61—82no—
211C2×C61#151911yescharacteristic
212C2×C61#212011yescharacteristic
213C2×C61#212711yescharacteristic
214D61—41no—
215D61—42no—
216D61—41no—
217C2×#121—1221no—
218C2×C4×D4—611no—
219C2×C4×D4#692311yescharacteristic, complemented
220C2×#99—1221no—
221C2×C22⋊C8—1221no—
222#397—611no—
223#397—611no—
224#392—611no—
225#392—611no—
226#386—1221no—
227#386—1221no—
228#382—611no—
229#380—1221no—
230#380—1221no—
231#342—612no—
232#324—1221no—
233#323—612no—
234C61#1187511yescharacteristic, complemented, Sylow
235C22×D4—611no—
236C22×D4—1221no—
237C22×D4—1221no—
238C22×D4C2×#692311yescharacteristic
239C23×C4—21no—
240C23×C4—611no—
241C23×C4—611no—
242C22×C8—2441no—
243#121—2442no—
244#121—2442no—
245C4×D4—41no—
246C4×D4—41no—
247C4×D4—2441no—
248C4×D4—2441no—
249C2×C4⋊C4—611no—
250C2×C4⋊C4—1221no—
251C2×C4⋊C4—2441no—
252C2×C4⋊C4C2×#692311yescharacteristic
253C2×C22⋊C4—21no—
254C2×C22⋊C4—611no—
255C2×C22⋊C4—611no—
256C2×C22⋊C4—1221no—
257C2×C22⋊C4—1221no—
258C2×C22⋊C4—1221no—
259C2×C22⋊C4—1221no—
260C2×C22⋊C4—1222no—
261C2×C42C2×#692311yescharacteristic
262#99—2442no—
263C22⋊C8—1222no—
264C22⋊C8—2441no—
265#95—611no—
266#95—611no—
267#95—1221no—
268#95—1221no—
269#95—1221no—
270#95—2441no—
271C24—21no—
272C24—611no—
273C24—611no—
274C2×D4—41no—
275C2×D4—42no—
276C2×D4—612no—
277C2×D4—612no—
278C2×D4—4881no—
279C2×D4—4881no—
280C2×D4—2441no—
281C2×D4—2441no—
282C2×D4—2442no—
283C2×D4—2442no—
284C2×D4—2442no—
285C2×D4—2441no—
286C2×D4—2441no—
287C22×C4—21no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—611no—
292C22×C4—611no—
293C22×C4—1221no—
294C22×C4—1221no—
295C22×C4—1221no—
296C22×C4—1221no—
297C22×C4—1221no—
298C22×C4—1221no—
299C22×C4—4881no—
300C22×C4—4881no—
301C22×C4—2442no—
302C22×C4C22⋊#692311yescharacteristic
303C22×C4C22⋊#692311yescharacteristic
304C22×C4C22×#692311yescharacteristic
305C2×C8—4882no—
306C2×C8—2442no—
307C4⋊C4—41no—
308C4⋊C4—4882no—
309C4⋊C4—2441no—
310C4⋊C4—2442no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—4881no—
314C22⋊C4—4881no—
315C22⋊C4—2441no—
316C22⋊C4—2441no—
317C22⋊C4—2442no—
318C22⋊C4—2441no—
319C22⋊C4—2441no—
320C22⋊C4—2442no—
321C22⋊C4—2442no—
322C42—21no—
323C42—21no—
324C23—22no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—612no—
332C23—611no—
333C23—611no—
334C23—1221no—
335C23—1221no—
336C23—1221no—
337C23—1221no—
338C23—2441no—
339C23—2441no—
340C23—2441no—
341C23—2441no—
342C23C2×C22⋊#692311yescharacteristic
343D4—42no—
344D4—42no—
345D4—4882no—
346D4—4884no—
347D4—4882no—
348D4—2442no—
349D4—2442no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—84no—
359C2×C4—1221no—
360C2×C4—1221no—
361C2×C4—4881no—
362C2×C4—4881no—
363C2×C4—4882no—
364C2×C4—4882no—
365C2×C4—4882no—
366C2×C4—2442no—
367C2×C4—2442no—
368C2×C4—2441no—
369C2×C4—2441no—
370C2×C4—2441no—
371C2×C4—2441no—
372C2×C4—2441no—
373C2×C4—2441no—
374C2×C4—2441no—
375C2×C4—2441no—
376C2×C4#1451983511yescharacteristic
377C2×C4#1451983511yescharacteristic
378C2×C4#1451983912yes—
379C8—4882no—
380C22—21no—
381C22—21no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—1221no—
390C22—1222no—
391C22—1221no—
392C22—4881no—
393C22—4881no—
394C22—2441no—
395C22—2442no—
396C22—2441no—
397C22—2441no—
398C22—2441no—
399C22—2441no—
400C22—2441no—
401C22C2×#1451983511yescharacteristic
402C22C2×#1451983911yescharacteristic
403C22#1684076011yescharacteristic
404C4—22no—
405C4—21no—
406C4—21no—
407C4—41no—
408C4—81no—
409C4—9762no—
410C4—4881no—
411C4—4881no—
412C4—2441no—
413C4—2441no—
414C2—21no—
415C2—21no—
416C2—82no—
417C2—2441no—
418C2—2441no—
419C2—2442no—
420C2#2146125511yescharacteristic
421C2#2146207011yescharacteristic
422C2#2146209411yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4×D4×C61C4
socleC22×C61C2×#99
prime radical#21450720C2
cosocleC2×#11758198C23
hypercenterC2×C4×D4#6923
hypocenterC61#11875
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #21461680C2semidirect
2. #21461056C2semidirect
3. #21460799C2semidirect
4. #21460797C2semidirect
14. #16839688C4semidirect
16. #16839670C4semidirect
20. #16839628C4semidirect
22. C2×C4×D4×C61C4semidirect
219. C2×C4×D4#6923semidirect
234. C61#11875semidirect

Maximal subgroups

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

subgroupindex
#214616802
#214610562
#214607992
#214607972
#214507202
#1187561