Finite Group Databasebeta

← #30604976

Subgroups — #30604976

Order 15616.

586 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 70 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#21461516C211yesmaximal, characteristic
2#21461065C211yesmaximal, characteristic
3#21460814C212yesmaximal, complemented
4#21460800C212yesmaximal, complemented
5#21450973C211yesmaximal, characteristic
6C22×C22⋊#6923—21no—
7#16840733—21no—
8#16840640—21no—
9#16840640C2212yes—
10#16840606C2212yes—
11#16840605—21no—
12C2×#14519753C411yescharacteristic, complemented
13C2×C22⋊C4×D61C2211yescharacteristic
14#16839689C2211yescharacteristic
15#16839687C411yescharacteristic, complemented
16#16839673C2211yescharacteristic
17#16839651—21no—
18#16839650—21no—
19#16839624—21no—
20#16839617—21no—
21#16839609C411yescharacteristic, complemented
22C61×#393C411yescharacteristic, complemented
23C23×#6923—41no—
24C2×#11758213—41no—
25C2×C22⋊#6923—24no—
26C2×C22⋊#6923—22no—
27C2×C22⋊#6923—41no—
28C2×C22⋊#6923—42no—
29#14519834—41no—
30C22⋊#92420—22no—
31C24×D61D411yescharacteristic
32C22×C4×D61C2×C411yescharacteristic
33C22×C4×D61D411yescharacteristic
34#14519753—41no—
35#14519753—41no—
36C2×C4⋊Dic61C2×C411yescharacteristic, complemented
37C2×#11758196—21no—
38C2×C22⋊Dic61—21no—
39C2×C22⋊Dic61D411yescharacteristic
40C2×#11758198—21no—
41C2×#11758198—21no—
42C2×#11758198—21no—
43C2×#11758198—21no—
44C2×#11758198D411yescharacteristic
45C2×#11758198C2311yescharacteristic
46C22⋊C4×D61—41no—
47C22⋊C4×D61—41no—
48C2×C4×Dic61—21no—
49#14519684—21no—
50#14519684—21no—
51#14519684—21no—
52#14519684C2×C411yescharacteristic, complemented
53C22×Q8×C61C2×C411yescharacteristic, complemented
54C2×C22⋊C4×C61—21no—
55C2×C22⋊C4×C61C2×C411yescharacteristic
56#95×C61—21no—
57#95×C61C2×C411yescharacteristic, complemented
58C24×C61C22⋊C411yescharacteristic
59C23×D61—22no—
60C23×D61—21no—
61C23×D61—21no—
62C23×D61—21no—
63C23×D61—21no—
64C23×D61—42no—
65C23×D61—81no—
66C23×D61C22⋊C411yescharacteristic
67C23×D61C2×D411yescharacteristic
68C22×#6923—44no—
69C22×#6923—42no—
70C22×#6923—82no—
71C22×#6923—82no—
72C2×Q8×C61—41no—
73C2×Q8×C61—81no—
74C22×C4×C61—21no—
75C22×C4×C61—21no—
76C22×C4×C61C22⋊C411yescharacteristic
77C22×C4×C61C22×C411yescharacteristic
78C22×Dic61—21no—
79C22×Dic61—21no—
80C22×Dic61C22⋊C411yescharacteristic
81C22×Dic61C2×D411yescharacteristic
82C2×C4×D61—21no—
83C2×C4×D61—21no—
84C2×C4×D61—21no—
85C2×C4×D61—21no—
86C2×C4×D61—42no—
87C2×C4×D61—81no—
88C22⋊#6923—24no—
89C22⋊#6923—44no—
90C22⋊#6923—42no—
91C2×#92420—42no—
92#11758213—82no—
93C22⋊C4×C61—41no—
94C22⋊C4×C61—81no—
95C22⋊Dic61—41no—
96C22⋊Dic61—42no—
97#11758198—41no—
98#11758198—41no—
99#11758198—42no—
100#11758198—41no—
101#11758198—41no—
102#11758198—41no—
103#11758198—41no—
104#11758198—81no—
105C4⋊Dic61—41no—
106#11758196—42no—
107C4×Dic61—81no—
108C23×C61—21no—
109C23×C61—21no—
110C23×C61—41no—
111C23×C61—41no—
112C23×C61#10412yes—
113C23×C61C2×C22⋊C411yescharacteristic
114C22×D61—22no—
115C22×D61—22no—
116C22×D61—21no—
117C22×D61—21no—
118C22×D61—21no—
119C22×D61—22no—
120C22×D61—22no—
121C22×D61—21no—
122C22×D61—22no—
123C22×D61—41no—
124C22×D61—42no—
125C22×D61—42no—
126C22×D61—41no—
127C22×D61—41no—
128C22×D61—81no—
129C22×D61—81no—
130C22×D61—82no—
131C22×D61—82no—
132C22×D61—84no—
133C22×D61#9911yescharacteristic
134C22×D61#9911yescharacteristic
135C22×D61AGL(1,ℤ/8)11yescharacteristic
136C22×D61#13711yescharacteristic
137C2×#6923—44no—
138C2×#6923—88no—
139C2×#6923—82no—
140C2×#6923—82no—
141Q8×C61—82no—
142C2×C4×C61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
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—41no—
161C2×Dic61—41no—
162C4×D61—42no—
163C4×D61—81no—
164C4×D61—81no—
165C4×D61—81no—
166#92420—82no—
167#11759—611nomaximal, Sylow
168C22×C61—21no—
169C22×C61—21no—
170C22×C61—22no—
171C22×C61—42no—
172C22×C61—41no—
173C22×C61—41no—
174C22×C61—81no—
175C22×C61C2×#9911yescharacteristic
176C22×C61#41711yescharacteristic
177C22×C61C2×#10411yescharacteristic
178C2×D61—21no—
179C2×D61—21no—
180C2×D61—21no—
181C2×D61—21no—
182C2×D61—21no—
183C2×D61—21no—
184C2×D61—21no—
185C2×D61—42no—
186C2×D61—42no—
187C2×D61—44no—
188C2×D61—44no—
189C2×D61—41no—
190C2×D61—41no—
191C2×D61—41no—
192C2×D61—41no—
193C2×D61—81no—
194C2×D61—82no—
195C2×D61—82no—
196C2×D61#35412yes—
197#6923—84no—
198C4×C61—41no—
199C4×C61—41no—
200C4×C61—81no—
201C4×C61—81no—
202Dic61—41no—
203Dic61—41no—
204Dic61—81no—
205Dic61—81no—
206#2561—611no—
207C2×#342—611no—
208#1462—611no—
209#1351—612no—
210#1344—612no—
211C2×C61—21no—
212C2×C61—21no—
213C2×C61—42no—
214C2×C61#150411yescharacteristic
215C2×C61#212111yescharacteristic
216C2×C61C2×#35411yescharacteristic
217D61—22no—
218D61—41no—
219D61—41no—
220D61—41no—
221C2×C22⋊Q8—611no—
222C22×C22⋊C4—611no—
223C22×C22⋊C4—1221no—
224#407—1221no—
225#397—611no—
226#397—611no—
227#393—611no—
228#393#692311yescharacteristic, complemented
229#388—1221no—
230#388—1221no—
231#381—1221no—
232#379—1221no—
233C4×C22⋊C4—611no—
234#342—612no—
235#342—1221no—
236#323—612no—
237#323—1221no—
238C61#1175911yescharacteristic, complemented, Sylow
239C25—611no—
240C22×Q8C2×#692311yescharacteristic, complemented
241C23×C4—611no—
242C23×C4—611no—
243C23×C4—2441no—
244C2×M16—2441no—
245C22⋊Q8—2441no—
246C22⋊Q8—2441no—
247C2×C4⋊C4—611no—
248C2×C4⋊C4—1221no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—611no—
251C2×C22⋊C4—611no—
252C2×C22⋊C4—611no—
253C2×C22⋊C4—1222no—
254C2×C22⋊C4—1224no—
255C2×C22⋊C4—1221no—
256C2×C22⋊C4—1221no—
257C2×C22⋊C4—1221no—
258C2×C22⋊C4—1221no—
259C2×C22⋊C4—1221no—
260C2×C22⋊C4—2441no—
261C2×C22⋊C4—2441no—
262C2×C22⋊C4—2442no—
263C2×C22⋊C4—2441no—
264C2×C22⋊C4C2×#692311yescharacteristic
265C2×C42—1221no—
266C22⋊C8—1222no—
267C22⋊C8—2441no—
268#95—21no—
269#95—611no—
270#95—1221no—
271#95—1221no—
272#95—1221no—
273#95C2×#692311yescharacteristic, complemented
274C24—611no—
275C24—611no—
276C24—1221no—
277C24—1221no—
278C24—1221no—
279C24—1221no—
280C24—1222no—
281C24—4881no—
282C24—2442no—
283C24C22⋊#692311yescharacteristic
284C2×Q8—41no—
285C2×Q8—81no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—611no—
289C22×C4—611no—
290C22×C4—1221no—
291C22×C4—1221no—
292C22×C4—1221no—
293C22×C4—1221no—
294C22×C4—1221no—
295C22×C4—1221no—
296C22×C4—4881no—
297C22×C4—4882no—
298C22×C4—4882no—
299C22×C4—2442no—
300C22×C4—2444no—
301C22×C4—2442no—
302C22×C4C22⋊#692311yescharacteristic
303C22×C4C22×#692311yescharacteristic
304M16—4882no—
305C2×C8—2442no—
306C4⋊C4—2441no—
307C4⋊C4—2442no—
308C22⋊C4—41no—
309C22⋊C4—81no—
310C22⋊C4—1224no—
311C22⋊C4—4881no—
312C22⋊C4—2441no—
313C22⋊C4—2441no—
314C22⋊C4—2441no—
315C22⋊C4—2441no—
316C22⋊C4—2441no—
317C22⋊C4—2441no—
318C22⋊C4—2442no—
319C22⋊C4—2441no—
320C22⋊C4—2442no—
321C22⋊C4—2444no—
322C22⋊C4—2442no—
323C42—4881no—
324C23—21no—
325C23—21no—
326C23—41no—
327C23—41no—
328C23—611no—
329C23—611no—
330C23—611no—
331C23—611no—
332C23—1222no—
333C23—1222no—
334C23—1222no—
335C23—1222no—
336C23—1222no—
337C23—1221no—
338C23—1221no—
339C23—1221no—
340C23—1221no—
341C23—4881no—
342C23—4881no—
343C23—4884no—
344C23—4882no—
345C23—4882no—
346C23—2441no—
347C23—2441no—
348C23—2441no—
349C23—2442no—
350C23—2442no—
351C23#1451984212yes—
352C23C2×C22⋊#692311yescharacteristic
353Q8—82no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—1221no—
365C2×C4—1221no—
366C2×C4—4881no—
367C2×C4—4881no—
368C2×C4—4881no—
369C2×C4—4888no—
370C2×C4—4882no—
371C2×C4—4882no—
372C2×C4—2442no—
373C2×C4—2441no—
374C2×C4—2441no—
375C2×C4—2444no—
376C2×C4—2441no—
377C2×C4—2441no—
378C2×C4—2441no—
379C2×C4—2441no—
380C2×C4—2441no—
381C2×C4—2441no—
382C8—4882no—
383C22—21no—
384C22—21no—
385C22—22no—
386C22—42no—
387C22—41no—
388C22—41no—
389C22—81no—
390C22—612no—
391C22—1221no—
392C22—1221no—
393C22—1221no—
394C22—1221no—
395C22—1221no—
396C22—1221no—
397C22—1221no—
398C22—4881no—
399C22—4882no—
400C22—4882no—
401C22—2442no—
402C22—2442no—
403C22—2441no—
404C22—2441no—
405C22—2444no—
406C22—2444no—
407C22—2441no—
408C22—2441no—
409C22D61⋊#9911yescharacteristic
410C22#1684075311yescharacteristic
411C22C2×#1451984211yescharacteristic
412C4—41no—
413C4—41no—
414C4—81no—
415C4—81no—
416C4—4881no—
417C4—4884no—
418C4—4881no—
419C4—2441no—
420C4—2441no—
421C2—21no—
422C2—21no—
423C2—42no—
424C2—1222no—
425C2—2441no—
426C2—2441no—
427C2—2441no—
428C2#2146120611yescharacteristic
429C2#2146207511yescharacteristic
430C2D61⋊#35411yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#393C4
socleC22×C61C2×#99
prime radicalC24×D61D4
cosocleC2×#11758198C23
hypercenter#393#6923
hypocenterC61#11759
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #393

Complements

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

NH ≅ G/Ntype
3. #21460814C2semidirect
4. #21460800C2semidirect
12. C2×#14519753C4semidirect
15. #16839687C4semidirect
21. #16839609C4semidirect
22. C61×#393C4semidirect
36. C2×C4⋊Dic61C2×C4semidirect
52. #14519684C2×C4semidirect
53. C22×Q8×C61C2×C4semidirect
57. #95×C61C2×C4semidirect
228. #393#6923semidirect
238. C61#11759semidirect
240. C22×Q8C2×#6923semidirect
273. #95C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214615162
#214610652
#214608142
#214608002
#214509732
#1175961