Finite Group Databasebeta

← #29568439

Subgroups — #29568439

Order 15616.

488 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 97 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#21454157C211yesmaximal, characteristic
2#21445580C211yesmaximal, characteristic
3#21445576C211yesmaximal, characteristic
4#21443644C211yesmaximal, characteristic, complemented
5#21443644C211yesmaximal, characteristic, complemented
6#21443642C211yesmaximal, characteristic, complemented
7C61×#1271C211yesmaximal, characteristic, complemented
8C2×#14519757C411yescharacteristic
9C2×C42×D61C2211yescharacteristic
10C2×#14519689—21no—
11C2×#14519689—21no—
12C2×#14519687—21no—
13C2×C8×Dic61—21no—
14#16839668C2211yescharacteristic
15#16839667C411yescharacteristic
16#16839658C411yescharacteristic
17#16839649—21no—
18#16839648—21no—
19#16839633—21no—
20#16839624—21no—
21#16839619C411yescharacteristic
22#16839517C2211yescharacteristic
23#16839517C2211yescharacteristic
24C61×#389C2211yescharacteristic
25C61×#336C2211yescharacteristic
26C61×#336C2211yescharacteristic
27C22×Dic122C2×C411yescharacteristic
28C22×C4×D61C2×C411yescharacteristic
29C22×C4×D61D411yescharacteristic
30C22×C4×D61D411yescharacteristic
31C22×#92418—21no—
32C22×#92418—21no—
33#14519757—42no—
34C2×C4⋊Dic61—21no—
35C2×C4⋊Dic61C2×C411yescharacteristic
36C2×#11758196—21no—
37C2×#11758198—21no—
38C2×#11758198—21no—
39C2×C4×Dic61C2×C411yescharacteristic
40C2×C4×Dic61D411yescharacteristic
41C2×C4×Dic61D411yescharacteristic
42C42×D61—42no—
43#14519689—24no—
44#14519689—24no—
45#14519687—42no—
46C8×Dic61—42no—
47#14519684—21no—
48#14519684—21no—
49#14519684—21no—
50#14519684—21no—
51#14519684—21no—
52#14519684—21no—
53C22×C8×C61—21no—
54C22×C8×C61—21no—
55C2×C4⋊C4×C61—21no—
56C2×C4⋊C4×C61C2×C411yescharacteristic
57C2×C4⋊C4×C61C2×C411yescharacteristic
58C2×C42×C61C2311yescharacteristic
59#95×C61—21no—
60C23×D61C22⋊C411yescharacteristic
61C22×C4×C61—21no—
62C22×C4×C61—21no—
63C22×C4×C61C22×C411yescharacteristic
64C22×C4×C61C2×D411yescharacteristic
65C22×C4×C61C2×D411yescharacteristic
66C22×Dic61—21no—
67C22×Dic61—21no—
68C22×Dic61C22⋊C411yescharacteristic
69C22×Dic61C22⋊C411yescharacteristic
70C22×Dic61C22⋊C411yescharacteristic
71C2×C4×D61—22no—
72C2×C4×D61—21no—
73C2×C4×D61—21no—
74C2×C4×D61—21no—
75C2×C4×D61—21no—
76C2×C4×D61—21no—
77C2×C4×D61—21no—
78C2×C4×D61—22no—
79C2×C4×D61—22no—
80C2×C4×D61—41no—
81C2×C4×D61—41no—
82C2×C4×D61—41no—
83C2×Dic122—21no—
84C2×Dic122—21no—
85C2×Dic122—41no—
86C2×Dic122—41no—
87C2×C8×C61—22no—
88C2×C8×C61—22no—
89C2×C8×C61—41no—
90C2×C8×C61—41no—
91C2×C8×C61—41no—
92C2×C8×C61—41no—
93C4⋊C4×C61—21no—
94C4⋊C4×C61—21no—
95C4⋊C4×C61—41no—
96C4⋊C4×C61—41no—
97C4⋊C4×C61—41no—
98C42×C61—41no—
99#11758198—42no—
100#11758198—42no—
101C4⋊Dic61—42no—
102C4⋊Dic61—41no—
103#11758196—42no—
104C4×Dic61—22no—
105C4×Dic61—22no—
106C4×Dic61—41no—
107C2×#92418—22no—
108C2×#92418—22no—
109C2×#92418—41no—
110C2×#92418—41no—
111C2×#92418—41no—
112C2×#92418—41no—
113C23×C61C2×C22⋊C411yescharacteristic
114C22×D61—21no—
115C22×D61—21no—
116C22×D61—21no—
117C22×D61—21no—
118C22×D61—21no—
119C22×D61—21no—
120C22×D61#10412yes—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61#13111yescharacteristic
134C2×C4×C61#13111yescharacteristic
135C2×C4×C61#13511yescharacteristic
136C2×C4×C61#13511yescharacteristic
137C2×C4×C61#13711yescharacteristic
138C2×C4×C61#13711yescharacteristic
139C2×Dic61—21no—
140C2×Dic61—21no—
141C2×Dic61—21no—
142C2×Dic61—21no—
143C2×Dic61—21no—
144C2×Dic61—21no—
145C2×Dic61—21no—
146C2×Dic61—41no—
147C2×Dic61—41no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61—41no—
152C2×Dic61#9912yes—
153C2×Dic61#10412yes—
154C4×D61—42no—
155C4×D61—42no—
156C4×D61—42no—
157C4×D61—42no—
158C4×D61—42no—
159C4×D61—42no—
160Dic122—81no—
161Dic122—81no—
162C8×C61—42no—
163C8×C61—42no—
164#92418—42no—
165#92418—42no—
166#7675—611nomaximal, Sylow
167C22×C61#40811yescharacteristic
168C22×C61C2×#9911yescharacteristic
169C22×C61#41311yescharacteristic
170C22×C61#41611yescharacteristic
171C22×C61#41911yescharacteristic
172C22×C61C2×#10411yescharacteristic
173C22×C61C2×#10411yescharacteristic
174C2×D61—22no—
175C2×D61—22no—
176C2×D61—22no—
177C2×D61—41no—
178C2×D61—41no—
179C2×D61—41no—
180C2×D61—41no—
181C4×C61—21no—
182C4×C61—21no—
183C4×C61—21no—
184C4×C61—21no—
185C4×C61—21no—
186C4×C61—21no—
187C4×C61—81no—
188C4×C61—81no—
189Dic61—22no—
190Dic61—22no—
191Dic61—41no—
192Dic61—81no—
193Dic61—81no—
194#2855—611no—
195#1749—611no—
196C4×C22⋊C8—611no—
197#1272—611no—
198#1272—611no—
199#1271—611no—
200#1271D6111yescharacteristic, complemented
201C2×C61#146811yescharacteristic
202C2×C61#148511yescharacteristic
203C2×C61#149411yescharacteristic
204C2×C61#150111yescharacteristic
205C2×C61#150811yescharacteristic
206C2×C61#152411yescharacteristic
207C2×C61#155411yescharacteristic
208D61—42no—
209C2×C22⋊Q8—611no—
210C22×C42—611no—
211C2×C22⋊C8—1221no—
212C2×C22⋊C8—1221no—
213C2×#97—1221no—
214C2×C4×C8—1221no—
215#391—611no—
216#391—611no—
217#389—611no—
218#389C2×D6111yescharacteristic
219#387—1221no—
220#387—1221no—
221#385—1221no—
222#381—1221no—
223#380—611no—
224#336—611no—
225#336—611no—
226#336C2×D6111yescharacteristic
227#336C2×D6111yescharacteristic
228C61#767511yescharacteristic, complemented, Sylow
229C22×Q8—611no—
230C23×C4—611no—
231C23×C4—611no—
232C23×C4—611no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—1221no—
236C22×C8—1221no—
237C22⋊Q8—2442no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—611no—
240C2×C4⋊C4—1221no—
241C2×C4⋊C4—1221no—
242C2×C4⋊C4C4×D6111yescharacteristic
243C2×C4⋊C4C4×D6111yescharacteristic
244C2×C22⋊C4—1221no—
245C2×C22⋊C4—1221no—
246C2×C42—611no—
247C2×C42—611no—
248C2×C42—611no—
249C2×C42—2442no—
250C2×C42C22×D6111yescharacteristic
251C22⋊C8—1224no—
252C22⋊C8—1224no—
253#97—2442no—
254C4×C8—2442no—
255#95—21no—
256#95—1221no—
257#95—1221no—
258#95—1221no—
259#95—1221no—
260#95—1221no—
261#95—1221no—
262C24—611no—
263C2×Q8—1221no—
264C2×Q8—1221no—
265C2×Q8—2441no—
266C2×Q8—2441no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—611no—
270C22×C4—611no—
271C22×C4—611no—
272C22×C4—1222no—
273C22×C4—1222no—
274C22×C4—1222no—
275C22×C4—1221no—
276C22×C4—1221no—
277C22×C4—1221no—
278C22×C4—1221no—
279C22×C4—1221no—
280C22×C4—1221no—
281C22×C4—1221no—
282C22×C4—1221no—
283C22×C4—2441no—
284C22×C4—2441no—
285C22×C4—2441no—
286C22×C4C2×C4×D6111yescharacteristic
287C22×C4D4×D6111yescharacteristic
288C22×C4D4×D6111yescharacteristic
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—1222no—
296C2×C8—1222no—
297C2×C8—2441no—
298C2×C8—2441no—
299C2×C8—2441no—
300C2×C8—2441no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—2441no—
307C4⋊C4—2442no—
308C4⋊C4—2442no—
309C22⋊C4—2442no—
310C22⋊C4—2442no—
311C42—41no—
312C42—1222no—
313C42—1222no—
314C42—2441no—
315C23—612no—
316C23—1221no—
317C23—1221no—
318C23—1221no—
319C23—1221no—
320C23—1221no—
321C23—1221no—
322C23C22⋊C4×D6111yescharacteristic
323Q8—4881no—
324Q8—4881no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—612no—
338C2×C4—612no—
339C2×C4—1221no—
340C2×C4—1221no—
341C2×C4—1221no—
342C2×C4—1221no—
343C2×C4—1221no—
344C2×C4—1221no—
345C2×C4—1221no—
346C2×C4—2442no—
347C2×C4—2442no—
348C2×C4—2442no—
349C2×C4—2442no—
350C2×C4—2442no—
351C2×C4—2442no—
352C2×C4—2441no—
353C2×C4—2441no—
354C2×C4—2441no—
355C2×C4—2441no—
356C2×C4—2441no—
357C2×C4—2441no—
358C2×C4#1451978311yescharacteristic
359C2×C4#1451978511yescharacteristic
360C2×C4#1451979611yescharacteristic
361C2×C4#1451979811yescharacteristic
362C2×C4#1451980611yescharacteristic
363C2×C4#1451980911yescharacteristic
364C8—42no—
365C8—42no—
366C8—2442no—
367C8—2442no—
368C22—1222no—
369C22—1222no—
370C22—1222no—
371C22—2441no—
372C22—2441no—
373C22—2441no—
374C22—2441no—
375C22#1683972711yescharacteristic
376C22#99×D6111yescharacteristic
377C22#1683974511yescharacteristic
378C22#1683975511yescharacteristic
379C22#1683976911yescharacteristic
380C22#104×D6111yescharacteristic
381C22#104×D6111yescharacteristic
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—81no—
389C4—81no—
390C4—1222no—
391C4—1222no—
392C4—4881no—
393C4—4881no—
394C4—2441no—
395C2—2442no—
396C2#2144416611yescharacteristic
397C2#2144423811yescharacteristic
398C2#2144427611yescharacteristic
399C2#2144430711yescharacteristic
400C2#2144433711yescharacteristic
401C2#2144440611yescharacteristic
402C2#2144452611yescharacteristic
subgroupG/H
centerC23C22⋊C4×D61
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1271C2
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC2×C42×C61C23
hypercenter#1271D61
hypocenterC61#7675
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
4. #21443644C2semidirect
5. #21443644C2semidirect
6. #21443642C2semidirect
7. C61×#1271C2semidirect
200. #1271D61semidirect
228. C61#7675semidirect

Maximal subgroups

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

subgroupindex
#214541572
#214455802
#214455762
#214436442
#214436422
C61×#12712
#767561