Finite Group Databasebeta

← #29577044

Subgroups — #29577044

Order 15616.

478 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 89 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#21456640C211yesmaximal, characteristic
2#21448982C211yesmaximal, characteristic
3#21444352C211yesmaximal, characteristic
4#21444338C211yesmaximal, characteristic
5#21443725C211yesmaximal, characteristic
6#21443672C211yesmaximal, characteristic
7C61×#1996C211yesmaximal, characteristic
8C2×#14519756C2211yescharacteristic
9#16840092C412yescomplemented
10#16840090—21no—
11#16840018—22no—
12#16839918C412yescomplemented
13#16839917—21no—
14#16839726C2211yescharacteristic
15#16839726C2211yescharacteristic
16#16839666—21no—
17#16839657—21no—
18#16839656—21no—
19#16839650—21no—
20#16839645—21no—
21#16839643—21no—
22#16839625C2211yescharacteristic
23#16839499—21no—
24#16839499—21no—
25#16839498—22no—
26C2×C22⋊Q8×C61C2211yescharacteristic
27C61×#380C2211yescharacteristic
28C4×C22⋊C4×C61C2211yescharacteristic
29C61×#342—21no—
30C61×#342—21no—
31C22×Dic122D411yescharacteristic
32C2×#11758202—22no—
33C2×#11758194—21no—
34C2×#11758194—21no—
35#14519756—21no—
36#14519756—21no—
37#14519756C2×C412yes—
38#14519756D412yes—
39C2×C4⋊Dic61D411yescharacteristic
40C2×#11758196—21no—
41C2×#11758196—21no—
42C2×#11758196—21no—
43C2×C22⋊Dic61—21no—
44C2×C22⋊Dic61—21no—
45C2×C4×Dic61—21no—
46C2×C4×Dic61—21no—
47#14519708—22no—
48#14519701—22no—
49#14519700—22no—
50C22⋊#92418—21no—
51C22⋊#92418—21no—
52C22⋊#92418—21no—
53C22⋊#92418C2×C412yes—
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57#14519684—21no—
58#14519684—21no—
59#14519684D411yescharacteristic
60#14519684D411yescharacteristic
61C22×Q8×C61C2×C411yescharacteristic
62C23×C4×C61C2311yescharacteristic
63C22⋊Q8×C61—21no—
64C22⋊Q8×C61—21no—
65C22⋊Q8×C61—21no—
66C22⋊Q8×C61—21no—
67C2×C4⋊C4×C61—21no—
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61C2×C411yescharacteristic
70C2×C22⋊C4×C61—21no—
71C2×C22⋊C4×C61—21no—
72C2×C22⋊C4×C61D411yescharacteristic
73C2×C22⋊C4×C61D411yescharacteristic
74C2×C42×C61—21no—
75#95×C61—21no—
76#95×C61—21no—
77C24×C61C2×D411yescharacteristic
78C2×Q8×C61—21no—
79C2×Q8×C61—21no—
80C2×Q8×C61—81no—
81C22×C4×C61—21no—
82C22×C4×C61—21no—
83C22×C4×C61—21no—
84C22×C4×C61—21no—
85C22×C4×C61—41no—
86C22×C4×C61—81no—
87C22×C4×C61C22×C411yescharacteristic
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61#4111yescharacteristic
92C22×C4×C61#4111yescharacteristic
93C22×Dic61—21no—
94C22×Dic61—21no—
95C22×Dic61—21no—
96C22×Dic61—21no—
97C2×Dic122—21no—
98C2×Dic122—81no—
99C2×Dic122C22⋊C412yes—
100C4⋊C4×C61—21no—
101C4⋊C4×C61—21no—
102C4⋊C4×C61—41no—
103C4⋊C4×C61—41no—
104C4⋊C4×C61—81no—
105C22⋊C4×C61—22no—
106C22⋊C4×C61—41no—
107C22⋊C4×C61—41no—
108C22⋊C4×C61—41no—
109C22⋊C4×C61—41no—
110C22⋊C4×C61—41no—
111C42×C61—81no—
112C22⋊Dic61—22no—
113C22⋊Dic61—22no—
114C22⋊Dic61—81no—
115#11758202—82no—
116C4⋊Dic61—21no—
117C4⋊Dic61C22⋊C412yes—
118#11758196—22no—
119#11758196—22no—
120#11758196—81no—
121#11758196—81no—
122C4×Dic61—81no—
123C4×Dic61—81no—
124#11758194—81no—
125#11758194—81no—
126C2×#92418—22no—
127C2×#92418—41no—
128C23×C61—21no—
129C23×C61—21no—
130C23×C61—41no—
131C23×C61—41no—
132C23×C61C4×D411yescharacteristic
133C23×C61#12011yescharacteristic
134C23×C61#12111yescharacteristic
135Q8×C61—42no—
136Q8×C61—81no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—42no—
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×C4×C61—81no—
153C2×C4×C61—81no—
154C2×C4×C61—81no—
155C2×C4×C61—81no—
156C2×C4×C61C2×C22⋊C411yescharacteristic
157C2×C4×C61C4×D411yescharacteristic
158C2×C4×C61#12111yescharacteristic
159C2×C4×C61#12311yescharacteristic
160C2×Dic61—22no—
161C2×Dic61—22no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61—41no—
171C2×Dic61—41no—
172Dic122—42no—
173Dic122—81no—
174#92418—42no—
175#92418—81no—
176#9342—611nomaximal, Sylow
177C22×C61—21no—
178C22×C61—21no—
179C22×C61—21no—
180C22×C61—21no—
181C22×C61—21no—
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—81no—
186C22×C61—81no—
187C22×C61#38611yescharacteristic
188C22×C61#45511yescharacteristic
189C22×C61#45611yescharacteristic
190C4×C61—21no—
191C4×C61—21no—
192C4×C61—21no—
193C4×C61—21no—
194C4×C61—81no—
195C4×C61—81no—
196C4×C61—82no—
197C4×C61—81no—
198C4×C61—81no—
199Dic61—42no—
200Dic61—42no—
201Dic61—81no—
202Dic61—81no—
203#3195—611no—
204#2378—611no—
205#1996D6111yescharacteristic
206#1511—611no—
207#1508—611no—
208#1305—611no—
209#1284—611no—
210C2×C61—21no—
211C2×C61—21no—
212C2×C61—41no—
213C2×C61—41no—
214C2×C61#188211yescharacteristic
215C2×C61#190511yescharacteristic
216C2×C61#191311yescharacteristic
217C2×C22⋊Q8—611no—
218C2×C22⋊Q8C2×D6111yescharacteristic
219#483—612no—
220#483—1221no—
221#470—1222no—
222#451—612no—
223#451—1221no—
224#407—611no—
225#407—611no—
226#391—1221no—
227#389—1221no—
228#388—1221no—
229#388—1221no—
230#387—1221no—
231#387—1221no—
232#381—611no—
233#380C2×D6111yescharacteristic
234C4×C22⋊C4C2×D6111yescharacteristic
235#342—21no—
236#342—21no—
237#328—1222no—
238#328—1221no—
239#328—1221no—
240C61#934211yescharacteristic, complemented, Sylow
241C22×Q8—611no—
242C22×Q8C4×D6111yescharacteristic
243C23×C4C22×D6111yescharacteristic
244C2×Q16—1222no—
245C2×M16—1221no—
246C2×M16—1221no—
247C22⋊Q8—21no—
248C22⋊Q8—21no—
249C22⋊Q8—21no—
250C22⋊Q8—21no—
251C22⋊Q8—612no—
252C22⋊Q8—612no—
253C22⋊Q8—1221no—
254C22⋊Q8—1221no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—611no—
258C2×C4⋊C4—1221no—
259C2×C4⋊C4—1221no—
260C2×C4⋊C4—1221no—
261C2×C4⋊C4C4×D6111yescharacteristic
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—1221no—
265C2×C22⋊C4—1221no—
266C2×C22⋊C4C22⋊D6111yescharacteristic
267C2×C22⋊C4C22⋊D6111yescharacteristic
268C2×C42—21no—
269C2×C42—1221no—
270C2×C42—1221no—
271#106—1222no—
272#103—1222no—
273#103—1222no—
274C22⋊C8—612no—
275C22⋊C8—1221no—
276C22⋊C8—1221no—
277C22⋊C8—1221no—
278#95—21no—
279#95—21no—
280#95—611no—
281#95—611no—
282#95—1221no—
283#95—1221no—
284#95—1221no—
285#95—1221no—
286#95—1221no—
287C24C2×C22⋊D6111yescharacteristic
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—81no—
291C2×Q8—612no—
292C2×Q8—1221no—
293C2×Q8—4881no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—41no—
299C22×C4—81no—
300C22×C4—1221no—
301C22×C4—1221no—
302C22×C4—1221no—
303C22×C4—1221no—
304C22×C4C2×C4×D6111yescharacteristic
305C22×C4#1175822211yescharacteristic
306C22×C4D4×D6111yescharacteristic
307C22×C4D4×D6111yescharacteristic
308C22×C4D4×D6111yescharacteristic
309C22×C4#1175822411yescharacteristic
310Q16—4882no—
311M16—4881no—
312M16—4881no—
313C2×C8—1222no—
314C2×C8—2441no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—81no—
320C4⋊C4—612no—
321C4⋊C4—1222no—
322C4⋊C4—1222no—
323C4⋊C4—1221no—
324C4⋊C4—4881no—
325C4⋊C4—4881no—
326C22⋊C4—22no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—1222no—
333C22⋊C4—1222no—
334C22⋊C4—4881no—
335C42—81no—
336C42—4881no—
337C42—4881no—
338C23—21no—
339C23—21no—
340C23—41no—
341C23—41no—
342C23C4×C22⋊D6111yescharacteristic
343C23#1451974511yescharacteristic
344C23#1451974811yescharacteristic
345Q8—42no—
346Q8—81no—
347Q8—4881no—
348Q8—2442no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—1222no—
369C2×C4—1222no—
370C2×C4—2441no—
371C2×C4—2441no—
372C2×C4—2441no—
373C2×C4—2441no—
374C2×C4—2441no—
375C2×C4—2441no—
376C2×C4—2441no—
377C2×C4—2441no—
378C2×C4—2441no—
379C2×C4—2441no—
380C2×C4C22⋊C4×D6111yescharacteristic
381C2×C4#1451973711yescharacteristic
382C2×C4#1451975011yescharacteristic
383C2×C4#1451975911yescharacteristic
384C8—4881no—
385C8—2442no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—81no—
395C22—81no—
396C22#1683963811yescharacteristic
397C22#1683993911yescharacteristic
398C22#1683994411yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—81no—
404C4—81no—
405C4—82no—
406C4—81no—
407C4—81no—
408C4—4881no—
409C4—4881no—
410C4—2442no—
411C4—2442no—
412C2—21no—
413C2—21no—
414C2—41no—
415C2—41no—
416C2#2144619111yescharacteristic
417C2#2144631611yescharacteristic
418C2#2144635811yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1996C2
socleC22×C61#386
prime radicalC24×C61C2×D4
cosocleC23×C4×C61C23
hypercenter#1996D61
hypocenterC61#9342
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
9. #16840092C4semidirect
12. #16839918C4semidirect
240. C61#9342semidirect

Maximal subgroups

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

subgroupindex
#214566402
#214489822
#214443522
#214443382
#214437252
#214436722
C61×#19962
#934261