Finite Group Databasebeta

← #29577744

Subgroups — #29577744

Order 15616.

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

normal subgroups only

This group has 700 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#21449464C211yesmaximal, characteristic
2#21448404C211yesmaximal, characteristic
3#21445742C211yesmaximal, characteristic
4#21444352C211yesmaximal, characteristic
5#21443673C211yesmaximal, characteristic
6#21443671C211yesmaximal, characteristic
7C61×#1511C211yesmaximal, characteristic
8C23×Dic122C2211yescharacteristic
9C2×#14519775C412yescomplemented
10C2×#14519733—21no—
11C2×#14519695—22no—
12#16839726C2211yescharacteristic
13#16839682—22no—
14#16839667C412yescomplemented
15#16839666—22no—
16#16839666—22no—
17#16839634—21no—
18#16839634—21no—
19#16839625C2211yescharacteristic
20#16839623—21no—
21C4×C22⋊Dic61C2211yescharacteristic
22#16839499—22no—
23C2×C22⋊Q8×C61C2211yescharacteristic
24C61×#407C2211yescharacteristic
25C61×#381C2211yescharacteristic
26C61×#328—22no—
27C22×Dic122—21no—
28C22×Dic122—21no—
29C22×Dic122—21no—
30C22×Dic122—21no—
31C22×Dic122—44no—
32C22×Dic122C2×C412yes—
33C22×Dic122D412yes—
34C23×Dic61D411yescharacteristic
35C23×Dic61D411yescharacteristic
36C2×#11758194—21no—
37#14519775—44no—
38#14519733—44no—
39C2×#11758196—22no—
40C2×#11758196—22no—
41C2×#11758196—22no—
42C2×C22⋊Dic61—21no—
43C2×C22⋊Dic61—21no—
44C2×C4×Dic61—22no—
45C2×C4×Dic61C2×C412yes—
46#14519695—44no—
47C22⋊#92418—21no—
48C22⋊#92418—21no—
49#14519684—24no—
50#14519684—21no—
51#14519684—21no—
52#14519684—22no—
53#14519684—22no—
54#14519684D412yes—
55C22×Q8×C61C2×C411yescharacteristic
56C23×C4×C61C2311yescharacteristic
57C2×M16×C61—21no—
58C22⋊Q8×C61—21no—
59C22⋊Q8×C61—21no—
60C22⋊Q8×C61—21no—
61C22⋊Q8×C61—21no—
62C2×C4⋊C4×C61—21no—
63C2×C4⋊C4×C61C2×C411yescharacteristic
64C2×C22⋊C4×C61—21no—
65C2×C22⋊C4×C61—21no—
66C22⋊C8×C61—21no—
67C22⋊C8×C61—21no—
68#95×C61—21no—
69#95×C61D412yes—
70C24×C61C2×D411yescharacteristic
71C2×Q8×C61—21no—
72C2×Q8×C61—21no—
73C2×Q8×C61—42no—
74C22×C4×C61—21no—
75C22×C4×C61—21no—
76C22×C4×C61—22no—
77C22×C4×C61—42no—
78C22×C4×C61C22×C411yescharacteristic
79C22×C4×C61C2×D411yescharacteristic
80C22×C4×C61C2×D411yescharacteristic
81C22×C4×C61C2×D411yescharacteristic
82C22×C4×C61#4111yescharacteristic
83C22×C4×C61#4111yescharacteristic
84C22×Dic61—21no—
85C22×Dic61—21no—
86C22×Dic61—24no—
87C22×Dic61—22no—
88C22×Dic61—21no—
89C22×Dic61—21no—
90C22×Dic61—22no—
91C22×Dic61—42no—
92C22×Dic61C22⋊C412yes—
93C22×Dic61C22⋊C412yes—
94C2×Dic122—22no—
95C2×Dic122—22no—
96C2×Dic122—22no—
97C2×Dic122—22no—
98C2×Dic122—24no—
99C2×Dic122—24no—
100C2×Dic122—42no—
101C2×Dic122—42no—
102C2×Dic122—42no—
103C2×Dic122—42no—
104C2×Dic122—42no—
105C2×Dic122—42no—
106C2×Dic122—44no—
107C2×Dic122—44no—
108M16×C61—42no—
109C2×C8×C61—41no—
110C4⋊C4×C61—21no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61—41no—
113C4⋊C4×C61—41no—
114C22⋊C4×C61—42no—
115C22⋊C4×C61—41no—
116C22⋊C4×C61—41no—
117C22⋊Dic61—24no—
118C22⋊Dic61—42no—
119#11758196—44no—
120#11758196—44no—
121#11758196—44no—
122C4×Dic61—44no—
123C4×Dic61—42no—
124#11758194—42no—
125C2×#92418—41no—
126C23×C61—22no—
127C23×C61—22no—
128C23×C61—41no—
129C23×C61C2×C22⋊C411yescharacteristic
130C23×C61#12011yescharacteristic
131C23×C61#12311yescharacteristic
132Q8×C61—81no—
133Q8×C61—81no—
134C2×C4×C61—22no—
135C2×C4×C61—22no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—42no—
141C2×C4×C61—42no—
142C2×C4×C61—42no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—42no—
146C2×C4×C61—42no—
147C2×C4×C61C4×D411yescharacteristic
148C2×C4×C61C4×D411yescharacteristic
149C2×C4×C61#12111yescharacteristic
150C2×C4×C61#12111yescharacteristic
151C2×Dic61—22no—
152C2×Dic61—22no—
153C2×Dic61—24no—
154C2×Dic61—22no—
155C2×Dic61—22no—
156C2×Dic61—24no—
157C2×Dic61—22no—
158C2×Dic61—22no—
159C2×Dic61—22no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162C2×Dic61—42no—
163C2×Dic61—42no—
164C2×Dic61—42no—
165C2×Dic61—42no—
166C2×Dic61—42no—
167C2×Dic61—42no—
168C2×Dic61#9914yes—
169Dic122—44no—
170Dic122—44no—
171Dic122—44no—
172Dic122—44no—
173C8×C61—81no—
174#92418—81no—
175#9462—611nomaximal, Sylow
176C22×C61—21no—
177C22×C61—21no—
178C22×C61—21no—
179C22×C61—21no—
180C22×C61—21no—
181C22×C61—21no—
182C22×C61—21no—
183C22×C61—44no—
184C22×C61#38611yescharacteristic
185C22×C61C2×#9912yes—
186C22×C61#45612yes—
187C4×C61—21no—
188C4×C61—21no—
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—82no—
192C4×C61—81no—
193C4×C61—81no—
194Dic61—24no—
195Dic61—44no—
196Dic61—42no—
197Dic61—82no—
198#2434—611no—
199#2310—611no—
200#1784—611no—
201#1511—611no—
202#1511D6111yescharacteristic
203#1284—611no—
204#1284—611no—
205C2×C61—22no—
206C2×C61—22no—
207C2×C61—41no—
208C2×C61#188611yescharacteristic
209C2×C61#189811yescharacteristic
210C2×C61#191211yescharacteristic
211C23×Q8—611no—
212C2×C4⋊Q8—612no—
213C2×C22⋊Q8C2×D6111yescharacteristic
214C2×#117—1221no—
215C2×#101—1222no—
216#407—611no—
217#407C2×D6111yescharacteristic
218#395—1222no—
219#391—612no—
220#391—1222no—
221#391—1222no—
222#385—1221no—
223#385—1221no—
224#381—611no—
225#381—1221no—
226#381C2×D6111yescharacteristic
227C4×C22⋊C4—611no—
228#328—22no—
229#328—1222no—
230C61#946211yescharacteristic, complemented, Sylow
231C22×Q8—612no—
232C22×Q8—612no—
233C22×Q8—1221no—
234C22×Q8—1221no—
235C22×Q8—1221no—
236C22×Q8—1221no—
237C22×Q8—2444no—
238C22×Q8C4×D6111yescharacteristic
239C23×C4—611no—
240C23×C4—611no—
241C23×C4C22×D6111yescharacteristic
242C2×M16—21no—
243C2×M16—1221no—
244C4⋊Q8—2444no—
245C22⋊Q8—21no—
246C22⋊Q8—21no—
247C22⋊Q8—21no—
248C22⋊Q8—21no—
249#117—2444no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—1222no—
252C2×C4⋊C4—1222no—
253C2×C4⋊C4—1222no—
254C2×C4⋊C4C4×D6111yescharacteristic
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—1221no—
258C2×C22⋊C4—1221no—
259C2×C42—612no—
260C2×C42—1222no—
261#101—2444no—
262C22⋊C8—21no—
263C22⋊C8—21no—
264C22⋊C8—1221no—
265C22⋊C8—1221no—
266#95—21no—
267#95—612no—
268#95—1222no—
269#95—1222no—
270#95—1224no—
271#95—1221no—
272#95—1221no—
273#95C22⋊D6112yes—
274C24D4×D6111yescharacteristic
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—42no—
278C2×Q8—1224no—
279C2×Q8—1224no—
280C2×Q8—1222no—
281C2×Q8—1222no—
282C2×Q8—1222no—
283C2×Q8—1222no—
284C2×Q8—2442no—
285C2×Q8—2442no—
286C2×Q8—2444no—
287C2×Q8—2442no—
288C2×Q8—2442no—
289C2×Q8—2444no—
290C2×Q8—2442no—
291C2×Q8—2442no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—42no—
296C22×C4—612no—
297C22×C4—612no—
298C22×C4—1224no—
299C22×C4—1221no—
300C22×C4—1221no—
301C22×C4—1222no—
302C22×C4—1221no—
303C22×C4—1221no—
304C22×C4—1222no—
305C22×C4—2442no—
306C22×C4C2×C4×D6111yescharacteristic
307C22×C4#1175822211yescharacteristic
308C22×C4D4×D6111yescharacteristic
309C22×C4D4×D6111yescharacteristic
310C22×C4#1175822411yescharacteristic
311C22×C4C2×C22⋊D6111yescharacteristic
312M16—42no—
313M16—2442no—
314C2×C8—41no—
315C2×C8—2441no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—2444no—
321C4⋊C4—2444no—
322C4⋊C4—2444no—
323C22⋊C4—42no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—1224no—
327C22⋊C4—2442no—
328C42—2442no—
329C42—2444no—
330C23—22no—
331C23—22no—
332C23—41no—
333C23C22⋊C4×D6111yescharacteristic
334C23#1451974511yescharacteristic
335C23#1451975911yescharacteristic
336Q8—81no—
337Q8—81no—
338Q8—2444no—
339Q8—2444no—
340Q8—2444no—
341Q8—2444no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—614no—
356C2×C4—1222no—
357C2×C4—1222no—
358C2×C4—1224no—
359C2×C4—1222no—
360C2×C4—1222no—
361C2×C4—1224no—
362C2×C4—1222no—
363C2×C4—1222no—
364C2×C4—1222no—
365C2×C4—2442no—
366C2×C4—2442no—
367C2×C4—2442no—
368C2×C4—2442no—
369C2×C4—2441no—
370C2×C4—2441no—
371C2×C4—2442no—
372C2×C4—2442no—
373C2×C4#1451973711yescharacteristic
374C2×C4C4×C22⋊D6111yescharacteristic
375C2×C4#1451974811yescharacteristic
376C2×C4#1451975011yescharacteristic
377C8—81no—
378C8—4881no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—44no—
387C22#1683963811yescharacteristic
388C22#99×D6112yes—
389C22#1683994312yes—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—82no—
395C4—81no—
396C4—81no—
397C4—1224no—
398C4—4882no—
399C4—2444no—
400C4—2442no—
401C2—22no—
402C2—22no—
403C2—41no—
404C2#2144620911yescharacteristic
405C2#2144626911yescharacteristic
406C2#2144635311yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1511C2
socleC22×C61#386
prime radicalC24×C61C2×D4
cosocleC23×C4×C61C23
hypercenter#1511D61
hypocenterC61#9462
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1511

Complements

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

NH ≅ G/Ntype
9. C2×#14519775C4semidirect
14. #16839667C4semidirect
230. C61#9462semidirect

Maximal subgroups

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

subgroupindex
#214494642
#214484042
#214457422
#214443522
#214436732
#214436712
C61×#15112
#946261