Finite Group Databasebeta

← #29577207

Subgroups — #29577207

Order 15616.

542 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 99 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#21455759C211yesmaximal, characteristic
2#21448712C211yesmaximal, characteristic, complemented
3#21446655C211yesmaximal, characteristic
4#21444336C211yesmaximal, characteristic
5#21443726C211yesmaximal, characteristic, complemented
6#21443705C211yesmaximal, characteristic, complemented
7C61×#1826C211yesmaximal, characteristic, complemented
8C2×#14519751C2211yescharacteristic
9#16840086—22no—
10#16840080—21no—
11#16840075—21no—
12#16839805C412yescomplemented
13#16839766C412yescomplemented
14C2×#14519691—22no—
15#16839726C2211yescharacteristic
16#16839661—21no—
17#16839655—22no—
18#16839630—21no—
19#16839622C2211yescharacteristic
20C4⋊C4×Dic61—21no—
21C4×C4⋊Dic61—21no—
22C4×C22⋊Dic61C2211yescharacteristic
23#16839497—21no—
24#16839497—21no—
25C2×#117×C61C2211yescharacteristic
26C61×#407C2211yescharacteristic
27C4×C22⋊C4×C61C2211yescharacteristic
28C61×#339—22no—
29C22×D244C2×C411yescharacteristic
30C22×C22⋊D61—21no—
31C2×#11758194C2×C412yes—
32#14519751—21no—
33#14519751—21no—
34#14519751—21no—
35#14519751—21no—
36C2×C4⋊Dic61—22no—
37C2×C4⋊Dic61—21no—
38C2×C4⋊Dic61C2×C411yescharacteristic
39C2×C22⋊Dic61—21no—
40C2×C22⋊Dic61D412yes—
41C2×#11758198—21no—
42C2×C4×Dic61—21no—
43C2×C4×Dic61—22no—
44#14519710—22no—
45#14519708—22no—
46#14519697—22no—
47#14519697—22no—
48#14519691—44no—
49C22⋊#92418—21no—
50C22⋊#92418—21no—
51#14519684—22no—
52#14519684—22no—
53#14519684—21no—
54#14519684D411yescharacteristic
55#14519684D411yescharacteristic
56C23×C4×C61C2311yescharacteristic
57C2×M16×C61—21no—
58#117×C61—21no—
59#117×C61—21no—
60#117×C61C2×C412yes—
61#117×C61D412yes—
62C2×C4⋊C4×C61D411yescharacteristic
63C2×C4⋊C4×C61D411yescharacteristic
64C2×C22⋊C4×C61—21no—
65C2×C22⋊C4×C61—21no—
66C2×C42×C61—21no—
67C2×C42×C61—21no—
68C22⋊C8×C61—21no—
69C22⋊C8×C61—21no—
70#95×C61—21no—
71C24×C61C2×D411yescharacteristic
72C23×D61—21no—
73C22×C4×C61—21no—
74C22×C4×C61—21no—
75C22×C4×C61—42no—
76C22×C4×C61—41no—
77C22×C4×C61C22×C411yescharacteristic
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61C2×D411yescharacteristic
80C22×C4×C61C2×D411yescharacteristic
81C22×C4×C61#4111yescharacteristic
82C22×C4×C61#4111yescharacteristic
83C2×C22⋊D61—42no—
84C2×C22⋊D61—42no—
85C2×C22⋊D61—41no—
86C2×C22⋊D61—41no—
87C22×Dic61—21no—
88C22×Dic61—21no—
89C22×Dic61—21no—
90C22×Dic61—22no—
91C2×D244—21no—
92C2×D244—21no—
93C2×D244—42no—
94M16×C61—42no—
95C2×C8×C61—41no—
96C4⋊C4×C61—21no—
97C4⋊C4×C61—21no—
98C4⋊C4×C61C22⋊C412yes—
99C4⋊C4×C61C22⋊C412yes—
100C22⋊C4×C61—22no—
101C22⋊C4×C61—22no—
102C22⋊C4×C61—22no—
103C22⋊C4×C61—41no—
104C42×C61—22no—
105C42×C61—22no—
106C42×C61—42no—
107C22⋊Dic61—42no—
108C22⋊Dic61—42no—
109#11758198—41no—
110#11758198—41no—
111C4⋊Dic61—21no—
112C4⋊Dic61—21no—
113C4⋊Dic61—44no—
114C4⋊Dic61—42no—
115C4×Dic61—42no—
116C4×Dic61—82no—
117#11758194—42no—
118C2×#92418—22no—
119C23×C61—21no—
120C23×C61—21no—
121C23×C61—41no—
122C23×C61—41no—
123C23×C61C4×D411yescharacteristic
124C23×C61#12011yescharacteristic
125C23×C61#12111yescharacteristic
126C22×D61—41no—
127C22×D61—42no—
128C22×D61—81no—
129C22×D61—81no—
130C2×C4×C61—22no—
131C2×C4×C61—22no—
132C2×C4×C61—22no—
133C2×C4×C61—22no—
134C2×C4×C61—22no—
135C2×C4×C61—41no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—42no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61—42no—
142C2×C4×C61—81no—
143C2×C4×C61—81no—
144C2×C4×C61C2×C22⋊C411yescharacteristic
145C2×C4×C61C4×D411yescharacteristic
146C2×C4×C61#12111yescharacteristic
147C2×C4×C61#12311yescharacteristic
148C22⋊D61—82no—
149C22⋊D61—81no—
150C22⋊D61—81no—
151C2×Dic61—22no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—42no—
156C2×Dic61—42no—
157C2×Dic61—41no—
158C2×Dic61—42no—
159C2×Dic61—41no—
160C2×Dic61—42no—
161C2×Dic61—41no—
162D244—42no—
163D244—81no—
164C8×C61—81no—
165#92418—42no—
166#9366—611nomaximal, Sylow
167C22×C61—21no—
168C22×C61—21no—
169C22×C61—21no—
170C22×C61—21no—
171C22×C61—21no—
172C22×C61—21no—
173C22×C61—21no—
174C22×C61—21no—
175C22×C61—81no—
176C22×C61—81no—
177C22×C61#38611yescharacteristic
178C22×C61#45311yescharacteristic
179C22×C61#45411yescharacteristic
180C2×D61—81no—
181C2×D61—81no—
182C2×D61—81no—
183C2×D61—162no—
184C4×C61—21no—
185C4×C61—21no—
186C4×C61—21no—
187C4×C61—42no—
188C4×C61—42no—
189C4×C61—82no—
190C4×C61#41012yes—
191Dic61—42no—
192Dic61—82no—
193Dic61—81no—
194Dic61—81no—
195#3094—611no—
196#2347—611no—
197#1964—611no—
198#1826D6111yescharacteristic, complemented
199#1507—611no—
200#1305—611no—
201#1297—611no—
202C2×C61—21no—
203C2×C61—21no—
204C2×C61—41no—
205C2×C61—41no—
206C2×C61#188311yescharacteristic
207C2×C61#189911yescharacteristic
208C2×C61#190611yescharacteristic
209D61—161no—
210C2×#121—611no—
211C2×#117C2×D6111yescharacteristic
212#482—1222no—
213#481—1221no—
214#480—1221no—
215#428—612no—
216#418—612no—
217C2×#99—1222no—
218#407—611no—
219#407C2×D6111yescharacteristic
220#389—1221no—
221#388—1222no—
222#383—1221no—
223#380—611no—
224C4×C4⋊C4—1221no—
225C4×C4⋊C4—1221no—
226C4×C22⋊C4—611no—
227C4×C22⋊C4C2×D6111yescharacteristic
228#339—22no—
229#327—1221no—
230#327—1221no—
231C61#936611yescharacteristic, complemented, Sylow
232C22×D4—611no—
233C22×D4—1221no—
234C23×C4C22×D6111yescharacteristic
235C2×M16—21no—
236C2×M16—612no—
237#121—1221no—
238#121—1221no—
239#121—1221no—
240#121—1221no—
241#117—21no—
242#117—21no—
243#117C4×D6112yes—
244#117C22⋊D6112yes—
245C2×C4⋊C4—611no—
246C2×C4⋊C4—1222no—
247C2×C4⋊C4—1221no—
248C2×C4⋊C4D24411yescharacteristic
249C2×C4⋊C4D24411yescharacteristic
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—612no—
253C2×C22⋊C4—1221no—
254C2×C22⋊C4—1221no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—1222no—
258C2×C42—1221no—
259#107—1222no—
260#106—1222no—
261#102—1222no—
262#102—1222no—
263#99—2444no—
264C22⋊C8—21no—
265C22⋊C8—21no—
266C22⋊C8—1221no—
267C22⋊C8—1221no—
268#95—21no—
269#95—611no—
270#95—611no—
271#95—1222no—
272#95—1222no—
273#95—1221no—
274C24—1221no—
275C24D4×D6111yescharacteristic
276C2×D4—1221no—
277C2×D4—1221no—
278C2×D4—2442no—
279C2×D4—2442no—
280C2×D4—2442no—
281C2×D4—2441no—
282C2×D4—2441no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—1221no—
288C22×C4—1221no—
289C22×C4—1221no—
290C22×C4—1222no—
291C22×C4C2×C4×D6111yescharacteristic
292C22×C4C2×D24411yescharacteristic
293C22×C4D4×D6111yescharacteristic
294C22×C4#1175822411yescharacteristic
295C22×C4#1175822411yescharacteristic
296C22×C4C2×C22⋊D6111yescharacteristic
297M16—42no—
298M16—2442no—
299C2×C8—41no—
300C2×C8—1222no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—1221no—
304C4⋊C4—1221no—
305C4⋊C4—2442no—
306C4⋊C4—2444no—
307C4⋊C4#1175819812yes—
308C4⋊C4#1175819812yes—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—41no—
313C22⋊C4—2442no—
314C22⋊C4—2441no—
315C22⋊C4—2441no—
316C22⋊C4—2442no—
317C42—22no—
318C42—22no—
319C42—42no—
320C42—4882no—
321C42—2442no—
322C23—21no—
323C23—21no—
324C23—41no—
325C23—41no—
326C23—4881no—
327C23—4881no—
328C23—2442no—
329C23—2441no—
330C23#1451973711yescharacteristic
331C23#1451974711yescharacteristic
332C23#1451974911yescharacteristic
333D4—4881no—
334D4—4882no—
335D4—4881no—
336D4—4881no—
337D4—2442no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—1222no—
353C2×C4—2441no—
354C2×C4—2441no—
355C2×C4—2442no—
356C2×C4—2442no—
357C2×C4—2442no—
358C2×C4—2441no—
359C2×C4—2441no—
360C2×C4—2442no—
361C2×C4—2441no—
362C2×C4—2441no—
363C2×C4C2×#1175819811yescharacteristic
364C2×C4#1451973711yescharacteristic
365C2×C4#1451974911yescharacteristic
366C2×C4#1451976111yescharacteristic
367C8—81no—
368C8—2442no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—81no—
378C22—81no—
379C22—9762no—
380C22—4881no—
381C22—4881no—
382C22—4881no—
383C22#1683963911yescharacteristic
384C22#1683993111yescharacteristic
385C22#1683993611yescharacteristic
386C4—21no—
387C4—21no—
388C4—21no—
389C4—42no—
390C4—42no—
391C4—82no—
392C4—4882no—
393C4—4881no—
394C4—4881no—
395C4—2442no—
396C4#1683973512yes—
397C2—21no—
398C2—21no—
399C2—41no—
400C2—41no—
401C2—9761no—
402C2#2144619311yescharacteristic
403C2#2144627511yescharacteristic
404C2#2144631811yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1826C2
socleC22×C61#386
prime radicalC2×#14519751C22
cosocleC23×C4×C61C23
hypercenter#1826D61
hypocenterC61#9366
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1826

Complements

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

NH ≅ G/Ntype
2. #21448712C2semidirect
5. #21443726C2semidirect
6. #21443705C2semidirect
7. C61×#1826C2semidirect
12. #16839805C4semidirect
13. #16839766C4semidirect
198. #1826D61semidirect
231. C61#9366semidirect

Maximal subgroups

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

subgroupindex
#214557592
#214487122
#214466552
#214443362
#214437262
#214437052
C61×#18262
#936661