Finite Group Databasebeta

← #29580468

Subgroups — #29580468

Order 15616.

610 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 109 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#21456572C211yesmaximal, characteristic
2C2×#16840080C211yesmaximal, characteristic
3#21446655C211yesmaximal, characteristic
4#21445909C211yesmaximal, characteristic, complemented
5#21444391C211yesmaximal, characteristic, complemented
6#21443667C211yesmaximal, characteristic, complemented
7C61×#2502C211yesmaximal, characteristic, complemented
8C2×#14519751C2211yescharacteristic
9#16840090—21no—
10#16840080—22no—
11#16840080—22no—
12#16840080—21no—
13#16839993—22no—
14#16839912—21no—
15#16839901—21no—
16C2×#14519708—21no—
17C2×#14519697—21no—
18C2×#14519691—22no—
19C2×C22⋊#92418C2211yescharacteristic
20C2×C22⋊#92418C2211yescharacteristic
21#16839622C2211yescharacteristic
22#16839524—22no—
23#16839499—22no—
24C2×C22⋊Q8×C61C2211yescharacteristic
25C22×C4⋊C4×C61C2211yescharacteristic
26C61×#398—22no—
27C61×#391—21no—
28C61×#385—22no—
29C61×#384—21no—
30C4×C22⋊C4×C61C2211yescharacteristic
31C22×D244D411yescharacteristic
32C22×C22⋊D61—21no—
33C2×#11758201—22no—
34C22×#92418—21no—
35C22×#92418D412yes—
36#14519751—21no—
37#14519751—21no—
38#14519751—21no—
39#14519751—21no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61D411yescharacteristic
42C2×C22⋊Dic61—21no—
43C2×#11758198—21no—
44#14519708—22no—
45#14519708—24no—
46#14519701—22no—
47#14519697—22no—
48#14519697—24no—
49#14519691—44no—
50C22⋊#92418—21no—
51C22⋊#92418—21no—
52C22⋊#92418—21no—
53C22⋊#92418D412yes—
54#14519684—21no—
55C22×Q8×C61D411yescharacteristic
56C23×C4×C61—21no—
57C23×C4×C61C2311yescharacteristic
58C22⋊Q8×C61—21no—
59C22⋊Q8×C61—21no—
60C22⋊Q8×C61—21no—
61C22⋊Q8×C61—21no—
62C2×C4⋊C4×C61—21no—
63C2×C4⋊C4×C61—21no—
64C2×C4⋊C4×C61—21no—
65C2×C4⋊C4×C61—21no—
66C2×C4⋊C4×C61—21no—
67C2×C4⋊C4×C61—21no—
68C2×C4⋊C4×C61—24no—
69C2×C4⋊C4×C61D411yescharacteristic
70C2×C4⋊C4×C61D412yes—
71C2×C22⋊C4×C61—21no—
72C2×C22⋊C4×C61D412yes—
73C2×C42×C61—21no—
74#95×C61—24no—
75#95×C61—21no—
76C24×C61C2×D411yescharacteristic
77C23×D61—21no—
78C2×Q8×C61—21no—
79C2×Q8×C61—21no—
80C2×Q8×C61—42no—
81C22×C4×C61—22no—
82C22×C4×C61—21no—
83C22×C4×C61—22no—
84C22×C4×C61—21no—
85C22×C4×C61—42no—
86C22×C4×C61—41no—
87C22×C4×C61—41no—
88C22×C4×C61—82no—
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61#4111yescharacteristic
95C2×C22⋊D61—41no—
96C2×C22⋊D61—41no—
97C2×C22⋊D61—42no—
98C2×C22⋊D61—42no—
99C22×Dic61—21no—
100C22×Dic61—41no—
101C2×D244—21no—
102C2×D244—21no—
103C2×D244—42no—
104C4⋊C4×C61—22no—
105C4⋊C4×C61—21no—
106C4⋊C4×C61—22no—
107C4⋊C4×C61—21no—
108C4⋊C4×C61—41no—
109C4⋊C4×C61—41no—
110C4⋊C4×C61—41no—
111C4⋊C4×C61—41no—
112C4⋊C4×C61—84no—
113C22⋊C4×C61—22no—
114C22⋊C4×C61—41no—
115C22⋊C4×C61—41no—
116C22⋊C4×C61SD1614yescomplemented
117C42×C61—42no—
118C22⋊Dic61—42no—
119#11758201—82no—
120#11758198—41no—
121#11758198—41no—
122C4⋊Dic61—21no—
123C4⋊Dic61—21no—
124C4⋊Dic61—42no—
125C2×#92418—22no—
126C2×#92418—22no—
127C2×#92418—42no—
128C2×#92418—42no—
129C23×C61—41no—
130C23×C61—41no—
131C23×C61#12011yescharacteristic
132C23×C61#12111yescharacteristic
133C23×C61#12711yescharacteristic
134C23×C61C2×SD1612yes—
135C23×C61#13712yes—
136C22×D61—41no—
137C22×D61—42no—
138C22×D61—81no—
139C22×D61—81no—
140Q8×C61—42no—
141Q8×C61—81no—
142C2×C4×C61—22no—
143C2×C4×C61—22no—
144C2×C4×C61—22no—
145C2×C4×C61—24no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61—42no—
150C2×C4×C61—42no—
151C2×C4×C61—42no—
152C2×C4×C61—44no—
153C2×C4×C61—42no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—42no—
157C2×C4×C61—82no—
158C2×C4×C61—82no—
159C2×C4×C61#12011yescharacteristic
160C2×C4×C61#12011yescharacteristic
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61#12111yescharacteristic
163C22⋊D61—82no—
164C22⋊D61—81no—
165C22⋊D61—81no—
166C2×Dic61—41no—
167C2×Dic61—42no—
168C2×Dic61—81no—
169C2×Dic61—81no—
170C2×Dic61—81no—
171D244—42no—
172D244—42no—
173#92418—42no—
174#92418—42no—
175#9951—611nomaximal, Sylow
176C22×C61—21no—
177C22×C61—21no—
178C22×C61—21no—
179C22×C61—21no—
180C22×C61—21no—
181C22×C61—21no—
182C22×C61—42no—
183C22×C61—42no—
184C22×C61#39211yescharacteristic
185C22×C61#44812yes—
186C22×C61#46112yes—
187C22×C61#49211yescharacteristic
188C22×C61#49711yescharacteristic
189C2×D61—81no—
190C2×D61—81no—
191C2×D61—81no—
192C2×D61—162no—
193C4×C61—21no—
194C4×C61—21no—
195C4×C61—21no—
196C4×C61—44no—
197C4×C61—81no—
198C4×C61—82no—
199C4×C61—81no—
200C4×C61#45812yes—
201Dic61—81no—
202Dic61—161no—
203#3187—611no—
204C2×#481—611no—
205#2502D6111yescharacteristic, complemented
206#1964—611no—
207#1824—611no—
208#1520—611no—
209#1282—611no—
210C2×C61—21no—
211C2×C61—22no—
212C2×C61—22no—
213C2×C61—21no—
214C2×C61#200811yescharacteristic
215C2×C61#201311yescharacteristic
216C2×C61#202211yescharacteristic
217D61—161no—
218C2×C22⋊Q8C2×D6111yescharacteristic
219C2×#121—611no—
220C22×C4⋊C4C2×D6111yescharacteristic
221#483—1221no—
222#481—1222no—
223#481—1222no—
224#481—1221no—
225#465—1222no—
226#450—1221no—
227#448—1221no—
228C2×#106—1221no—
229C2×#102—1221no—
230C2×#99—1222no—
231C2×C22⋊C8—611no—
232C2×C22⋊C8—611no—
233#398—22no—
234#391—21no—
235#385—22no—
236#384—21no—
237#380—611no—
238C4×C22⋊C4C2×D6111yescharacteristic
239#340—1222no—
240#328—1222no—
241C61#995111yescharacteristic, complemented, Sylow
242C22×Q8C22⋊D6111yescharacteristic
243C22×D4—611no—
244C22×D4—1221no—
245C23×C4—21no—
246C23×C4C22×D6111yescharacteristic
247C2×SD16—1222no—
248C22×C8—612no—
249C22×C8—1221no—
250C22⋊Q8—21no—
251C22⋊Q8—21no—
252C22⋊Q8—21no—
253C22⋊Q8—21no—
254#121—1221no—
255#121—1221no—
256#121—1221no—
257#121—1221no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—611no—
266C2×C4⋊C4—1221no—
267C2×C4⋊C4C22⋊D6112yes—
268C2×C4⋊C4C22⋊D6111yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—1221no—
271C2×C22⋊C4—1221no—
272C2×C22⋊C4C22⋊D6112yes—
273C2×C42—21no—
274#106—1224no—
275#106—1222no—
276#103—1222no—
277#102—1224no—
278#102—1222no—
279#99—2444no—
280C22⋊C8—612no—
281C22⋊C8—1221no—
282C22⋊C8—1221no—
283C22⋊C8—1221no—
284#95—24no—
285#95—21no—
286#95—1221no—
287C24—1221no—
288C24C2×C22⋊D6111yescharacteristic
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—42no—
292C2×D4—1221no—
293C2×D4—1221no—
294C2×D4—2442no—
295C2×D4—2442no—
296C2×D4—2441no—
297C2×D4—2441no—
298C2×D4—2442no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—82no—
307C22×C4—1221no—
308C22×C4—2441no—
309C22×C4D4×D6111yescharacteristic
310C22×C4D4×D6111yescharacteristic
311C22×C4D4×D6111yescharacteristic
312C22×C4#1175822411yescharacteristic
313C22×C4C2×C22⋊D6111yescharacteristic
314C22×C4C2×C22⋊D6111yescharacteristic
315SD16—4882no—
316C2×C8—1222no—
317C2×C8—1222no—
318C2×C8—2442no—
319C2×C8—2442no—
320C4⋊C4—22no—
321C4⋊C4—21no—
322C4⋊C4—22no—
323C4⋊C4—21no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—84no—
329C4⋊C4—1221no—
330C4⋊C4—1221no—
331C4⋊C4—2442no—
332C22⋊C4—22no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—2442no—
336C22⋊C4—2441no—
337C22⋊C4—2441no—
338C22⋊C4#1175820114yescomplemented
339C42—42no—
340C23—41no—
341C23—41no—
342C23—4881no—
343C23—4881no—
344C23—2442no—
345C23—2441no—
346C23#1451974611yescharacteristic
347C23#1451974811yescharacteristic
348C23#1451977311yescharacteristic
349C23C2×#1175820112yes—
350C23#1451980812yes—
351Q8—42no—
352Q8—81no—
353D4—4881no—
354D4—4881no—
355D4—4882no—
356D4—2442no—
357D4—2442no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—4881no—
376C2×C4—4881no—
377C2×C4—4881no—
378C2×C4—2442no—
379C2×C4—2441no—
380C2×C4#1451974511yescharacteristic
381C2×C4#1451974511yescharacteristic
382C2×C4#1451974911yescharacteristic
383C2×C4#1451974911yescharacteristic
384C8—2442no—
385C8—2442no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—42no—
393C22—42no—
394C22—9762no—
395C22—4881no—
396C22—4881no—
397C22—4881no—
398C22#1683967011yescharacteristic
399C22#1683990112yes—
400C22#1683996412yes—
401C22#1684013611yescharacteristic
402C22#1684015411yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—44no—
407C4—81no—
408C4—81no—
409C4—82no—
410C4—9761no—
411C4—4881no—
412C4#1683995112yes—
413C2—22no—
414C2—21no—
415C2—22no—
416C2—21no—
417C2—9761no—
418C2#2144687211yescharacteristic
419C2#2144690211yescharacteristic
420C2#2144694711yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#2502C2
socleC22×C61#392
prime radicalC2×#14519751C22
hypercenter#2502D61
hypocenterC61#9951
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2502

Complements

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

NH ≅ G/Ntype
4. #21445909C2semidirect
5. #21444391C2semidirect
6. #21443667C2semidirect
7. C61×#2502C2semidirect
116. C22⋊C4×C61SD16semidirect
205. #2502D61semidirect
241. C61#9951semidirect
338. C22⋊C4#11758201semidirect

Maximal subgroups

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

subgroupindex
#214565722
C2×#168400802
#214466552
#214459092
#214443912
#214436672
C61×#25022
#995161