Finite Group Databasebeta

← #29580467

Subgroups — #29580467

Order 15616.

610 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 107 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#21456576C211yesmaximal, characteristic
2C2×#16840078C211yesmaximal, characteristic
3#21450297C211yesmaximal, characteristic, complemented
4#21445907C211yesmaximal, characteristic, complemented
5#21444390C211yesmaximal, characteristic, complemented
6#21443667C211yesmaximal, characteristic, complemented
7C61×#1964C211yesmaximal, characteristic
8C2×#14519756C2211yescharacteristic
9C22×C4⋊Dic61C2211yescharacteristic
10#16840092—21no—
11#16840078—22no—
12#16840078—21no—
13#16840078—22no—
14#16839989—22no—
15#16839913—21no—
16#16839900—21no—
17C2×#14519708—21no—
18C2×#14519698—21no—
19C2×C22⋊#92418C2211yescharacteristic
20C2×C22⋊#92418C2211yescharacteristic
21#16839694—22no—
22#16839668—21no—
23#16839637—22no—
24#16839632—21no—
25#16839524—22no—
26#16839498—22no—
27C2×#121×C61C2211yescharacteristic
28C2×#99×C61—22no—
29C61×#380C2211yescharacteristic
30C4×C22⋊C4×C61C2211yescharacteristic
31C22×Dic122D411yescharacteristic
32C23×Dic61—21no—
33C2×#11758200—22no—
34C22×#92418—21no—
35C22×#92418D412yes—
36#14519756—21no—
37#14519756—21no—
38#14519756—21no—
39#14519756—21no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61—21no—
42C2×C4⋊Dic61—21no—
43C2×C4⋊Dic61—21no—
44C2×C4⋊Dic61—21no—
45C2×C4⋊Dic61—24no—
46C2×C4⋊Dic61D412yes—
47C2×C4⋊Dic61D411yescharacteristic
48C2×#11758196—21no—
49C2×C22⋊Dic61—21no—
50#14519708—22no—
51#14519708—24no—
52#14519700—22no—
53#14519698—22no—
54#14519698—24no—
55C22⋊#92418—21no—
56C22⋊#92418—21no—
57C22⋊#92418—21no—
58C22⋊#92418D412yes—
59#14519684—24no—
60C22×D4×C61—21no—
61C22×D4×C61D411yescharacteristic
62C23×C4×C61C2311yescharacteristic
63#121×C61—21no—
64#121×C61—21no—
65#121×C61—21no—
66#121×C61—21no—
67C2×C4⋊C4×C61—21no—
68C2×C4⋊C4×C61D411yescharacteristic
69C2×C22⋊C4×C61—21no—
70C2×C22⋊C4×C61—21no—
71C2×C22⋊C4×C61D412yes—
72C2×C42×C61—21no—
73#99×C61—44no—
74#95×C61—21no—
75#95×C61—21no—
76C24×C61—21no—
77C24×C61C2×D411yescharacteristic
78C2×D4×C61—21no—
79C2×D4×C61—21no—
80C2×D4×C61—42no—
81C2×D4×C61—41no—
82C2×D4×C61—41no—
83C2×D4×C61—42no—
84C2×D4×C61—42no—
85C22×C4×C61—22no—
86C22×C4×C61—21no—
87C22×C4×C61—42no—
88C22×C4×C61—41no—
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61#4111yescharacteristic
95C22×Dic61—22no—
96C22×Dic61—21no—
97C22×Dic61—21no—
98C22×Dic61—41no—
99C22×Dic61—41no—
100C22×Dic61—82no—
101C2×Dic122—21no—
102C2×Dic122—21no—
103C2×Dic122—42no—
104C4⋊C4×C61—21no—
105C4⋊C4×C61—21no—
106C4⋊C4×C61—42no—
107C22⋊C4×C61—22no—
108C22⋊C4×C61—41no—
109C22⋊C4×C61—41no—
110C22⋊C4×C61—42no—
111C22⋊C4×C61SD1614yescomplemented
112C42×C61—42no—
113C22⋊Dic61—41no—
114C22⋊Dic61—41no—
115#11758200—82no—
116C4⋊Dic61—22no—
117C4⋊Dic61—21no—
118C4⋊Dic61—22no—
119C4⋊Dic61—21no—
120C4⋊Dic61—41no—
121C4⋊Dic61—41no—
122C4⋊Dic61—84no—
123#11758196—41no—
124#11758196—41no—
125C2×#92418—22no—
126C2×#92418—22no—
127C2×#92418—42no—
128C2×#92418—42no—
129C23×C61—41no—
130C23×C61—41no—
131C23×C61—41no—
132C23×C61—42no—
133C23×C61—81no—
134C23×C61—81no—
135C23×C61#12011yescharacteristic
136C23×C61#12111yescharacteristic
137C23×C61#12711yescharacteristic
138C23×C61C2×SD1612yes—
139C23×C61#13712yes—
140D4×C61—42no—
141D4×C61—42no—
142D4×C61—82no—
143D4×C61—81no—
144D4×C61—81no—
145C2×C4×C61—22no—
146C2×C4×C61—22no—
147C2×C4×C61—22no—
148C2×C4×C61—24no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—42no—
153C2×C4×C61—42no—
154C2×C4×C61—41no—
155C2×C4×C61—42no—
156C2×C4×C61—81no—
157C2×C4×C61—81no—
158C2×C4×C61—81no—
159C2×C4×C61#12011yescharacteristic
160C2×C4×C61#12011yescharacteristic
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61#12111yescharacteristic
163C2×Dic61—42no—
164C2×Dic61—44no—
165C2×Dic61—42no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—42no—
169C2×Dic61—82no—
170C2×Dic61—82no—
171Dic122—42no—
172Dic122—81no—
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—81no—
185C22×C61—81no—
186C22×C61—81no—
187C22×C61—162no—
188C22×C61#39211yescharacteristic
189C22×C61#44812yes—
190C22×C61#46112yes—
191C22×C61#49211yescharacteristic
192C22×C61#49711yescharacteristic
193C4×C61—21no—
194C4×C61—21no—
195C4×C61—21no—
196C4×C61—44no—
197C4×C61—81no—
198C4×C61—161no—
199C4×C61#45812yes—
200Dic61—81no—
201Dic61—82no—
202Dic61—81no—
203#3187—611no—
204C2×#481—611no—
205#2502—611no—
206#1964D6111yescharacteristic
207#1824—611no—
208#1520—611no—
209#1282—611no—
210C2×C61—22no—
211C2×C61—21no—
212C2×C61—22no—
213C2×C61—21no—
214C2×C61—161no—
215C2×C61#200811yescharacteristic
216C2×C61#201311yescharacteristic
217C2×C61#202211yescharacteristic
218C2×C22⋊Q8—611no—
219C2×#121C2×D6111yescharacteristic
220C22×C4⋊C4—611no—
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—22no—
231C2×C22⋊C8—611no—
232C2×C22⋊C8—611no—
233#398—1222no—
234#391—1221no—
235#385—1222no—
236#384—1221no—
237#380C2×D6111yescharacteristic
238C4×C22⋊C4C2×D6111yescharacteristic
239#340—1222no—
240#328—1222no—
241C61#995111yescharacteristic, complemented, Sylow
242C22×Q8—611no—
243C22×D4—21no—
244C22×D4D24411yescharacteristic
245C23×C4—1221no—
246C23×C4C22×D6111yescharacteristic
247C2×SD16—1222no—
248C22×C8—612no—
249C22×C8—1221no—
250C22⋊Q8—1221no—
251C22⋊Q8—1221no—
252C22⋊Q8—1221no—
253C22⋊Q8—1221no—
254#121—21no—
255#121—21no—
256#121—21no—
257#121—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—612no—
260C2×C4⋊C4—611no—
261C2×C4⋊C4—1221no—
262C2×C4⋊C4—1221no—
263C2×C4⋊C4—1221no—
264C2×C4⋊C4—1221no—
265C2×C4⋊C4—1224no—
266C2×C4⋊C4—1221no—
267C2×C4⋊C4—1221no—
268C2×C4⋊C4D24411yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
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—44no—
280C22⋊C8—612no—
281C22⋊C8—1221no—
282C22⋊C8—1221no—
283C22⋊C8—1221no—
284#95—21no—
285#95—21no—
286#95—1224no—
287C24—21no—
288C24C2×C22⋊D6111yescharacteristic
289C2×Q8—1221no—
290C2×Q8—1221no—
291C2×Q8—2442no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—42no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—42no—
298C2×D4—42no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—1221no—
304C22×C4—1222no—
305C22×C4—1221no—
306C22×C4—4882no—
307C22×C4—2441no—
308C22×C4—2441no—
309C22×C4C2×D24411yescharacteristic
310C22×C4#1175822211yescharacteristic
311C22×C4D4×D6111yescharacteristic
312C22×C4D4×D6111yescharacteristic
313C22×C4D4×D6111yescharacteristic
314C22×C4D4×D6111yescharacteristic
315SD16—4882no—
316C2×C8—1222no—
317C2×C8—1222no—
318C2×C8—2442no—
319C2×C8—2442no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—42no—
323C4⋊C4—1222no—
324C4⋊C4—1221no—
325C4⋊C4—1222no—
326C4⋊C4—1221no—
327C4⋊C4—4884no—
328C4⋊C4—2441no—
329C4⋊C4—2441no—
330C4⋊C4—2441no—
331C4⋊C4—2441no—
332C22⋊C4—22no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—42no—
336C22⋊C4—2441no—
337C22⋊C4—2441no—
338C22⋊C4#1175820014yescomplemented
339C42—42no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—42no—
344C23—81no—
345C23—81no—
346C23#1451974511yescharacteristic
347C23#1451975111yescharacteristic
348C23#1451977311yescharacteristic
349C23C2×#1175820012yes—
350C23#1451980712yes—
351Q8—4881no—
352Q8—2442no—
353D4—42no—
354D4—42no—
355D4—81no—
356D4—82no—
357D4—81no—
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—41no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—4882no—
373C2×C4—4882no—
374C2×C4—2442no—
375C2×C4—2444no—
376C2×C4—2442no—
377C2×C4—2441no—
378C2×C4—2441no—
379C2×C4—2442no—
380C2×C4#1451974711yescharacteristic
381C2×C4#1451974711yescharacteristic
382C2×C4#1451975011yescharacteristic
383C2×C4#1451975011yescharacteristic
384C8—2442no—
385C8—2442no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—42no—
393C22—42no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22—162no—
398C22#1683967111yescharacteristic
399C22#1683990012yes—
400C22#1683996712yes—
401C22#1684013411yescharacteristic
402C22#1684015711yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—44no—
407C4—81no—
408C4—161no—
409C4—4881no—
410C4—4882no—
411C4—4881no—
412C4#1683995312yes—
413C2—22no—
414C2—21no—
415C2—22no—
416C2—21no—
417C2—161no—
418C2#2144687111yescharacteristic
419C2#2144690111yescharacteristic
420C2#2144694911yescharacteristic
subgroupG/H
centerC22#16839671
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#1964C2
socleC22×C61#392
prime radicalC2×#121×C61C22
hypercenter#1964D61
hypocenterC61#9951
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1964

Complements

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

NH ≅ G/Ntype
3. #21450297C2semidirect
4. #21445907C2semidirect
5. #21444390C2semidirect
6. #21443667C2semidirect
111. C22⋊C4×C61SD16semidirect
241. C61#9951semidirect
338. C22⋊C4#11758200semidirect

Maximal subgroups

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

subgroupindex
#214565762
C2×#168400782
#214502972
#214459072
#214443902
#214436672
C61×#19642
#995161