Finite Group Databasebeta

← #29580452

Subgroups — #29580452

Order 15616.

786 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C2×#16839900C211yesmaximal, characteristic
2#21449464C211yesmaximal, characteristic, complemented
3#21446656C211yesmaximal, characteristic
4#21445755C211yesmaximal, characteristic, complemented
5#21444391C211yesmaximal, characteristic, complemented
6#21443666C211yesmaximal, characteristic, complemented
7C61×#3187C211yesmaximal, characteristic
8C23×Dic122C2211yescharacteristic
9C22×#11758200—21no—
10C2×#14519775—21no—
11#16839900—22no—
12#16839900—22no—
13C2×#14519702—22no—
14C2×#14519700—21no—
15C2×#14519692—22no—
16C2×C22⋊#92418C2211yescharacteristic
17#16839682—22no—
18#16839667—21no—
19#16839634—22no—
20#16839622C2211yescharacteristic
21C4×C22⋊Dic61C2211yescharacteristic
22#16839499—22no—
23C2×C22⋊Q8×C61C2211yescharacteristic
24C2×#121×C61C2211yescharacteristic
25C61×#483—21no—
26C61×#481—21no—
27C61×#465—22no—
28C61×#450—21no—
29C61×#448—21no—
30C2×C22⋊C8×C61C2211yescharacteristic
31C22×Dic122—21no—
32C22×Dic122—21no—
33C22×Dic122—21no—
34C22×Dic122—21no—
35C22×Dic122—21no—
36C22×Dic122—44no—
37C22×Dic122D412yes—
38C23×Dic61—21no—
39C2×#11758200—24no—
40C2×#11758200—44no—
41C22×#92418—21no—
42#14519775—44no—
43C2×C4⋊Dic61—21no—
44C2×#11758196—24no—
45C2×C22⋊Dic61—21no—
46C2×C22⋊Dic61D412yes—
47C2×C4×Dic61—21no—
48#14519702—44no—
49#14519700—24no—
50#14519692—44no—
51C22⋊#92418—21no—
52C22⋊#92418D412yes—
53#14519684—24no—
54#14519684—21no—
55#14519684—21no—
56C22×Q8×C61D411yescharacteristic
57C22×D4×C61—21no—
58C22×D4×C61D411yescharacteristic
59C23×C4×C61C2311yescharacteristic
60C2×SD16×C61—22no—
61C22×C8×C61D412yes—
62C22⋊Q8×C61—21no—
63C22⋊Q8×C61—21no—
64C22⋊Q8×C61—21no—
65C22⋊Q8×C61—21no—
66#121×C61—21no—
67#121×C61—21no—
68#121×C61—21no—
69#121×C61—21no—
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61D411yescharacteristic
72C2×C4⋊C4×C61D411yescharacteristic
73C2×C22⋊C4×C61—21no—
74C2×C22⋊C4×C61—21no—
75#106×C61—22no—
76#103×C61—22no—
77#102×C61—22no—
78C22⋊C8×C61—21no—
79C22⋊C8×C61—21no—
80C24×C61—21no—
81C24×C61C2×D411yescharacteristic
82C2×Q8×C61—21no—
83C2×Q8×C61—21no—
84C2×Q8×C61—42no—
85C2×D4×C61—21no—
86C2×D4×C61—21no—
87C2×D4×C61—42no—
88C2×D4×C61—41no—
89C2×D4×C61—41no—
90C2×D4×C61—42no—
91C2×D4×C61—42no—
92C22×C4×C61—21no—
93C22×C4×C61—21no—
94C22×C4×C61—21no—
95C22×C4×C61—42no—
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61#4111yescharacteristic
102C22×Dic61—22no—
103C22×Dic61—22no—
104C22×Dic61—44no—
105C22×Dic61—41no—
106C22×Dic61—41no—
107C22×Dic61—41no—
108C2×Dic122—24no—
109C2×Dic122—24no—
110C2×Dic122—22no—
111C2×Dic122—22no—
112C2×Dic122—24no—
113C2×Dic122—44no—
114C2×Dic122—44no—
115C2×Dic122—44no—
116C2×Dic122—42no—
117C2×Dic122—42no—
118C2×Dic122—42no—
119C2×Dic122—41no—
120C2×Dic122—41no—
121SD16×C61—82no—
122C2×C8×C61—22no—
123C2×C8×C61—42no—
124C4⋊C4×C61—21no—
125C4⋊C4×C61—21no—
126C4⋊C4×C61—21no—
127C4⋊C4×C61—21no—
128C4⋊C4×C61—41no—
129C4⋊C4×C61—41no—
130C22⋊C4×C61—41no—
131C22⋊C4×C61—41no—
132C22⋊C4×C61—41no—
133C22⋊C4×C61—41no—
134C22⋊Dic61—22no—
135C22⋊Dic61—42no—
136C22⋊Dic61SD1614yescomplemented
137#11758200—48no—
138C4⋊Dic61—42no—
139#11758196—48no—
140C4×Dic61—42no—
141C2×#92418—22no—
142C2×#92418—42no—
143C23×C61—41no—
144C23×C61—41no—
145C23×C61—41no—
146C23×C61—42no—
147C23×C61—81no—
148C23×C61—81no—
149C23×C61#12011yescharacteristic
150C23×C61#12111yescharacteristic
151C23×C61#12711yescharacteristic
152C23×C61C2×SD1612yes—
153C23×C61#13712yes—
154Q8×C61—42no—
155Q8×C61—81no—
156D4×C61—42no—
157D4×C61—42no—
158D4×C61—82no—
159D4×C61—81no—
160D4×C61—81no—
161C2×C4×C61—22no—
162C2×C4×C61—22no—
163C2×C4×C61—22no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—42no—
168C2×C4×C61—42no—
169C2×C4×C61—41no—
170C2×C4×C61—41no—
171C2×C4×C61—41no—
172C2×C4×C61—42no—
173C2×C4×C61#12011yescharacteristic
174C2×C4×C61#12011yescharacteristic
175C2×C4×C61#12111yescharacteristic
176C2×C4×C61#12111yescharacteristic
177C2×Dic61—24no—
178C2×Dic61—24no—
179C2×Dic61—42no—
180C2×Dic61—44no—
181C2×Dic61—44no—
182C2×Dic61—42no—
183C2×Dic61—42no—
184C2×Dic61—42no—
185C2×Dic61—81no—
186C2×Dic61—81no—
187C2×Dic61—81no—
188Dic122—24no—
189Dic122—24no—
190Dic122—42no—
191Dic122—42no—
192Dic122—44no—
193Dic122—44no—
194C8×C61—42no—
195#92418—42no—
196#9949—611nomaximal, Sylow
197C22×C61—21no—
198C22×C61—21no—
199C22×C61—21no—
200C22×C61—21no—
201C22×C61—21no—
202C22×C61—21no—
203C22×C61—42no—
204C22×C61—42no—
205C22×C61—81no—
206C22×C61—81no—
207C22×C61—81no—
208C22×C61—162no—
209C22×C61#39211yescharacteristic
210C22×C61#44812yes—
211C22×C61#46112yes—
212C22×C61#49211yescharacteristic
213C22×C61#49711yescharacteristic
214C4×C61—21no—
215C4×C61—21no—
216C4×C61—21no—
217C4×C61—81no—
218C4×C61—81no—
219C4×C61—81no—
220C4×C61B(4,𝔽2)12yes—
221Dic61—44no—
222Dic61—44no—
223Dic61—161no—
224#3187D6111yescharacteristic
225C2×#448—611no—
226#2434—611no—
227#1964—611no—
228#1787—611no—
229#1520—611no—
230#1282—611no—
231C2×C61—22no—
232C2×C61—21no—
233C2×C61—22no—
234C2×C61—21no—
235C2×C61—161no—
236C2×C61#200911yescharacteristic
237C2×C61#201311yescharacteristic
238C2×C61#202111yescharacteristic
239C23×Q8—611no—
240C22×SD16—1221no—
241C2×C4⋊Q8—1221no—
242C2×C22⋊Q8C2×D6111yescharacteristic
243C2×#121C2×D6111yescharacteristic
244#483—21no—
245#481—21no—
246#465—22no—
247#450—21no—
248#448—21no—
249#448—1222no—
250#448—1222no—
251C2×#103—1221no—
252C2×#103—1222no—
253C2×#99—1222no—
254C2×C22⋊C8—611no—
255C2×C22⋊C8C2×D6111yescharacteristic
256#395—1222no—
257#391—1221no—
258#385—1222no—
259#380—611no—
260C4×C22⋊C4—611no—
261#328—1222no—
262C61#994911yescharacteristic, complemented, Sylow
263C22×Q8—612no—
264C22×Q8—1221no—
265C22×Q8—1221no—
266C22×Q8—1221no—
267C22×Q8—1221no—
268C22×Q8—1221no—
269C22×Q8—2444no—
270C22×Q8C22⋊D6111yescharacteristic
271C22×D4—21no—
272C22×D4C22⋊D6111yescharacteristic
273C23×C4—1221no—
274C23×C4C22×D6111yescharacteristic
275C2×SD16—22no—
276C2×SD16—1224no—
277C2×SD16—2444no—
278C22×C8—1221no—
279C22×C8C22⋊D6112yes—
280C4⋊Q8—2444no—
281C22⋊Q8—21no—
282C22⋊Q8—21no—
283C22⋊Q8—21no—
284C22⋊Q8—21no—
285#121—21no—
286#121—21no—
287#121—21no—
288#121—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—1224no—
291C2×C4⋊C4—1221no—
292C2×C4⋊C4C22⋊D6111yescharacteristic
293C2×C4⋊C4C22⋊D6111yescharacteristic
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—612no—
297C2×C22⋊C4—1221no—
298C2×C42—1221no—
299#106—22no—
300#103—22no—
301#103—1224no—
302#103—2444no—
303#102—22no—
304#99—2444no—
305C22⋊C8—21no—
306C22⋊C8—21no—
307C22⋊C8—612no—
308C22⋊C8—1221no—
309#95—1224no—
310#95—1221no—
311#95—1221no—
312C24—21no—
313C24D4×D6111yescharacteristic
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—42no—
317C2×Q8—1224no—
318C2×Q8—1224no—
319C2×Q8—1224no—
320C2×Q8—1222no—
321C2×Q8—1222no—
322C2×Q8—2442no—
323C2×Q8—2442no—
324C2×Q8—2442no—
325C2×Q8—2444no—
326C2×Q8—2444no—
327C2×Q8—2444no—
328C2×Q8—2441no—
329C2×Q8—2441no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—42no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—42no—
336C2×D4—42no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—42no—
341C22×C4—1222no—
342C22×C4—1222no—
343C22×C4—2444no—
344C22×C4—2441no—
345C22×C4—2441no—
346C22×C4—2441no—
347C22×C4D4×D6111yescharacteristic
348C22×C4D4×D6111yescharacteristic
349C22×C4#1175822411yescharacteristic
350C22×C4C2×C22⋊D6111yescharacteristic
351C22×C4C2×C22⋊D6111yescharacteristic
352C22×C4C2×C22⋊D6111yescharacteristic
353SD16—82no—
354SD16—2448no—
355C2×C8—22no—
356C2×C8—42no—
357C2×C8—1222no—
358C2×C8—2442no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—2442no—
366C4⋊C4—2448no—
367C22⋊C4—41no—
368C22⋊C4—41no—
369C22⋊C4—41no—
370C22⋊C4—41no—
371C22⋊C4—614no—
372C22⋊C4—1222no—
373C22⋊C4—2442no—
374C42—2442no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—42no—
379C23—81no—
380C23—81no—
381C23#1451974511yescharacteristic
382C23#1451974911yescharacteristic
383C23#1451977311yescharacteristic
384C23SD16×D6112yes—
385C23#1451980612yes—
386Q8—42no—
387Q8—81no—
388Q8—1224no—
389Q8—1224no—
390Q8—2442no—
391Q8—2442no—
392Q8—2444no—
393Q8—2444no—
394D4—42no—
395D4—42no—
396D4—81no—
397D4—81no—
398D4—82no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—1224no—
412C2×C4—1224no—
413C2×C4—4881no—
414C2×C4—4881no—
415C2×C4—4881no—
416C2×C4—2442no—
417C2×C4—2442no—
418C2×C4—2442no—
419C2×C4—2444no—
420C2×C4—2444no—
421C2×C4—2442no—
422C2×C4#1451974511yescharacteristic
423C2×C4#1451974611yescharacteristic
424C2×C4#1451974811yescharacteristic
425C2×C4#1451974911yescharacteristic
426C8—42no—
427C8—2442no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—42no—
435C22—42no—
436C22—81no—
437C22—81no—
438C22—81no—
439C22—162no—
440C22#1683967011yescharacteristic
441C22#1683989912yes—
442C22#1683996512yes—
443C22#1684013311yescharacteristic
444C22#1684015311yescharacteristic
445C4—21no—
446C4—21no—
447C4—21no—
448C4—81no—
449C4—81no—
450C4—81no—
451C4—9761no—
452C4—2444no—
453C4—2444no—
454C4#1683994912yes—
455C2—21no—
456C2—22no—
457C2—22no—
458C2—21no—
459C2—161no—
460C2#2144687611yescharacteristic
461C2#2144690011yescharacteristic
462C2#2144694311yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#3187C2
socleC22×C61#392
prime radicalC2×#121×C61C22
hypercenter#3187D61
hypocenterC61#9949
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3187

Complements

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

NH ≅ G/Ntype
2. #21449464C2semidirect
4. #21445755C2semidirect
5. #21444391C2semidirect
6. #21443666C2semidirect
136. C22⋊Dic61SD16semidirect
262. C61#9949semidirect

Maximal subgroups

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

subgroupindex
C2×#168399002
#214494642
#214466562
#214457552
#214443912
#214436662
C61×#31872
#994961