Finite Group Databasebeta

← #29580454

Subgroups — #29580454

Order 15616.

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

normal subgroups only

This group has 786 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×#16839901C211yesmaximal, characteristic
3#21446655C211yesmaximal, characteristic
4#21445754C211yesmaximal, characteristic, complemented
5#21444391C211yesmaximal, characteristic, complemented
6#21443667C211yesmaximal, characteristic, complemented
7C61×#2434C211yesmaximal, characteristic, complemented
8C22×#11758201—21no—
9C2×#14519751C2211yescharacteristic
10#16840090—21no—
11#16840080—21no—
12#16839993—22no—
13#16839912—21no—
14#16839901—22no—
15#16839901—22no—
16#16839901—21no—
17C2×#14519701—21no—
18C2×#14519701—22no—
19C2×#14519691—22no—
20C2×C22⋊#92418C2211yescharacteristic
21C2×C22⋊#92418C2211yescharacteristic
22#16839622C2211yescharacteristic
23#16839499—22no—
24C23×Q8×C61C2211yescharacteristic
25C2×C4⋊Q8×C61—21no—
26C2×C22⋊Q8×C61C2211yescharacteristic
27C61×#395—22no—
28C61×#391—21no—
29C61×#385—22no—
30C4×C22⋊C4×C61C2211yescharacteristic
31C22×D244D411yescharacteristic
32C22×C22⋊D61—21no—
33C2×#11758201—24no—
34C2×#11758201—22no—
35C2×#11758201—44no—
36C22×#92418—21no—
37C22×#92418D412yes—
38#14519751—21no—
39#14519751—21no—
40#14519751—21no—
41#14519751—21no—
42C2×C4⋊Dic61—21no—
43C2×C4⋊Dic61D411yescharacteristic
44C2×C22⋊Dic61—21no—
45C2×#11758198—21no—
46#14519708—22no—
47#14519701—22no—
48#14519701—24no—
49#14519701—44no—
50#14519697—22no—
51#14519691—44no—
52C22⋊#92418—21no—
53C22⋊#92418—21no—
54C22⋊#92418—21no—
55C22⋊#92418D412yes—
56#14519684—21no—
57C22×Q8×C61—21no—
58C22×Q8×C61—21no—
59C22×Q8×C61—21no—
60C22×Q8×C61—21no—
61C22×Q8×C61—21no—
62C22×Q8×C61—44no—
63C22×Q8×C61D411yescharacteristic
64C22×Q8×C61D412yes—
65C23×C4×C61—21no—
66C23×C4×C61C2311yescharacteristic
67C4⋊Q8×C61—44no—
68C22⋊Q8×C61—21no—
69C22⋊Q8×C61—21no—
70C22⋊Q8×C61—21no—
71C22⋊Q8×C61—21no—
72C2×C4⋊C4×C61—24no—
73C2×C4⋊C4×C61—21no—
74C2×C4⋊C4×C61D411yescharacteristic
75C2×C22⋊C4×C61—21no—
76C2×C22⋊C4×C61D412yes—
77C2×C42×C61—21no—
78#95×C61—24no—
79#95×C61—21no—
80C24×C61C2×D411yescharacteristic
81C23×D61—21no—
82C2×Q8×C61—24no—
83C2×Q8×C61—24no—
84C2×Q8×C61—24no—
85C2×Q8×C61—22no—
86C2×Q8×C61—21no—
87C2×Q8×C61—22no—
88C2×Q8×C61—21no—
89C2×Q8×C61—44no—
90C2×Q8×C61—42no—
91C2×Q8×C61—42no—
92C2×Q8×C61—42no—
93C2×Q8×C61—42no—
94C2×Q8×C61—44no—
95C2×Q8×C61—44no—
96C2×Q8×C61—41no—
97C2×Q8×C61—41no—
98C22×C4×C61—22no—
99C22×C4×C61—22no—
100C22×C4×C61—21no—
101C22×C4×C61—21no—
102C22×C4×C61—42no—
103C22×C4×C61—44no—
104C22×C4×C61—41no—
105C22×C4×C61—41no—
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61#4111yescharacteristic
112C2×C22⋊D61—41no—
113C2×C22⋊D61—41no—
114C2×C22⋊D61—42no—
115C2×C22⋊D61—42no—
116C22×Dic61—21no—
117C22×Dic61—41no—
118C2×D244—21no—
119C2×D244—21no—
120C2×D244—42no—
121C4⋊C4×C61—21no—
122C4⋊C4×C61—21no—
123C4⋊C4×C61—48no—
124C4⋊C4×C61—41no—
125C4⋊C4×C61—41no—
126C22⋊C4×C61—22no—
127C22⋊C4×C61—41no—
128C22⋊C4×C61—41no—
129C22⋊C4×C61SD1614yescomplemented
130C42×C61—42no—
131C22⋊Dic61—42no—
132#11758201—48no—
133#11758201—82no—
134#11758198—41no—
135#11758198—41no—
136C4⋊Dic61—21no—
137C4⋊Dic61—21no—
138C4⋊Dic61—42no—
139C2×#92418—22no—
140C2×#92418—22no—
141C2×#92418—42no—
142C2×#92418—42no—
143C23×C61—41no—
144C23×C61—41no—
145C23×C61#12011yescharacteristic
146C23×C61#12111yescharacteristic
147C23×C61#12711yescharacteristic
148C23×C61C2×SD1612yes—
149C23×C61#13712yes—
150C22×D61—41no—
151C22×D61—42no—
152C22×D61—81no—
153C22×D61—81no—
154Q8×C61—24no—
155Q8×C61—24no—
156Q8×C61—42no—
157Q8×C61—42no—
158Q8×C61—44no—
159Q8×C61—44no—
160Q8×C61—42no—
161Q8×C61—81no—
162C2×C4×C61—22no—
163C2×C4×C61—22no—
164C2×C4×C61—22no—
165C2×C4×C61—24no—
166C2×C4×C61—24no—
167C2×C4×C61—41no—
168C2×C4×C61—41no—
169C2×C4×C61—41no—
170C2×C4×C61—42no—
171C2×C4×C61—42no—
172C2×C4×C61—42no—
173C2×C4×C61—44no—
174C2×C4×C61—44no—
175C2×C4×C61—42no—
176C2×C4×C61—42no—
177C2×C4×C61—42no—
178C2×C4×C61—41no—
179C2×C4×C61—41no—
180C2×C4×C61#12011yescharacteristic
181C2×C4×C61#12011yescharacteristic
182C2×C4×C61#12111yescharacteristic
183C2×C4×C61#12111yescharacteristic
184C22⋊D61—82no—
185C22⋊D61—81no—
186C22⋊D61—81no—
187C2×Dic61—41no—
188C2×Dic61—42no—
189C2×Dic61—81no—
190C2×Dic61—81no—
191C2×Dic61—81no—
192D244—42no—
193D244—42no—
194#92418—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#39211yescharacteristic
206C22×C61#44812yes—
207C22×C61#46112yes—
208C22×C61#49211yescharacteristic
209C22×C61#49711yescharacteristic
210C2×D61—81no—
211C2×D61—81no—
212C2×D61—81no—
213C2×D61—162no—
214C4×C61—21no—
215C4×C61—21no—
216C4×C61—21no—
217C4×C61—44no—
218C4×C61—44no—
219C4×C61—81no—
220C4×C61—81no—
221C4×C61B(4,𝔽2)12yes—
222Dic61—81no—
223Dic61—161no—
224#3187—611no—
225C2×#448—611no—
226#2434D6111yescharacteristic, complemented
227#1964—611no—
228#1787—611no—
229#1520—611no—
230#1282—611no—
231C2×C61—21no—
232C2×C61—22no—
233C2×C61—22no—
234C2×C61—21no—
235C2×C61#200911yescharacteristic
236C2×C61#201311yescharacteristic
237C2×C61#202111yescharacteristic
238D61—161no—
239C23×Q8C2×D6111yescharacteristic
240C22×SD16—1221no—
241C2×C4⋊Q8—21no—
242C2×C22⋊Q8C2×D6111yescharacteristic
243C2×#121—611no—
244#483—1221no—
245#481—1221no—
246#465—1222no—
247#450—1221no—
248#448—1222no—
249#448—1221no—
250#448—1222no—
251C2×#103—1221no—
252C2×#103—1222no—
253C2×#99—1222no—
254C2×C22⋊C8—611no—
255C2×C22⋊C8—611no—
256#395—22no—
257#391—21no—
258#385—22no—
259#380—611no—
260C4×C22⋊C4C2×D6111yescharacteristic
261#328—1222no—
262C61#994911yescharacteristic, complemented, Sylow
263C22×Q8—21no—
264C22×Q8—21no—
265C22×Q8—21no—
266C22×Q8—21no—
267C22×Q8—21no—
268C22×Q8—44no—
269C22×Q8C22⋊D6111yescharacteristic
270C22×Q8C22⋊D6112yes—
271C22×D4—611no—
272C22×D4—1221no—
273C23×C4—21no—
274C23×C4C22×D6111yescharacteristic
275C2×SD16—1222no—
276C2×SD16—1224no—
277C2×SD16—2444no—
278C22×C8—612no—
279C22×C8—1221no—
280C4⋊Q8—44no—
281C22⋊Q8—21no—
282C22⋊Q8—21no—
283C22⋊Q8—21no—
284C22⋊Q8—21no—
285#121—1221no—
286#121—1221no—
287#121—1221no—
288#121—1221no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—611no—
292C2×C4⋊C4—1221no—
293C2×C4⋊C4C22⋊D6111yescharacteristic
294C2×C22⋊C4—21no—
295C2×C22⋊C4—1221no—
296C2×C22⋊C4—1221no—
297C2×C22⋊C4C22⋊D6112yes—
298C2×C42—21no—
299#106—1222no—
300#103—1224no—
301#103—1222no—
302#103—2444no—
303#102—1222no—
304#99—2444no—
305C22⋊C8—612no—
306C22⋊C8—1221no—
307C22⋊C8—1221no—
308C22⋊C8—1221no—
309#95—24no—
310#95—21no—
311#95—1221no—
312C24—1221no—
313C24C2×C22⋊D6111yescharacteristic
314C2×Q8—24no—
315C2×Q8—24no—
316C2×Q8—24no—
317C2×Q8—22no—
318C2×Q8—21no—
319C2×Q8—22no—
320C2×Q8—21no—
321C2×Q8—44no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—44no—
327C2×Q8—44no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×D4—1221no—
331C2×D4—1221no—
332C2×D4—2442no—
333C2×D4—2442no—
334C2×D4—2441no—
335C2×D4—2441no—
336C2×D4—2442no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—42no—
342C22×C4—44no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—1221no—
346C22×C4—2441no—
347C22×C4D4×D6111yescharacteristic
348C22×C4D4×D6111yescharacteristic
349C22×C4D4×D6111yescharacteristic
350C22×C4#1175822411yescharacteristic
351C22×C4C2×C22⋊D6111yescharacteristic
352C22×C4C2×C22⋊D6111yescharacteristic
353SD16—4882no—
354SD16—2448no—
355C2×C8—1222no—
356C2×C8—1222no—
357C2×C8—2442no—
358C2×C8—2442no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—48no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—1221no—
365C4⋊C4—1221no—
366C4⋊C4—2442no—
367C22⋊C4—22no—
368C22⋊C4—41no—
369C22⋊C4—41no—
370C22⋊C4—2442no—
371C22⋊C4—2441no—
372C22⋊C4—2441no—
373C22⋊C4#1175820114yescomplemented
374C42—42no—
375C23—41no—
376C23—41no—
377C23—4881no—
378C23—4881no—
379C23—2442no—
380C23—2441no—
381C23#1451974611yescharacteristic
382C23#1451974811yescharacteristic
383C23#1451977311yescharacteristic
384C23C2×#1175820112yes—
385C23#1451980812yes—
386Q8—24no—
387Q8—24no—
388Q8—42no—
389Q8—42no—
390Q8—44no—
391Q8—44no—
392Q8—42no—
393Q8—81no—
394D4—4881no—
395D4—4881no—
396D4—4882no—
397D4—2442no—
398D4—2442no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—4881no—
418C2×C4—4881no—
419C2×C4—4881no—
420C2×C4—2442no—
421C2×C4—2441no—
422C2×C4#1451974511yescharacteristic
423C2×C4#1451974511yescharacteristic
424C2×C4#1451974911yescharacteristic
425C2×C4#1451974911yescharacteristic
426C8—2442no—
427C8—2442no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—42no—
435C22—42no—
436C22—9762no—
437C22—4881no—
438C22—4881no—
439C22—4881no—
440C22#1683967011yescharacteristic
441C22#1683990112yes—
442C22#1683996412yes—
443C22#1684013611yescharacteristic
444C22#1684015411yescharacteristic
445C4—21no—
446C4—21no—
447C4—21no—
448C4—44no—
449C4—44no—
450C4—81no—
451C4—81no—
452C4—9761no—
453C4—4881no—
454C4#1683994812yes—
455C2—22no—
456C2—21no—
457C2—22no—
458C2—21no—
459C2—9761no—
460C2#2144687911yescharacteristic
461C2#2144690211yescharacteristic
462C2#2144694211yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#2434C2
socleC22×C61#392
prime radicalC2×#14519751C22
hypercenter#2434D61
hypocenterC61#9949
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2434

Complements

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

NH ≅ G/Ntype
4. #21445754C2semidirect
5. #21444391C2semidirect
6. #21443667C2semidirect
7. C61×#2434C2semidirect
129. C22⋊C4×C61SD16semidirect
226. #2434D61semidirect
262. C61#9949semidirect
373. 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×#168399012
#214466552
#214457542
#214443912
#214436672
C61×#24342
#994961