Finite Group Databasebeta

← #29662425

Subgroups — #29662425

Order 15616.

628 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456144C211yesmaximal, characteristic
2#21455850C211yesmaximal, characteristic
3C2×Q8×#92418C211yesmaximal, characteristic, complemented
4#21445279C212yesmaximal, complemented
5#21445265C212yesmaximal, complemented
6#21445251C212yesmaximal, complemented
7#21445237C212yesmaximal, complemented
8#21444565C211yesmaximal, characteristic, complemented
9#21444476C211yesmaximal, characteristic, complemented
10#21444466C211yesmaximal, characteristic
11C61×#3104C211yesmaximal, characteristic, complemented
12#16840312C2211yescharacteristic
13#16840119—21no—
14#16840119—21no—
15#16840114C2211yescharacteristic
16#16840114C2211yescharacteristic
17#16840110C2211yescharacteristic
18#16840110C2211yescharacteristic
19#16840080—21no—
20#16840047C2212yes—
21#16839962C2212yes—
22#16839901—21no—
23Q8×#92418—21no—
24Q8×#92418—21no—
25Q8×#92418C2212yes—
26Q8×#92418C2212yes—
27C4×#11758201—22no—
28#16839815C2211yescharacteristic
29C2×C4⋊#92418—21no—
30C2×C4⋊#92418C2211yescharacteristic
31#16839756—21no—
32#16839756—21no—
33C2×C4×#92418—21no—
34C2×C4×#92418C2211yescharacteristic
35#16839514C2212yes—
36#16839512C2212yes—
37#16839511C2212yes—
38#16839508C2212yes—
39#16839507C2212yes—
40#16839505C2212yes—
41C61×#533C2211yescharacteristic
42C2×C4×Q8×C61C2211yescharacteristic
43C61×#484—21no—
44C61×#477C2212yes—
45C61×#462C2212yes—
46C61×#451—21no—
47C4×Q16×C61—22no—
48C61×#431C2211yescharacteristic
49C2×#11758222—21no—
50C2×#11758201—22no—
51C22×#92418—21no—
52C22×#92418D411yescharacteristic
53C22×#92418D411yescharacteristic
54#14519777C2311yescharacteristic
55#14519774C2311yescharacteristic
56#14519765—21no—
57#14519751—21no—
58#14519751—21no—
59C4×D244—21no—
60C4×D244—21no—
61#14519708—22no—
62C4⋊#92418—21no—
63C4⋊#92418—22no—
64C4⋊#92418—22no—
65C4⋊#92418C2312yes—
66C4⋊#92418C2312yes—
67#14519701—22no—
68#14519700—21no—
69#14519700—21no—
70#14519700—21no—
71#14519700—21no—
72#14519697—22no—
73#14519697—21no—
74#14519697—21no—
75#14519697—21no—
76#14519697—21no—
77C22⋊#92418—21no—
78C4×#92418—21no—
79C4×#92418—22no—
80C4×#92418—22no—
81C4×#92418D411yescharacteristic
82C4×#92418D411yescharacteristic
83C4×#92418C2311yescharacteristic
84C4×#92418C2311yescharacteristic
85C22×Q8×C61D411yescharacteristic
86C2×Q16×C61—22no—
87C4⋊Q8×C61C2311yescharacteristic
88C4⋊Q8×C61C2311yescharacteristic
89#125×C61—21no—
90C22⋊Q8×C61—21no—
91C22⋊Q8×C61—21no—
92C4×Q8×C61—21no—
93C4×Q8×C61—21no—
94C4×Q8×C61—21no—
95C4×Q8×C61—21no—
96C4×Q8×C61D412yes—
97C4×Q8×C61C2312yes—
98#117×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61D411yescharacteristic
101C2×C42×C61—21no—
102C2×C42×C61C2311yescharacteristic
103#107×C61—22no—
104C4⋊C8×C61C2312yes—
105#103×C61—22no—
106#103×C61—22no—
107C22⋊C8×C61—21no—
108C4×C8×C61—21no—
109C2×Q8×C61—24no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61C2×D412yes—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61#4111yescharacteristic
119C2×C22⋊D61—41no—
120#11758222—42no—
121#11758222—42no—
122C2×D244—21no—
123C2×D244—21no—
124C2×D244—21no—
125C2×C4×D61—41no—
126C2×Dic122—21no—
127Q16×C61—44no—
128C2×C8×C61—22no—
129C4⋊C4×C61—22no—
130C4⋊C4×C61—21no—
131C4⋊C4×C61—22no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—41no—
137C4⋊C4×C61—41no—
138C4⋊C4×C61C2×D412yes—
139C22⋊C4×C61—41no—
140C42×C61—22no—
141C42×C61—22no—
142C42×C61—41no—
143C42×C61C2×D411yescharacteristic
144C42×C61C2×D411yescharacteristic
145C42×C61#4111yescharacteristic
146C42×C61C2411yescharacteristic
147#11758201—44no—
148#11758198—41no—
149#11758198—41no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152C2×#92418—22no—
153C2×#92418—22no—
154C2×#92418—22no—
155C2×#92418—22no—
156C2×#92418—42no—
157C2×#92418C2×D411yescharacteristic
158C2×#92418C2×D411yescharacteristic
159C2×#92418C2×D411yescharacteristic
160C2×#92418C2×D411yescharacteristic
161C23×C61#12111yescharacteristic
162C22×D61—41no—
163Q8×C61—24no—
164Q8×C61—22no—
165Q8×C61—42no—
166Q8×C61—81no—
167Q8×C61#13514yes—
168C2×C4×C61—24no—
169C2×C4×C61—22no—
170C2×C4×C61—22no—
171C2×C4×C61—22no—
172C2×C4×C61—41no—
173C2×C4×C61—42no—
174C2×C4×C61—41no—
175C2×C4×C61—41no—
176C2×C4×C61#12111yescharacteristic
177C2×C4×C61#12111yescharacteristic
178C2×C4×C61#12111yescharacteristic
179C2×C4×C61#13512yes—
180C2×C4×C61#13712yes—
181C2×C4×C61C22×D411yescharacteristic
182C2×C4×C61C22×D411yescharacteristic
183C2×C4×C61C2×#4111yescharacteristic
184C22⋊D61—82no—
185C2×Dic61—41no—
186D244—42no—
187D244—41no—
188D244—41no—
189D244—81no—
190C4×D61—82no—
191Dic122—41no—
192Dic122—41no—
193C8×C61—42no—
194#92418—21no—
195#92418—21no—
196#92418—21no—
197#92418—21no—
198#92418—42no—
199#92418—42no—
200#19123—611nomaximal, Sylow
201C22×C61—21no—
202C22×C61—21no—
203C22×C61#46412yes—
204C22×C61C2×#12111yescharacteristic
205C2×D61—81no—
206C2×D61—81no—
207C2×D61—81no—
208C4×C61—24no—
209C4×C61—21no—
210C4×C61—21no—
211C4×C61—21no—
212C4×C61—42no—
213C4×C61—81no—
214C4×C61—81no—
215C4×C61#46412yes—
216C4×C61C2×#13512yes—
217C4×C61C2×#13511yescharacteristic
218C4×C61C2×#13711yescharacteristic
219Dic61—81no—
220#3131—611no—
221#3104D6111yescharacteristic, complemented
222#3103—611no—
223C2×C8×Q8—611no—
224#1684—612no—
225#1682—612no—
226#1680—612no—
227#1678—612no—
228#1564—611no—
229#1542—611no—
230#1540—611no—
231C2×C61—22no—
232C2×C61C2×#46411yescharacteristic
233C2×C61#305011yescharacteristic
234C2×C61#306711yescharacteristic
235D61—161no—
236#533C2×D6111yescharacteristic
237#532—611no—
238C2×C4×Q8C2×D6111yescharacteristic
239#488—1221no—
240#488—1221no—
241#487—611no—
242#487—611no—
243#486—611no—
244#486—611no—
245#484—21no—
246#481—1221no—
247#477C2×D6112yes—
248#475—612no—
249#462C2×D6112yes—
250#460—612no—
251#451—21no—
252#448—1221no—
253C8×Q8—612no—
254C8×Q8—612no—
255C8×Q8—1221no—
256C8×Q8—1221no—
257C4×Q16—22no—
258C4×SD16—1222no—
259#431—611no—
260#431C2×D6111yescharacteristic
261C2×C4⋊C8—611no—
262C2×C4⋊C8—1221no—
263#416—1221no—
264#416—1221no—
265C2×C4×C8—611no—
266C2×C4×C8—1221no—
267#335—612no—
268#334—612no—
269#333—612no—
270#332—612no—
271#332—612no—
272#331—612no—
273C61#1912311yescharacteristic, complemented, Sylow
274C2×#41—1221no—
275C22×Q8C22⋊D6111yescharacteristic
276C2×Q16—22no—
277C2×SD16—1222no—
278C22×C8—611no—
279C22×C8—611no—
280C22×C8—1221no—
281C4⋊Q8—611no—
282C4⋊Q8C22×D6111yescharacteristic
283C4⋊Q8C22×D6111yescharacteristic
284#127—611no—
285#125—21no—
286#124—1221no—
287C22⋊Q8—21no—
288C22⋊Q8—21no—
289#121—1221no—
290#121—1221no—
291C4×Q8—21no—
292C4×Q8—21no—
293C4×Q8—21no—
294C4×Q8—21no—
295C4×Q8C22⋊D6112yes—
296C4×Q8C22×D6112yes—
297C4×D4—1221no—
298C4×D4—1221no—
299#117—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4C22⋊D6111yescharacteristic
302C2×C42—21no—
303C2×C42C22×D6111yescharacteristic
304#107—22no—
305#106—1222no—
306C4⋊C8—612no—
307C4⋊C8—612no—
308C4⋊C8—1222no—
309C4⋊C8—1222no—
310C4⋊C8—1221no—
311C4⋊C8C22×D6112yes—
312#103—22no—
313#103—22no—
314#103—1221no—
315#103—1221no—
316#103—1221no—
317#103—1221no—
318#103—1222no—
319#102—1221no—
320#102—1221no—
321#102—1221no—
322#102—1221no—
323#102—1222no—
324C22⋊C8—21no—
325C22⋊C8—1221no—
326C4×C8—21no—
327C4×C8—611no—
328C4×C8—611no—
329C4×C8—611no—
330C4×C8—611no—
331C4×C8—1222no—
332C4×C8—1222no—
333C4×C8—1221no—
334#41—2442no—
335#41—2442no—
336C2×Q8—24no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—1221no—
341C2×Q8C2×C22⋊D6112yes—
342C2×D4—1221no—
343C2×D4—1221no—
344C2×D4—1221no—
345C2×D4—2441no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—2441no—
349C22×C4D4×D6111yescharacteristic
350C22×C4#1175822411yescharacteristic
351C22×C4C2×C22⋊D6111yescharacteristic
352Q16—44no—
353SD16—2444no—
354C2×C8—22no—
355C2×C8—611no—
356C2×C8—611no—
357C2×C8—611no—
358C2×C8—611no—
359C2×C8—1222no—
360C2×C8—1222no—
361C2×C8—1222no—
362C2×C8—1222no—
363C2×C8—2442no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—1221no—
374C4⋊C4—1221no—
375C4⋊C4C2×C22⋊D6112yes—
376C22⋊C4—41no—
377C22⋊C4—2441no—
378C22⋊C4—2441no—
379C42—22no—
380C42—22no—
381C42—41no—
382C42D4×D6111yescharacteristic
383C42#1175822411yescharacteristic
384C42C2×C22⋊D6111yescharacteristic
385C42C23×D6111yescharacteristic
386C23—2441no—
387C23#1451974911yescharacteristic
388Q8—24no—
389Q8—22no—
390Q8—42no—
391Q8—81no—
392Q8—2441no—
393Q8—2441no—
394Q8#1451979714yes—
395D4—4881no—
396D4—4882no—
397D4—2442no—
398D4—2441no—
399D4—2441no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—4882no—
409C2×C4—2441no—
410C2×C4#1451974911yescharacteristic
411C2×C4#1451974911yescharacteristic
412C2×C4#1451974911yescharacteristic
413C2×C4#1451980012yes—
414C2×C4#1451980912yes—
415C2×C4C2×D4×D6111yescharacteristic
416C2×C4C22×C22⋊D6111yescharacteristic
417C2×C4C2×#1175822411yescharacteristic
418C8—42no—
419C8—1221no—
420C8—1221no—
421C8—1221no—
422C8—1221no—
423C8—2442no—
424C8—2442no—
425C22—21no—
426C22—21no—
427C22—4881no—
428C22—4881no—
429C22—4881no—
430C22#1683998612yes—
431C22C2×#1451974911yescharacteristic
432C4—24no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—42no—
437C4—81no—
438C4—81no—
439C4—4881no—
440C4#1683998612yes—
441C4C2×#1451979712yes—
442C4C2×#1451980011yescharacteristic
443C4C2×#1451980911yescharacteristic
444C2—22no—
445C2—9761no—
446C2C2×#1683998611yescharacteristic
447C2#2145539711yescharacteristic
448C2#2145553611yescharacteristic
subgroupG/H
centerC2×C4#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3104C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenter#3104D61
hypocenterC61#19123
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3104

Complements

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

NH ≅ G/Ntype
3. C2×Q8×#92418C2semidirect
4. #21445279C2semidirect
5. #21445265C2semidirect
6. #21445251C2semidirect
7. #21445237C2semidirect
8. #21444565C2semidirect
9. #21444476C2semidirect
11. C61×#3104C2semidirect
221. #3104D61semidirect
273. C61#19123semidirect

Maximal subgroups

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

subgroupindex
#214561442
#214558502
C2×Q8×#924182
#214452792
#214452652
#214452512
#214452372
#214445652
#214444762
#214444662
C61×#31042
#1912361