Finite Group Databasebeta

← #29663874

Subgroups — #29663874

Order 15616.

628 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456378C211yesmaximal, characteristic
2#21455950C211yesmaximal, characteristic, complemented
3#21455942C211yesmaximal, characteristic
4C2×C8×Dic122C211yesmaximal, characteristic, complemented
5#21445278C212yesmaximal, complemented
6#21445264C212yesmaximal, complemented
7#21445202C212yesmaximal, complemented
8#21445168C212yesmaximal, complemented
9#21444520C211yesmaximal, characteristic, complemented
10#21444515C211yesmaximal, characteristic
11C61×#1568C211yesmaximal, characteristic, complemented
12#16840319C2211yescharacteristic
13#16840312C2211yescharacteristic
14C2×C4×Dic122C2211yescharacteristic
15#16840160C2211yescharacteristic
16#16840135C2211yescharacteristic
17#16840113C2211yescharacteristic
18#16840109C2211yescharacteristic
19#16840093—21no—
20#16840076—21no—
21#16840058C2212yes—
22#16840048C2212yes—
23#16839991—21no—
24#16839991—21no—
25#16839973C2212yes—
26#16839963C2212yes—
27#16839920—21no—
28#16839903—21no—
29C8×Dic122—21no—
30C8×Dic122—21no—
31C8×Dic122C2212yes—
32C8×Dic122C2212yes—
33#16839870—22no—
34#16839864—22no—
35C4×#11758190—21no—
36C4×#11758190—21no—
37C2×C4⋊#92418C2211yescharacteristic
38C2×#14519706—21no—
39C2×C8×Dic61—21no—
40#16839510C2212yes—
41#16839509C2212yes—
42#16839507C2212yes—
43#16839505C2212yes—
44C61×#432C2211yescharacteristic
45C61×#432C2211yescharacteristic
46C2×C4×C8×C61C2211yescharacteristic
47C61×#335C2212yes—
48C61×#334C2212yes—
49C2×#11758222—21no—
50C22×Dic122D411yescharacteristic
51C2×Dic244—22no—
52C2×#11758190—22no—
53C2×#11758190—21no—
54C2×#11758190—21no—
55C22×#92418—21no—
56#14519777C2311yescharacteristic
57#14519777C2311yescharacteristic
58#14519777C2311yescharacteristic
59#14519774C2311yescharacteristic
60#14519769—21no—
61#14519765—21no—
62#14519756—21no—
63#14519756—21no—
64#14519751—21no—
65#14519751—21no—
66C4×Dic122—21no—
67C4×Dic122—21no—
68C4×Dic122—21no—
69C4×Dic122—21no—
70C4×Dic122D412yes—
71C4×Dic122C2312yes—
72C4×D244—21no—
73C4×D244—21no—
74#14519736—21no—
75C2×C4⋊Dic61D411yescharacteristic
76C2×#11758196—21no—
77C2×C4×Dic61—21no—
78#14519711—22no—
79#14519709—21no—
80#14519709—21no—
81#14519709—22no—
82C4⋊#92418—21no—
83C4⋊#92418C2312yes—
84#14519706—22no—
85#14519706—22no—
86#14519702—22no—
87#14519702—22no—
88#14519702—21no—
89#14519702—21no—
90#14519702—22no—
91#14519699—21no—
92#14519699—21no—
93#14519699—22no—
94C8×Dic61—22no—
95C8×Dic61—22no—
96C22×C8×C61D411yescharacteristic
97C22×C8×C61D411yescharacteristic
98C2×C42×C61C2311yescharacteristic
99C4⋊C8×C61C2312yes—
100C4⋊C8×C61C2312yes—
101C22⋊C8×C61—21no—
102C22⋊C8×C61—21no—
103#97×C61—21no—
104#97×C61—21no—
105C4×C8×C61D411yescharacteristic
106C4×C8×C61D411yescharacteristic
107C4×C8×C61C2311yescharacteristic
108C4×C8×C61C2311yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61#4111yescharacteristic
112C2×C22⋊D61—41no—
113C22×Dic61—21no—
114C22×Dic61—21no—
115#11758222—82no—
116C2×D244—21no—
117C2×D244—21no—
118C2×D244—21no—
119C2×C4×D61—41no—
120C2×Dic122—21no—
121C2×Dic122—24no—
122C2×Dic122—21no—
123C2×Dic122—21no—
124C2×Dic122—21no—
125C2×Dic122C2×D412yes—
126C2×C8×C61—22no—
127C2×C8×C61—22no—
128C2×C8×C61—22no—
129C2×C8×C61—41no—
130C2×C8×C61C2×D411yescharacteristic
131C2×C8×C61C2×D411yescharacteristic
132C2×C8×C61C2×D411yescharacteristic
133C2×C8×C61C2×D411yescharacteristic
134C42×C61C2×D411yescharacteristic
135C42×C61C2×D411yescharacteristic
136C42×C61#4111yescharacteristic
137C42×C61C2411yescharacteristic
138C22⋊Dic61—41no—
139#11758198—41no—
140#11758198—41no—
141C4⋊Dic61—21no—
142C4⋊Dic61—21no—
143C4⋊Dic61—21no—
144C4⋊Dic61—21no—
145C4⋊Dic61—21no—
146C4⋊Dic61—21no—
147C4⋊Dic61—21no—
148C4⋊Dic61C2×D412yes—
149#11758196—22no—
150#11758196—22no—
151#11758196—41no—
152#11758196—41no—
153C4×Dic61—22no—
154C4×Dic61—22no—
155C4×Dic61—41no—
156C2×#92418—22no—
157C2×#92418—42no—
158Dic244—44no—
159#11758190—42no—
160#11758190—44no—
161#11758190—42no—
162C23×C61#12111yescharacteristic
163C22×D61—41no—
164C2×C4×C61—22no—
165C2×C4×C61—22no—
166C2×C4×C61—21no—
167C2×C4×C61—21no—
168C2×C4×C61#12111yescharacteristic
169C2×C4×C61#12111yescharacteristic
170C2×C4×C61#12111yescharacteristic
171C2×C4×C61C22×D411yescharacteristic
172C2×C4×C61C22×D411yescharacteristic
173C2×C4×C61C2×#4111yescharacteristic
174C22⋊D61—82no—
175C2×Dic61—24no—
176C2×Dic61—22no—
177C2×Dic61—22no—
178C2×Dic61—42no—
179C2×Dic61—41no—
180C2×Dic61—41no—
181C2×Dic61—41no—
182D244—42no—
183D244—41no—
184D244—41no—
185D244—81no—
186C4×D61—82no—
187Dic122—24no—
188Dic122—22no—
189Dic122—42no—
190Dic122—41no—
191Dic122—41no—
192Dic122—81no—
193Dic122#13514yes—
194C8×C61—21no—
195C8×C61—21no—
196C8×C61—21no—
197C8×C61—42no—
198C8×C61—42no—
199C8×C61#13712yes—
200#92418—42no—
201#19263—611nomaximal, Sylow
202C22×C61—21no—
203C22×C61—21no—
204C22×C61#47112yes—
205C22×C61C2×#12111yescharacteristic
206C2×D61—81no—
207C2×D61—81no—
208C2×D61—81no—
209C4×C61—21no—
210C4×C61—21no—
211C4×C61—21no—
212C4×C61—21no—
213C4×C61C2×#13512yes—
214C4×C61C2×#13711yescharacteristic
215C4×C61#57511yescharacteristic
216Dic61—24no—
217Dic61—42no—
218Dic61—81no—
219Dic61—81no—
220Dic61—81no—
221#3167—611no—
222#3113—611no—
223#3112—611no—
224C2×C8×Q8—611no—
225#1684—612no—
226#1682—612no—
227#1672—612no—
228#1666—612no—
229#1568D6111yescharacteristic, complemented
230#1552—611no—
231#1551—611no—
232C2×C61—22no—
233C2×C61#303711yescharacteristic
234C2×C61#305011yescharacteristic
235C2×C61C2×#47111yescharacteristic
236D61—161no—
237#533—611no—
238#532—611no—
239C2×C4×Q8—611no—
240#498—611no—
241#492—611no—
242#487—611no—
243#486—611no—
244#483—1221no—
245#480—1221no—
246#477—612no—
247#475—612no—
248#465—1221no—
249#465—1221no—
250#462—612no—
251#460—612no—
252#451—1221no—
253#448—1221no—
254C8×Q8—612no—
255C8×Q8—612no—
256C8×Q8—1221no—
257C8×Q8—1221no—
258#441—1222no—
259#440—1222no—
260C4×SD16—1221no—
261C4×SD16—1221no—
262#432C2×D6111yescharacteristic
263#432C2×D6111yescharacteristic
264C2×C4⋊C8—611no—
265C2×C4⋊C8—1221no—
266C2×C4×C8—1221no—
267C2×C4×C8C2×D6111yescharacteristic
268#335C2×D6112yes—
269#334C2×D6112yes—
270#333—612no—
271#332—612no—
272#332—612no—
273#331—612no—
274C61#1926311yescharacteristic, complemented, Sylow
275C2×#41—1221no—
276C22×Q8—611no—
277C2×Q16—1222no—
278C2×SD16—1222no—
279C2×SD16—1221no—
280C2×SD16—1221no—
281C22×C8—1221no—
282C22×C8D24411yescharacteristic
283C22×C8D24411yescharacteristic
284C4⋊Q8—611no—
285C4⋊Q8—611no—
286C4⋊Q8—611no—
287#127—611no—
288#125—1221no—
289#124—1221no—
290C22⋊Q8—1221no—
291C22⋊Q8—1221no—
292#121—1221no—
293#121—1221no—
294C4×Q8—612no—
295C4×Q8—612no—
296C4×Q8—1221no—
297C4×Q8—1221no—
298C4×Q8—1221no—
299C4×Q8—1221no—
300C4×D4—1221no—
301C4×D4—1221no—
302#117—1221no—
303C2×C4⋊C4—611no—
304C2×C4⋊C4—1221no—
305C2×C42—1221no—
306C2×C42C22×D6111yescharacteristic
307#107—1222no—
308#106—1222no—
309#106—1221no—
310#106—1221no—
311C4⋊C8—612no—
312C4⋊C8—1222no—
313C4⋊C8—1222no—
314C4⋊C8—1221no—
315C4⋊C8C22×D6112yes—
316C4⋊C8C22×D6112yes—
317#103—1222no—
318#103—1222no—
319#103—1222no—
320#103—1221no—
321#103—1221no—
322#102—1222no—
323#102—1221no—
324#102—1221no—
325C22⋊C8—21no—
326C22⋊C8—21no—
327#97—21no—
328#97—21no—
329C4×C8—1222no—
330C4×C8—1222no—
331C4×C8D24411yescharacteristic
332C4×C8D24411yescharacteristic
333C4×C8C22×D6111yescharacteristic
334C4×C8C22×D6111yescharacteristic
335#41—4882no—
336C2×Q8—612no—
337C2×Q8—1224no—
338C2×Q8—1221no—
339C2×Q8—1221no—
340C2×Q8—1221no—
341C2×Q8—1221no—
342C2×D4—1221no—
343C2×D4—1221no—
344C2×D4—1221no—
345C2×D4—2441no—
346C22×C4—1221no—
347C22×C4—1221no—
348C22×C4—2441no—
349C22×C4C2×D24411yescharacteristic
350C22×C4D4×D6111yescharacteristic
351C22×C4#1175822611yescharacteristic
352Q16—2444no—
353SD16—2442no—
354SD16—2444no—
355SD16—2442no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—41no—
360C2×C8—1222no—
361C2×C8—2442no—
362C2×C8C2×D24411yescharacteristic
363C2×C8C2×D24411yescharacteristic
364C2×C8C2×D24411yescharacteristic
365C2×C8C2×D24411yescharacteristic
366C4⋊C4—612no—
367C4⋊C4—1222no—
368C4⋊C4—1221no—
369C4⋊C4—1222no—
370C4⋊C4—1221no—
371C4⋊C4—1221no—
372C4⋊C4—1221no—
373C4⋊C4—1221no—
374C4⋊C4—1221no—
375C4⋊C4—1221no—
376C4⋊C4—2441no—
377C4⋊C4—2441no—
378C22⋊C4—2441no—
379C22⋊C4—2441no—
380C22⋊C4—2441no—
381C42—1222no—
382C42—1222no—
383C42—2441no—
384C42C2×D24411yescharacteristic
385C42D4×D6111yescharacteristic
386C42#1175822611yescharacteristic
387C42C23×D6111yescharacteristic
388C23—2441no—
389C23#1451975211yescharacteristic
390Q8—614no—
391Q8—1224no—
392Q8—1222no—
393Q8—4881no—
394Q8—2442no—
395Q8—2441no—
396Q8—2441no—
397D4—4881no—
398D4—4882no—
399D4—2442no—
400D4—2441no—
401D4—2441no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—1224no—
407C2×C4—1222no—
408C2×C4—1222no—
409C2×C4—4882no—
410C2×C4—2441no—
411C2×C4—2441no—
412C2×C4—2441no—
413C2×C4—2442no—
414C2×C4#1451975211yescharacteristic
415C2×C4#1451975211yescharacteristic
416C2×C4#1451975211yescharacteristic
417C2×C4C2×D4×D6111yescharacteristic
418C2×C4C22×D24411yescharacteristic
419C2×C4C2×#1175822611yescharacteristic
420C8—21no—
421C8—21no—
422C8—21no—
423C8—42no—
424C8—42no—
425C8—2442no—
426C8#1451981012yes—
427C22—21no—
428C22—21no—
429C22—4881no—
430C22—4881no—
431C22—4881no—
432C22#1684002612yes—
433C22C2×#1451975211yescharacteristic
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—1224no—
439C4—4881no—
440C4—4881no—
441C4—4881no—
442C4—2442no—
443C4#135×D6112yes—
444C4C2×#1451981011yescharacteristic
445C4#1684056211yescharacteristic
446C2—22no—
447C2—9761no—
448C2#2145529311yescharacteristic
449C2#2145540111yescharacteristic
450C2C2×#1684002611yescharacteristic
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1568C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenter#1568D61
hypocenterC61#19263
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1568

Complements

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

NH ≅ G/Ntype
2. #21455950C2semidirect
4. C2×C8×Dic122C2semidirect
5. #21445278C2semidirect
6. #21445264C2semidirect
7. #21445202C2semidirect
8. #21445168C2semidirect
9. #21444520C2semidirect
11. C61×#1568C2semidirect
229. #1568D61semidirect
274. C61#19263semidirect

Maximal subgroups

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

subgroupindex
#214563782
#214559502
#214559422
C2×C8×Dic1222
#214452782
#214452642
#214452022
#214451682
#214445202
#214445152
C61×#15682
#1926361