Finite Group Databasebeta

← #29663472

Subgroups — #29663472

Order 15616.

584 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457157C212yesmaximal
2#21456405C211yesmaximal, characteristic, complemented
3#21454794C211yesmaximal, characteristic
4#21445338C211yesmaximal, characteristic, complemented
5#21445333C211yesmaximal, characteristic, complemented
6#21445314C211yesmaximal, characteristic, complemented
7#21445309C211yesmaximal, characteristic, complemented
8#21445250C212yesmaximal, complemented
9#21445243C212yesmaximal, complemented
10#21444517C212yesmaximal, complemented
11C2×C61×#334C211yesmaximal, characteristic, complemented
12#16840319C2212yes—
13#16840296C2211yescharacteristic
14#16840167C2211yescharacteristic
15#16840142C2211yescharacteristic
16#16840113C2212yes—
17#16840070—22no—
18#16840048C2212yes—
19#16840043C2212yes—
20#16840003—22no—
21#16840003—21no—
22#16840003—21no—
23#16839991—22no—
24#16839985—22no—
25#16839973C2212yes—
26#16839968C2212yes—
27#16839892C2211yescharacteristic
28C8×Dic122C2211yescharacteristic
29C4×Dic244—21no—
30C4×Dic244—21no—
31#16839841C2211yescharacteristic
32#16839833—21no—
33C8×D244C2211yescharacteristic
34C8×C22⋊D61—21no—
35#16839821—21no—
36#16839819C2211yescharacteristic
37#16839716—21no—
38#16839509C2212yes—
39#16839509C2212yes—
40#16839507C2212yes—
41#16839507C2212yes—
42C2×C4⋊C8×C61C2212yes—
43C2×C4×C8×C61C2211yescharacteristic
44C61×#334C2212yes—
45C61×#334C2211yescharacteristic
46C61×#334C2211yescharacteristic
47C2×Dic244—21no—
48C2×Dic244—21no—
49C2×Dic244—22no—
50C2×#11758190—22no—
51C2×#11758189—21no—
52C2×C8×D61—21no—
53#14519777C2312yes—
54#14519777C2312yes—
55#14519771—21no—
56#14519769—22no—
57#14519769D411yescharacteristic
58#14519765D411yescharacteristic
59#14519760—21no—
60#14519756—22no—
61#14519756—22no—
62#14519756D411yescharacteristic
63#14519751D411yescharacteristic
64C4×Dic122—22no—
65C4×Dic122—22no—
66C4×Dic122C2311yescharacteristic
67C4×D244C2311yescharacteristic
68C4×C22⋊D61—21no—
69#14519736—22no—
70#14519734—21no—
71#14519711—21no—
72#14519711—21no—
73#14519711—22no—
74#14519709—22no—
75C4⋊#92418C2311yescharacteristic
76C4⋊#92418C2311yescharacteristic
77#14519706—21no—
78#14519706—21no—
79#14519702—22no—
80#14519702—22no—
81#14519702—22no—
82#14519702—22no—
83#14519702—22no—
84#14519699—22no—
85C22⋊#92418—21no—
86#14519689—21no—
87#14519689—21no—
88#14519688—21no—
89#14519687—21no—
90C8×Dic61—21no—
91C22×C8×C61—22no—
92C22×C8×C61D411yescharacteristic
93C22×C8×C61D411yescharacteristic
94C2×C42×C61C2311yescharacteristic
95C4⋊C8×C61—22no—
96C4⋊C8×C61C2312yes—
97C4⋊C8×C61C2312yes—
98C4×C8×C61D411yescharacteristic
99C4×C8×C61D411yescharacteristic
100C4×C8×C61C2311yescharacteristic
101C4×C8×C61C2311yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61#4111yescharacteristic
105C2×C22⋊D61—21no—
106C2×D244C2×D411yescharacteristic
107C2×C4×D61—21no—
108C2×Dic122—22no—
109C2×Dic122—22no—
110C2×Dic122C2×D411yescharacteristic
111C2×C8×C61—22no—
112C2×C8×C61—22no—
113C2×C8×C61—22no—
114C2×C8×C61—41no—
115C2×C8×C61—82no—
116C2×C8×C61C2×D411yescharacteristic
117C2×C8×C61C2×D411yescharacteristic
118C2×C8×C61C2×D411yescharacteristic
119C2×C8×C61C2×D411yescharacteristic
120C42×C61C2×D411yescharacteristic
121C42×C61C2×D411yescharacteristic
122C42×C61#4111yescharacteristic
123C42×C61C2411yescharacteristic
124C22⋊Dic61—21no—
125C22⋊Dic61—42no—
126#11758198—21no—
127#11758198—21no—
128#11758198—21no—
129#11758198—21no—
130C4⋊Dic61—22no—
131C4⋊Dic61—22no—
132C4⋊Dic61—22no—
133C4⋊Dic61—22no—
134C4⋊Dic61C2×D411yescharacteristic
135C4⋊Dic61C2×D411yescharacteristic
136#11758196—21no—
137#11758196—21no—
138#11758196—21no—
139#11758196—21no—
140#11758196—42no—
141#11758196—42no—
142C4×Dic61—21no—
143C4×Dic61—42no—
144C2×#92418—21no—
145C2×#92418—21no—
146Dic244—42no—
147Dic244—42no—
148Dic244—82no—
149#11758190—82no—
150#11758189—42no—
151C8×D61—42no—
152C23×C61#12111yescharacteristic
153C22×D61—21no—
154C2×C4×C61—22no—
155C2×C4×C61—21no—
156C2×C4×C61—21no—
157C2×C4×C61—22no—
158C2×C4×C61#12111yescharacteristic
159C2×C4×C61#12111yescharacteristic
160C2×C4×C61#12111yescharacteristic
161C2×C4×C61C22×D411yescharacteristic
162C2×C4×C61C22×D411yescharacteristic
163C2×C4×C61C2×#4111yescharacteristic
164C22⋊D61—42no—
165C2×Dic61—21no—
166C2×Dic61—21no—
167C2×Dic61—21no—
168C2×Dic61—42no—
169C2×Dic61—42no—
170D244—22no—
171C4×D61—41no—
172C4×D61—41no—
173Dic122—22no—
174Dic122—42no—
175Dic122—42no—
176Dic122—42no—
177Dic122—42no—
178C8×C61—21no—
179C8×C61—21no—
180C8×C61—21no—
181C8×C61—42no—
182C8×C61—42no—
183C8×C61#13512yes—
184#92418—41no—
185#92418—41no—
186#19228—611nomaximal, Sylow
187C22×C61—21no—
188C22×C61—21no—
189C22×C61#47312yes—
190C22×C61C2×#12111yescharacteristic
191C2×D61—41no—
192C2×D61—41no—
193C2×D61—41no—
194C4×C61—21no—
195C4×C61—21no—
196C4×C61—21no—
197C4×C61—21no—
198C4×C61C2×#13511yescharacteristic
199C4×C61#57511yescharacteristic
200C4×C61#57712yes—
201Dic61—41no—
202Dic61—41no—
203Dic61—41no—
204Dic61—82no—
205Dic61—82no—
206#3248—612no—
207#3170—611no—
208#2966—611no—
209#1694—611no—
210#1693—611no—
211#1690—611no—
212#1689—611no—
213#1680—612no—
214#1679—612no—
215C2×#334D6111yescharacteristic, complemented
216#1552—612no—
217C2×C61—22no—
218C2×C61#303511yescharacteristic
219C2×C61#306011yescharacteristic
220C2×C61C2×#47311yescharacteristic
221D61—81no—
222#533—612no—
223#529—611no—
224#500—611no—
225#494—611no—
226#487—612no—
227#479—1222no—
228#475—612no—
229#474—612no—
230#467—1222no—
231#467—1221no—
232#467—1221no—
233#465—1222no—
234#464—1222no—
235#462—612no—
236#461—612no—
237#446—611no—
238C8×Q8—611no—
239C4×Q16—1221no—
240C4×Q16—1221no—
241#436—611no—
242#435—1221no—
243C8×D4—611no—
244C8×D4—1221no—
245#433—1221no—
246#432—611no—
247C2×C4⋊C8C2×D6112yes—
248#405—1221no—
249C2×C4×C8C2×D6111yescharacteristic
250#334C2×D6112yes—
251#334C2×D6111yescharacteristic
252#334C2×D6111yescharacteristic
253#332—612no—
254#332—612no—
255#332—612no—
256#332—612no—
257C61#1922811yescharacteristic, complemented, Sylow
258C2×Q16—1222no—
259C2×Q16—1221no—
260C2×Q16—1221no—
261C2×SD16—1222no—
262C2×M16—1221no—
263C22×C8—22no—
264C22×C8—1221no—
265C22×C8D24411yescharacteristic
266C22×C8D24411yescharacteristic
267C4⋊Q8—612no—
268C4⋊Q8—612no—
269#126—1221no—
270#125—611no—
271#125—1222no—
272#124—611no—
273#123—1221no—
274C22⋊Q8—611no—
275C22⋊Q8—1222no—
276C22⋊Q8—1222no—
277#121—611no—
278C4×Q8—611no—
279C4×Q8—1222no—
280C4×Q8—1222no—
281C4×D4—611no—
282C4×D4—1221no—
283#117—1221no—
284#117—1222no—
285C2×C42C22×D6111yescharacteristic
286#107—1222no—
287#107—1221no—
288#107—1221no—
289#106—1222no—
290C4⋊C8—22no—
291C4⋊C8—611no—
292C4⋊C8—611no—
293C4⋊C8—1221no—
294C4⋊C8—1221no—
295C4⋊C8C22×D6112yes—
296C4⋊C8C22×D6112yes—
297#103—1222no—
298#103—1222no—
299#103—1222no—
300#103—1222no—
301#103—1222no—
302#102—1222no—
303C22⋊C8—1221no—
304C22⋊C8—1221no—
305C22⋊C8—1221no—
306#97—1221no—
307#97—1221no—
308C4×C8—1221no—
309C4×C8D24411yescharacteristic
310C4×C8D24411yescharacteristic
311C4×C8C22×D6111yescharacteristic
312C4×C8C22×D6111yescharacteristic
313C2×Q8—611no—
314C2×Q8—1222no—
315C2×Q8—1222no—
316C2×D4—611no—
317C2×D4—1221no—
318C22×C4—1221no—
319C22×C4C2×D24411yescharacteristic
320C22×C4D4×D6111yescharacteristic
321C22×C4#1175822611yescharacteristic
322Q16—4882no—
323Q16—2442no—
324Q16—2442no—
325SD16—4882no—
326M16—2442no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—41no—
331C2×C8—82no—
332C2×C8—1221no—
333C2×C8—1221no—
334C2×C8—2442no—
335C2×C8C2×D24411yescharacteristic
336C2×C8C2×D24411yescharacteristic
337C2×C8C2×D24411yescharacteristic
338C2×C8C2×D24411yescharacteristic
339C4⋊C4—611no—
340C4⋊C4—611no—
341C4⋊C4—1221no—
342C4⋊C4—1221no—
343C4⋊C4—1221no—
344C4⋊C4—1221no—
345C4⋊C4—1222no—
346C4⋊C4—1222no—
347C4⋊C4—1222no—
348C4⋊C4—1222no—
349C4⋊C4—2442no—
350C4⋊C4—2442no—
351C22⋊C4—1221no—
352C22⋊C4—1221no—
353C22⋊C4—1221no—
354C22⋊C4—1221no—
355C22⋊C4—1221no—
356C22⋊C4—2442no—
357C42—1221no—
358C42—2442no—
359C42C2×D24411yescharacteristic
360C42D4×D6111yescharacteristic
361C42#1175822611yescharacteristic
362C42C23×D6111yescharacteristic
363C23—1221no—
364C23#1451975211yescharacteristic
365Q8—1222no—
366Q8—2442no—
367Q8—2442no—
368Q8—2442no—
369Q8—2442no—
370D4—1222no—
371D4—2442no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—1221no—
377C2×C4—1221no—
378C2×C4—1221no—
379C2×C4—2442no—
380C2×C4—2442no—
381C2×C4—2441no—
382C2×C4—2441no—
383C2×C4#1451975211yescharacteristic
384C2×C4#1451975211yescharacteristic
385C2×C4#1451975211yescharacteristic
386C2×C4C2×D4×D6111yescharacteristic
387C2×C4C22×D24411yescharacteristic
388C2×C4C2×#1175822611yescharacteristic
389C8—21no—
390C8—21no—
391C8—21no—
392C8—42no—
393C8—42no—
394C8—2441no—
395C8—2441no—
396C8#1451979912yes—
397C22—21no—
398C22—21no—
399C22—2441no—
400C22—2441no—
401C22—2441no—
402C22#1684003812yes—
403C22C2×#1451975211yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—4882no—
409C4—4882no—
410C4—2441no—
411C4—2441no—
412C4—2441no—
413C4C2×#1451979911yescharacteristic
414C4#1684056211yescharacteristic
415C4#1684056812yes—
416C2—22no—
417C2—4881no—
418C2#2145527611yescharacteristic
419C2#2145548211yescharacteristic
420C2C2×#1684003811yescharacteristic
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#334C2
socleC22×C61C2×#121
prime radical#14519751D4
hypercenterC2×#334D61
hypocenterC61#19228
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#334

Complements

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

NH ≅ G/Ntype
2. #21456405C2semidirect
4. #21445338C2semidirect
5. #21445333C2semidirect
6. #21445314C2semidirect
7. #21445309C2semidirect
8. #21445250C2semidirect
9. #21445243C2semidirect
10. #21444517C2semidirect
11. C2×C61×#334C2semidirect
215. C2×#334D61semidirect
257. C61#19228semidirect

Maximal subgroups

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

subgroupindex
#214571572
#214564052
#214547942
#214453382
#214453332
#214453142
#214453092
#214452502
#214452432
#214445172
C2×C61×#3342
#1922861