Finite Group Databasebeta

← #29834813

Subgroups — #29834813

Order 15616.

590 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455402C211yesmaximal, characteristic
2C2×#16839799C211yesmaximal, characteristic, complemented
3#21448025C212yesmaximal, complemented
4#21448019C211yesmaximal, characteristic, complemented
5#21448012C212yesmaximal, complemented
6#21448005C211yesmaximal, characteristic, complemented
7#21447999C211yesmaximal, characteristic, complemented
8#21447602C212yesmaximal, complemented
9#21447525C212yesmaximal, complemented
10C22⋊C16×D61C211yesmaximal, characteristic, complemented
11C61×#2235C211yesmaximal, characteristic, complemented
12C2×#14519800C2211yescharacteristic
13C22×C8×D61C2211yescharacteristic
14#16840275—21no—
15C2×#14519735—21no—
16C2×C16×D61C2212yes—
17#16840000C2211yescharacteristic
18#16839999C2211yescharacteristic
19#16839998C2211yescharacteristic
20#16839994—21no—
21#16839993—21no—
22#16839799—21no—
23#16839799—21no—
24#16839799C2211yescharacteristic
25#16839799C2212yes—
26#16839799C2212yes—
27C2×#14519711C2211yescharacteristic
28C2×#14519708—21no—
29#16839757—21no—
30#16839590C2212yescomplemented
31#16839587C2212yescomplemented
32#16839587C2212yescomplemented
33#16839566C2212yescomplemented
34#16839564C2212yescomplemented
35#16839562C2212yescomplemented
36C22⋊#11758185C2211yescharacteristic
37#16839541C2212yes—
38C61×#467C2211yescharacteristic
39C2×#107×C61C2211yescharacteristic
40C61×#367C2212yescomplemented
41C61×#358C2212yescomplemented
42C61×C22⋊C16C2211yescharacteristic
43C2×#11758226—21no—
44C22×C4×D61D411yescharacteristic
45#14519800—24no—
46C2×#11758201—21no—
47C2×D488C2311yescharacteristic
48C22×#92418D411yescharacteristic
49C2×C8×D61—24no—
50C2×C8×D61D411yescharacteristic
51C2×C8×D61D411yescharacteristic
52C2×C8×D61C2311yescharacteristic
53C2×C8×D61C2311yescharacteristic
54#14519762—41no—
55#14519753—21no—
56#14519752—21no—
57#14519751—21no—
58#14519747—41no—
59#14519739—41no—
60#14519737—41no—
61#14519735—22no—
62#14519735—22no—
63#14519735—21no—
64#14519735—41no—
65#14519735—41no—
66C2×C4⋊Dic61—21no—
67C2×#11758198—41no—
68C2×C4×Dic61—41no—
69C2×#11758185C2312yes—
70C16×D61—24no—
71#14519711—21no—
72#14519711C2311yescharacteristic
73#14519711C2312yes—
74#14519708—21no—
75#14519708—22no—
76#14519708—22no—
77#14519701—21no—
78#14519699—21no—
79#14519697—21no—
80C2×Q16×C61C2311yescharacteristic
81C22×C8×C61C2311yescharacteristic
82C22⋊Q8×C61—21no—
83C2×C4⋊C4×C61—21no—
84C2×C16×C61C2312yes—
85#107×C61—21no—
86#107×C61C2311yescharacteristic
87#107×C61C2312yes—
88#103×C61—21no—
89C23×D61D811yescharacteristic
90C2×Q8×C61—21no—
91C22×C4×C61—41no—
92C22×C4×C61C2×D411yescharacteristic
93C2×C22⋊D61—41no—
94C22×Dic61—41no—
95C22×Dic61D811yescharacteristic
96#11758226—42no—
97#11758226—42no—
98C2×D244—21no—
99C2×D244—41no—
100C2×D244—41no—
101C2×C4×D61—22no—
102C2×C4×D61—22no—
103C2×C4×D61—41no—
104C2×C4×D61D811yescharacteristic
105C2×C4×D61D811yescharacteristic
106C2×C4×D61C2×D411yescharacteristic
107C2×C4×D61C2×D411yescharacteristic
108Q16×C61—22no—
109C2×C8×C61—22no—
110C2×C8×C61C2×D411yescharacteristic
111C2×C8×C61C2411yescharacteristic
112C4⋊C4×C61—21no—
113C4⋊C4×C61—22no—
114C4⋊C4×C61—41no—
115C4⋊C4×C61—41no—
116C22⋊C4×C61—41no—
117#11758201—42no—
118#11758198—42no—
119#11758198—42no—
120#11758198—41no—
121#11758198—41no—
122#11758198—41no—
123#11758198—41no—
124C4⋊Dic61—21no—
125C4⋊Dic61—22no—
126C4⋊Dic61—41no—
127#11758196—41no—
128C4×Dic61—42no—
129C4×Dic61—42no—
130C4×Dic61—41no—
131C2×#92418—22no—
132C2×#92418C2×D411yescharacteristic
133C2×#92418C2×D411yescharacteristic
134D488—22no—
135C8×D61—22no—
136C8×D61—22no—
137C8×D61—22no—
138C8×D61#4114yes—
139C16×C61—22no—
140#11758185—22no—
141C23×C61C2×D811yescharacteristic
142C22×D61—22no—
143C22×D61—22no—
144C22×D61—41no—
145C22×D61—41no—
146C22×D61C2×D811yescharacteristic
147C22×D61C2×D811yescharacteristic
148Q8×C61—41no—
149Q8×C61—41no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—42no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—82no—
156C2×C4×C61C2×D811yescharacteristic
157C2×C4×C61C22×D411yescharacteristic
158C22⋊D61—82no—
159C2×Dic61—22no—
160C2×Dic61—42no—
161C2×Dic61—41no—
162C2×Dic61—82no—
163C2×Dic61C2×D811yescharacteristic
164C2×Dic61C2×D811yescharacteristic
165D244—41no—
166D244—41no—
167D244—82no—
168D244—82no—
169C4×D61—22no—
170C4×D61—21no—
171C4×D61—21no—
172C4×D61—22no—
173C4×D61—82no—
174C4×D61#12312yes—
175C4×D61#13712yes—
176C8×C61—21no—
177C8×C61C2×#4112yes—
178#92418—21no—
179#92418#12312yes—
180#33645—611nomaximal, Sylow
181C22×C61—21no—
182C22×C61—21no—
183C22×C61#50812yes—
184C22×C61C22×D811yescharacteristic
185C2×D61—21no—
186C2×D61—21no—
187C2×D61—21no—
188C2×D61—21no—
189C2×D61—21no—
190C2×D61—44no—
191C2×D61—81no—
192C2×D61—81no—
193C2×D61—81no—
194C2×D61#48012yes—
195C4×C61—21no—
196C4×C61—81no—
197C4×C61—82no—
198C4×C61—81no—
199C4×C61C2×#12311yescharacteristic
200C4×C61C2×#13711yescharacteristic
201Dic61—21no—
202Dic61—81no—
203Dic61—82no—
204Dic61#48012yes—
205#3050—611no—
206C2×#427—611no—
207#2237—612no—
208#2236—611no—
209#2235D6111yescharacteristic, complemented
210#2234—612no—
211#2233—611no—
212#2232—611no—
213#2158—612no—
214#2138—612no—
215C2×C22⋊C16—611no—
216C2×C61—22no—
217C2×C61C2×#48011yescharacteristic
218C2×C61C2×#50811yescharacteristic
219C2×C61#341411yescharacteristic
220D61—22no—
221D61—41no—
222D61—161no—
223C2×#135—611no—
224C23×C8—611no—
225#525—1221no—
226C2×#117—1221no—
227C22×C16—612no—
228#467—611no—
229#467C2×D6111yescharacteristic
230#466—611no—
231#466—611no—
232#465—1221no—
233#465—1221no—
234#427—612no—
235#427—612no—
236#427—611no—
237#427—1221no—
238#427—1221no—
239C2×#107—611no—
240C2×#107C2×D6111yescharacteristic
241C2×#106—1221no—
242#416—1221no—
243#367—612no—
244#367C2×D6112yescomplemented
245#366—612no—
246#366—612no—
247#358—612no—
248#358C2×D6112yescomplemented
249#357—612no—
250#357—612no—
251C22⋊C16—611no—
252C22⋊C16—612no—
253C22⋊C16C2×D6111yescharacteristic
254C61#3364511yescharacteristic, complemented, Sylow
255C2×#41—1221no—
256C23×C4—611no—
257#135—1224no—
258C2×Q16C22×D6111yescharacteristic
259C2×SD16—1221no—
260C2×D8—611no—
261C22×C8—611no—
262C22×C8—611no—
263C22×C8—611no—
264C22×C8—611no—
265C22×C8—611no—
266C22×C8—1224no—
267C22×C8C22×D6111yescharacteristic
268#123—2441no—
269C22⋊Q8—21no—
270C22⋊Q8—1221no—
271#121—1221no—
272#121—1221no—
273#120—2441no—
274C4×D4—2441no—
275C4×D4—2441no—
276#117—1222no—
277#117—1222no—
278#117—1221no—
279#117—2441no—
280#117—2441no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—1221no—
283C2×C22⋊C4—2441no—
284C2×C42—2441no—
285C2×C16—612no—
286C2×C16—1224no—
287C2×C16C22×D6112yes—
288#107—21no—
289#107—612no—
290#107—611no—
291#107—1221no—
292#107C22×D6112yes—
293#107C22×D6111yescharacteristic
294#106—1222no—
295#106—1222no—
296#106—1221no—
297#103—21no—
298#103—1221no—
299#102—1221no—
300#102—1221no—
301C24—611no—
302#41—2442no—
303#41—2442no—
304C2×Q8—21no—
305C2×D4—1221no—
306C2×D4—2441no—
307C2×D4—2441no—
308C2×D4—2441no—
309C22×C4—41no—
310C22×C4—611no—
311C22×C4—611no—
312C22×C4—611no—
313C22×C4—611no—
314C22×C4—611no—
315C22×C4—1222no—
316C22×C4—1222no—
317C22×C4—2441no—
318C22×C4—2441no—
319C22×C4D4×D6111yescharacteristic
320Q16—22no—
321SD16—2442no—
322D8—1222no—
323C2×C8—22no—
324C2×C8—611no—
325C2×C8—614no—
326C2×C8—611no—
327C2×C8—1222no—
328C2×C8—1222no—
329C2×C8—1222no—
330C2×C8—1222no—
331C2×C8D4×D6111yescharacteristic
332C2×C8C23×D6111yescharacteristic
333C4⋊C4—21no—
334C4⋊C4—22no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—1221no—
338C4⋊C4—1222no—
339C4⋊C4—2441no—
340C4⋊C4—2441no—
341C22⋊C4—41no—
342C22⋊C4—2441no—
343C22⋊C4—2442no—
344C22⋊C4—2442no—
345C22⋊C4—2441no—
346C22⋊C4—2441no—
347C22⋊C4—2441no—
348C42—2442no—
349C42—2442no—
350C42—2441no—
351C16—22no—
352C16—1222no—
353C23—611no—
354C23—611no—
355C23—1222no—
356C23—1222no—
357C23—2441no—
358C23—2441no—
359C23D8×D6111yescharacteristic
360Q8—41no—
361Q8—41no—
362D4—4882no—
363D4—4882no—
364D4—4882no—
365D4—2441no—
366D4—2441no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—82no—
373C2×C4—611no—
374C2×C4—612no—
375C2×C4—612no—
376C2×C4—611no—
377C2×C4—1222no—
378C2×C4—1222no—
379C2×C4—1222no—
380C2×C4—1221no—
381C2×C4—1221no—
382C2×C4—4882no—
383C2×C4—4882no—
384C2×C4—2442no—
385C2×C4—2441no—
386C2×C4D8×D6111yescharacteristic
387C2×C4C2×D4×D6111yescharacteristic
388C8—21no—
389C8—612no—
390C8—1221no—
391C8#41×D6112yes—
392C22—21no—
393C22—21no—
394C22—612no—
395C22—1221no—
396C22—1221no—
397C22—1221no—
398C22—1221no—
399C22—1221no—
400C22—4881no—
401C22—4881no—
402C22—4881no—
403C22—2444no—
404C22#1684019312yes—
405C22C2×D8×D6111yescharacteristic
406C4—21no—
407C4—81no—
408C4—81no—
409C4—82no—
410C4—612no—
411C4—1221no—
412C4—4882no—
413C4—4881no—
414C4#123×D6111yescharacteristic
415C4#137×D6111yescharacteristic
416C2—22no—
417C2—1222no—
418C2—9761no—
419C2—2441no—
420C2#480×D6111yescharacteristic
421C2C2×#1684019311yescharacteristic
422C2#2145895411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2235C2
socleC22×C61C22×D8
prime radical#21455402C2
hypercenter#2235D61
hypocenterC61#33645
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2235

Complements

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

NH ≅ G/Ntype
2. C2×#16839799C2semidirect
3. #21448025C2semidirect
4. #21448019C2semidirect
5. #21448012C2semidirect
6. #21448005C2semidirect
7. #21447999C2semidirect
8. #21447602C2semidirect
9. #21447525C2semidirect
10. C22⋊C16×D61C2semidirect
11. C61×#2235C2semidirect
30. #16839590C22semidirect
31. #16839587C22semidirect
32. #16839587C22semidirect
33. #16839566C22semidirect
34. #16839564C22semidirect
35. #16839562C22semidirect
40. C61×#367C22semidirect
41. C61×#358C22semidirect
209. #2235D61semidirect
244. #367C2×D61semidirect
248. #358C2×D61semidirect
254. C61#33645semidirect

Maximal subgroups

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

subgroupindex
#214554022
C2×#168397992
#214480252
#214480192
#214480122
#214480052
#214479992
#214476022
#214475252
C22⋊C16×D612
C61×#22352
#3364561