Finite Group Databasebeta

← #29834844

Subgroups — #29834844

Order 15616.

580 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#467×D61C211yesmaximal, characteristic
2C2×#16839799C211yesmaximal, characteristic
3#21448029C212yesmaximal, complemented
4#21448025C212yesmaximal, complemented
5#21448021C211yesmaximal, characteristic, complemented
6#21448017C211yesmaximal, characteristic, complemented
7#21448014C211yesmaximal, characteristic, complemented
8#21447604C212yesmaximal, complemented
9#21447537C212yesmaximal, complemented
10#21447428C211yesmaximal, characteristic
11C61×#2236C211yesmaximal, characteristic, complemented
12C2×Q16×D61C2211yescharacteristic
13C22×C8×D61C2211yescharacteristic
14C22⋊Q8×D61—21no—
15C2×#14519735—21no—
16C2×#14519716C2212yes—
17#16840004C2211yescharacteristic
18#16840003C2211yescharacteristic
19#16840002—21no—
20#16840001—21no—
21#16840000C2211yescharacteristic
22#16839799—21no—
23#16839799—21no—
24#16839799C2212yes—
25#16839799C2212yes—
26C2×#14519711C2211yescharacteristic
27#107×D61C2211yescharacteristic
28C2×#14519708—21no—
29#103×D61—21no—
30#16839590C2212yescomplemented
31#16839589C2212yescomplemented
32#16839587C2212yescomplemented
33#16839567C2212yescomplemented
34#16839566C2212yescomplemented
35#16839565C2212yescomplemented
36C22⋊#11758185C2211yescharacteristic
37#16839541C2212yes—
38C61×#467C2211yescharacteristic
39C2×#107×C61C2211yescharacteristic
40C61×#366C2212yescomplemented
41C61×#358C2212yescomplemented
42C61×C22⋊C16C2211yescharacteristic
43C2×Q8×D61—21no—
44C22×C4×D61D411yescharacteristic
45C2×Dic244C2311yescharacteristic
46C2×#11758202—21no—
47Q16×D61—24no—
48C22×#92418D411yescharacteristic
49C2×C8×D61—24no—
50C2×C8×D61D411yescharacteristic
51C2×C8×D61D411yescharacteristic
52C2×C8×D61C2311yescharacteristic
53C2×C8×D61C2311yescharacteristic
54#14519757—21no—
55#14519756—21no—
56#14519755—41no—
57#14519754—41no—
58#14519753—21no—
59#14519735—22no—
60#14519735—22no—
61#14519735—41no—
62#14519735—41no—
63C2×C4⋊Dic61—21no—
64C4⋊C4×D61—21no—
65C4⋊C4×D61—41no—
66C2×#11758198—41no—
67C22⋊C4×D61—41no—
68C2×C4×Dic61—41no—
69#14519716—24no—
70C2×#11758185C2312yes—
71#14519711—21no—
72#14519711C2311yescharacteristic
73#14519711C2312yes—
74#14519710—21no—
75#14519708—22no—
76#14519708—22no—
77#14519702—21no—
78#14519701—21no—
79#14519700—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
93C22×Dic61—41no—
94C22×Dic61D811yescharacteristic
95Q8×D61—42no—
96Q8×D61—42no—
97C2×C4×D61—22no—
98C2×C4×D61—22no—
99C2×C4×D61—41no—
100C2×C4×D61—41no—
101C2×C4×D61D811yescharacteristic
102C2×C4×D61D811yescharacteristic
103C2×C4×D61C2×D411yescharacteristic
104C2×C4×D61C2×D411yescharacteristic
105C2×Dic122—21no—
106C2×Dic122—41no—
107Q16×C61—22no—
108C2×C8×C61—22no—
109C2×C8×C61C2×D411yescharacteristic
110C2×C8×C61C2411yescharacteristic
111C4⋊C4×C61—21no—
112C4⋊C4×C61—22no—
113C4⋊C4×C61—41no—
114C4⋊C4×C61—41no—
115C22⋊C4×C61—41no—
116C22⋊Dic61—41no—
117#11758202—42no—
118#11758198—41no—
119#11758198—41no—
120#11758198—42no—
121#11758198—42no—
122C4⋊Dic61—21no—
123C4⋊Dic61—22no—
124C4⋊Dic61—41no—
125C4⋊Dic61—41no—
126#11758196—41no—
127#11758196—41no—
128#11758196—41no—
129C4×Dic61—42no—
130C4×Dic61—42no—
131C2×#92418—22no—
132C2×#92418C2×D411yescharacteristic
133C2×#92418C2×D411yescharacteristic
134Dic244—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×D61C2×D811yescharacteristic
146C22×D61C2×D811yescharacteristic
147Q8×C61—41no—
148Q8×C61—41no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—42no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—82no—
155C2×C4×C61C2×D811yescharacteristic
156C2×C4×C61C22×D411yescharacteristic
157C2×Dic61—22no—
158C2×Dic61—42no—
159C2×Dic61—41no—
160C2×Dic61—41no—
161C2×Dic61—82no—
162C2×Dic61C2×D811yescharacteristic
163C2×Dic61C2×D811yescharacteristic
164C4×D61—22no—
165C4×D61—21no—
166C4×D61—21no—
167C4×D61—22no—
168C4×D61—82no—
169C4×D61—82no—
170C4×D61#12312yes—
171C4×D61#13712yes—
172Dic122—41no—
173Dic122—41no—
174Dic122—82no—
175C8×C61—21no—
176C8×C61C2×#4112yes—
177#92418—21no—
178#92418#12312yes—
179#33649—611nomaximal, Sylow
180C22×C61—21no—
181C22×C61—21no—
182C22×C61#51012yes—
183C22×C61C22×D811yescharacteristic
184C2×D61—21no—
185C2×D61—21no—
186C2×D61—21no—
187C2×D61—21no—
188C2×D61—21no—
189C2×D61—44no—
190C2×D61#48012yes—
191C4×C61—21no—
192C4×C61—81no—
193C4×C61—81no—
194C4×C61—82no—
195C4×C61C2×#12311yescharacteristic
196C4×C61C2×#13711yescharacteristic
197Dic61—21no—
198Dic61—81no—
199Dic61—82no—
200Dic61—81no—
201Dic61#48012yes—
202C2×#467—611no—
203C2×#427—611no—
204#2237—612no—
205#2237—612no—
206#2236—611no—
207#2236D6111yescharacteristic, complemented
208#2235—611no—
209#2235—611no—
210#2159—612no—
211#2141—612no—
212#2112—611no—
213C2×C61—22no—
214C2×C61C2×#48011yescharacteristic
215C2×C61C2×#51011yescharacteristic
216C2×C61#341411yescharacteristic
217D61—22no—
218D61—41no—
219C22×Q16—611no—
220C23×C8—611no—
221C2×C22⋊Q8—1221no—
222C2×#117—1221no—
223C2×M32—612no—
224#467—611no—
225#467—611no—
226#467—611no—
227#467—1221no—
228#467—1221no—
229#467C2×D6111yescharacteristic
230#427—612no—
231#427—612no—
232#427—1221no—
233#427—1221no—
234C2×#107—611no—
235C2×#107—611no—
236C2×#107C2×D6111yescharacteristic
237C2×#106—1221no—
238C2×#103—1221no—
239#367—612no—
240#367—612no—
241#366—612no—
242#366C2×D6112yescomplemented
243#358—612no—
244#358—612no—
245#358—612no—
246#358C2×D6112yescomplemented
247C22⋊C16—611no—
248C22⋊C16—612no—
249C22⋊C16C2×D6111yescharacteristic
250C61#3364911yescharacteristic, complemented, Sylow
251C22×Q8—1221no—
252C23×C4—611no—
253C2×Q16—611no—
254C2×Q16—1224no—
255C2×Q16—1221no—
256C2×Q16C22×D6111yescharacteristic
257C22×C8—611no—
258C22×C8—611no—
259C22×C8—611no—
260C22×C8—611no—
261C22×C8—611no—
262C22×C8—1224no—
263C22×C8C22×D6111yescharacteristic
264C22⋊Q8—21no—
265C22⋊Q8—1221no—
266C22⋊Q8—1221no—
267C22⋊Q8—1221no—
268C22⋊Q8—2441no—
269C22⋊Q8—2441no—
270#117—1222no—
271#117—1222no—
272#117—2441no—
273#117—2441no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—1221no—
276C2×C4⋊C4—1221no—
277C2×C4⋊C4—2441no—
278C2×C22⋊C4—2441no—
279C2×C22⋊C4—2441no—
280C2×C42—2441no—
281M32—1224no—
282C2×C16—612no—
283C2×C16C22×D6112yes—
284#107—21no—
285#107—612no—
286#107—611no—
287#107—1221no—
288#107—1221no—
289#107C22×D6112yes—
290#107C22×D6111yescharacteristic
291#106—1222no—
292#106—1222no—
293#103—21no—
294#103—1221no—
295#103—1221no—
296#103—1221no—
297C24—611no—
298C2×Q8—21no—
299C2×Q8—1221no—
300C2×Q8—2441no—
301C2×Q8—2442no—
302C2×Q8—2442no—
303C22×C4—41no—
304C22×C4—611no—
305C22×C4—611no—
306C22×C4—611no—
307C22×C4—611no—
308C22×C4—611no—
309C22×C4—1222no—
310C22×C4—1222no—
311C22×C4—2441no—
312C22×C4—2441no—
313C22×C4—2441no—
314C22×C4D4×D6111yescharacteristic
315Q16—22no—
316Q16—1222no—
317Q16—2442no—
318C2×C8—22no—
319C2×C8—611no—
320C2×C8—614no—
321C2×C8—611no—
322C2×C8—1222no—
323C2×C8—1222no—
324C2×C8—1222no—
325C2×C8—1222no—
326C2×C8D4×D6111yescharacteristic
327C2×C8C23×D6111yescharacteristic
328C4⋊C4—21no—
329C4⋊C4—22no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—1221no—
333C4⋊C4—1222no—
334C4⋊C4—2441no—
335C4⋊C4—2441no—
336C4⋊C4—2441no—
337C4⋊C4—2441no—
338C4⋊C4—2441no—
339C22⋊C4—41no—
340C22⋊C4—2442no—
341C22⋊C4—2442no—
342C22⋊C4—2441no—
343C22⋊C4—2441no—
344C22⋊C4—2441no—
345C42—2442no—
346C42—2442no—
347C16—22no—
348C16—1222no—
349C23—611no—
350C23—611no—
351C23—1222no—
352C23—1222no—
353C23—2441no—
354C23D8×D6111yescharacteristic
355Q8—41no—
356Q8—41no—
357Q8—4882no—
358Q8—2441no—
359Q8—2441no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—82no—
366C2×C4—611no—
367C2×C4—612no—
368C2×C4—612no—
369C2×C4—611no—
370C2×C4—1222no—
371C2×C4—1222no—
372C2×C4—1222no—
373C2×C4—1221no—
374C2×C4—1221no—
375C2×C4—4882no—
376C2×C4—4882no—
377C2×C4—4882no—
378C2×C4—2442no—
379C2×C4—2441no—
380C2×C4—2441no—
381C2×C4D8×D6111yescharacteristic
382C2×C4C2×D4×D6111yescharacteristic
383C8—21no—
384C8—612no—
385C8—1221no—
386C8#41×D6112yes—
387C22—21no—
388C22—21no—
389C22—612no—
390C22—1221no—
391C22—1221no—
392C22—1221no—
393C22—1221no—
394C22—1221no—
395C22—2444no—
396C22#1684020412yes—
397C22C2×D8×D6111yescharacteristic
398C4—21no—
399C4—82no—
400C4—81no—
401C4—81no—
402C4—612no—
403C4—1221no—
404C4—4882no—
405C4—4881no—
406C4—4881no—
407C4#123×D6111yescharacteristic
408C4#137×D6111yescharacteristic
409C2—22no—
410C2—1222no—
411C2—2441no—
412C2#480×D6111yescharacteristic
413C2C2×#1684020411yescharacteristic
414C2#2145895111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2236C2
socleC22×C61C22×D8
prime radicalC23×D61D8
hypercenter#2236D61
hypocenterC61#33649
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2236

Complements

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

NH ≅ G/Ntype
3. #21448029C2semidirect
4. #21448025C2semidirect
5. #21448021C2semidirect
6. #21448017C2semidirect
7. #21448014C2semidirect
8. #21447604C2semidirect
9. #21447537C2semidirect
11. C61×#2236C2semidirect
30. #16839590C22semidirect
31. #16839589C22semidirect
32. #16839587C22semidirect
33. #16839567C22semidirect
34. #16839566C22semidirect
35. #16839565C22semidirect
40. C61×#366C22semidirect
41. C61×#358C22semidirect
207. #2236D61semidirect
242. #366C2×D61semidirect
246. #358C2×D61semidirect
250. C61#33649semidirect

Maximal subgroups

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

subgroupindex
#467×D612
C2×#168397992
#214480292
#214480252
#214480212
#214480172
#214480142
#214476042
#214475372
#214474282
C61×#22362
#3364961