Finite Group Databasebeta

← #29709132

Subgroups — #29709132

Order 15616.

766 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#21454406C212yesmaximal
2C2×#16839799C211yesmaximal, characteristic
3#424×D61C211yesmaximal, characteristic
4#21447333C212yesmaximal, complemented
5#21447332C212yesmaximal, complemented
6#21447326C211yesmaximal, characteristic, complemented
7#21447325C211yesmaximal, characteristic, complemented
8#21447324C211yesmaximal, characteristic, complemented
9#21445556C212yesmaximal
10#341×D61C211yesmaximal, characteristic
11C61×#2089C211yesmaximal, characteristic, complemented
12C2×M16×D61C2211yescharacteristic
13C2×M16×D61C2212yes—
14C22×C8×D61C2211yescharacteristic
15C2×#14519735C2212yes—
16#117×D61C2211yescharacteristic
17#16839805C2212yes—
18#16839804C2212yes—
19#16839803—22no—
20#16839802C2212yes—
21#16839801—22no—
22#16839799—22no—
23#16839799—22no—
24#16839799—22no—
25#16839797—22no—
26C2×#14519711C2211yescharacteristic
27C2×#14519708C2212yes—
28#16839790C2211yescharacteristic
29#16839789C2211yescharacteristic
30#16839788C2211yescharacteristic
31#16839787—22no—
32C4⋊C8×D61—22no—
33#16839531C2212yes—
34#16839530C2212yes—
35#16839530C2212yes—
36#16839527C2211yescharacteristic
37#16839526C2212yes—
38C61×#428C2212yes—
39C2×#107×C61C2211yescharacteristic
40C61×#424C2211yescharacteristic
41C61×#343C2212yes—
42C61×#341C2211yescharacteristic
43C22×C4×D61C2311yescharacteristic
44C2×#11758194C2311yescharacteristic
45C2×#11758194C2312yes—
46C2×#11758189—22no—
47C2×#11758189—21no—
48M16×D61—24no—
49M16×D61—24no—
50M16×D61Q818yes—
51C22×#92418C2311yescharacteristic
52C2×C8×D61—22no—
53C2×C8×D61—24no—
54C2×C8×D61—21no—
55C2×C8×D61D412yes—
56C2×C8×D61D412yes—
57#14519736C2311yescharacteristic
58#14519735—22no—
59#14519735—22no—
60#14519735—22no—
61#14519735—22no—
62#14519735D411yescharacteristic
63#14519735D411yescharacteristic
64#14519734—21no—
65#14519733—21no—
66C2×C4⋊Dic61C2312yes—
67C4⋊C4×D61D411yescharacteristic
68C4⋊C4×D61D411yescharacteristic
69C2×#11758198—22no—
70C22⋊C4×D61—21no—
71C2×C4×Dic61—22no—
72C42×D61—21no—
73#14519711—22no—
74#14519711—22no—
75#14519710—22no—
76#14519710—22no—
77#14519709—22no—
78#14519708—22no—
79#14519708—22no—
80#14519708—24no—
81C4⋊#92418—22no—
82#14519706—22no—
83#14519706—22no—
84C2×M16×C61C2311yescharacteristic
85C2×M16×C61C2312yes—
86C22×C8×C61C2311yescharacteristic
87#117×C61C2311yescharacteristic
88C2×C4⋊C4×C61C2312yes—
89#107×C61—22no—
90#107×C61—22no—
91#106×C61—22no—
92C4⋊C8×C61—22no—
93C23×D61#4111yescharacteristic
94C22×C4×C61—22no—
95C22×C4×C61C2411yescharacteristic
96C22×Dic61—22no—
97C22×Dic61#4111yescharacteristic
98C2×C4×D61—21no—
99C2×C4×D61—21no—
100C2×C4×D61C2×D411yescharacteristic
101C2×C4×D61C2×D411yescharacteristic
102C2×C4×D61C2×D411yescharacteristic
103C2×C4×D61C2×D411yescharacteristic
104C2×C4×D61C2×Q814yes—
105C2×C4×D61#4114yes—
106M16×C61—22no—
107M16×C61—22no—
108M16×C61C2×Q814yes—
109C2×C8×C61—22no—
110C2×C8×C61—22no—
111C2×C8×C61—21no—
112C2×C8×C61C2×D412yes—
113C4⋊C4×C61—22no—
114C4⋊C4×C61—22no—
115C4⋊C4×C61C2×D411yescharacteristic
116C4⋊C4×C61C2×D411yescharacteristic
117C22⋊C4×C61—21no—
118C42×C61—21no—
119C22⋊Dic61—21no—
120#11758198—21no—
121#11758198—21no—
122#11758198—42no—
123#11758198—42no—
124C4⋊Dic61—22no—
125C4⋊Dic61—22no—
126C4⋊Dic61C2×D411yescharacteristic
127C4⋊Dic61C2×D411yescharacteristic
128#11758196—21no—
129#11758196—21no—
130C4×Dic61—21no—
131C4×Dic61—21no—
132C4×Dic61—21no—
133C4×Dic61—42no—
134C4×Dic61—42no—
135#11758194—22no—
136#11758194—22no—
137#11758194C2×Q814yes—
138C2×#92418—22no—
139C2×#92418—22no—
140C2×#92418—21no—
141C2×#92418C2×D412yes—
142#11758189—28no—
143#11758189—42no—
144C8×D61—28no—
145C8×D61—24no—
146C8×D61—24no—
147C8×D61—42no—
148C23×C61C2×#4111yescharacteristic
149C22×D61—41no—
150C22×D61—41no—
151C22×D61#12111yescharacteristic
152C22×D61#12111yescharacteristic
153C22×D61#12514yes—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—42no—
157C2×C4×C61—82no—
158C2×C4×C61C22×D411yescharacteristic
159C2×C4×C61C22×D411yescharacteristic
160C2×C4×C61C22×Q812yes—
161C2×C4×C61C2×#4112yes—
162C2×Dic61—21no—
163C2×Dic61—21no—
164C2×Dic61—42no—
165C2×Dic61—82no—
166C2×Dic61#12111yescharacteristic
167C2×Dic61#12111yescharacteristic
168C2×Dic61#12514yes—
169C4×D61—42no—
170C4×D61—42no—
171C4×D61C22⋊Q814yes—
172C4×D61C22⋊Q814yes—
173C4×D61#12312yes—
174C4×D61#12312yes—
175C4×D61C4⋊Q812yes—
176C4×D61C4⋊Q812yes—
177C8×C61—24no—
178C8×C61—22no—
179C8×C61—41no—
180#92418—24no—
181#92418—22no—
182#92418—41no—
183#23674—611nomaximal, Sylow
184C22×C61—21no—
185C22×C61—21no—
186C22×C61C2×#12111yescharacteristic
187C22×C61C2×#12512yes—
188C2×D61—21no—
189C2×D61—21no—
190C2×D61—21no—
191C2×D61—21no—
192C2×D61—44no—
193C2×D61#39812yes—
194C2×D61#39812yes—
195C4×C61—41no—
196C4×C61—41no—
197C4×C61—82no—
198C4×C61C2×C22⋊Q812yes—
199C4×C61C2×#12311yescharacteristic
200C4×C61C2×C4⋊Q811yescharacteristic
201Dic61—41no—
202Dic61—41no—
203Dic61—82no—
204Dic61#39812yes—
205Dic61#39812yes—
206#2911—612no—
207C2×#427—611no—
208C2×#424—611no—
209#2090—612no—
210#2090—612no—
211#2089—611no—
212#2089—611no—
213#2089—611no—
214#2089D6111yescharacteristic, complemented
215#1742—612no—
216C2×#341—611no—
217C2×C61—22no—
218C2×C61C2×#39811yescharacteristic
219C2×C61#306711yescharacteristic
220C2×C61#306711yescharacteristic
221D61—22no—
222D61—22no—
223C22×M16—612no—
224C22×M16—611no—
225C23×C8—611no—
226C2×#117—611no—
227C2×#117—612no—
228#428—612no—
229#428—612no—
230#428—612no—
231#428—1222no—
232#428—1222no—
233#428C2×D6112yes—
234#427—1222no—
235#427—1222no—
236#427—1222no—
237#427—1222no—
238C2×#107—611no—
239C2×#107C2×D6111yescharacteristic
240C2×#106—612no—
241#424—611no—
242#424—611no—
243#424—611no—
244#424—1222no—
245#424C2×D6111yescharacteristic
246C2×C4⋊C8—1222no—
247#343—612no—
248#343—612no—
249#343—612no—
250#343C2×D6112yes—
251#341—612no—
252#341—611no—
253#341C2×D6111yescharacteristic
254C61#2367411yescharacteristic, complemented, Sylow
255C23×C4—611no—
256C2×M16—612no—
257C2×M16—618no—
258C2×M16—611no—
259C2×M16—1222no—
260C2×M16—1221no—
261C2×M16—1224no—
262C2×M16—1224no—
263C2×M16C22×D6111yescharacteristic
264C2×M16C22×D6112yes—
265C22×C8—611no—
266C22×C8—612no—
267C22×C8—612no—
268C22×C8—1224no—
269C22×C8—1222no—
270C22×C8—1221no—
271C22×C8C22×D6111yescharacteristic
272#117—611no—
273#117—611no—
274#117—611no—
275#117—1221no—
276#117—1221no—
277#117—1222no—
278#117—1222no—
279#117—1222no—
280#117—1222no—
281#117C22×D6111yescharacteristic
282C2×C4⋊C4—611no—
283C2×C4⋊C4—611no—
284C2×C4⋊C4—612no—
285C2×C4⋊C4C22×D6112yes—
286C2×C22⋊C4—1221no—
287C2×C22⋊C4—1222no—
288C2×C42—1221no—
289C2×C42—1222no—
290#107—22no—
291#107—22no—
292#107—1222no—
293#107—1222no—
294#107—1222no—
295#107—1222no—
296#106—22no—
297#106—1222no—
298#106—1224no—
299#106—1222no—
300#106—1222no—
301C4⋊C8—22no—
302C4⋊C8—1222no—
303C4⋊C8—1222no—
304C4⋊C8—1222no—
305C24—611no—
306C22×C4—22no—
307C22×C4—611no—
308C22×C4—611no—
309C22×C4—611no—
310C22×C4—611no—
311C22×C4—611no—
312C22×C4—614no—
313C22×C4—614no—
314C22×C4—1221no—
315C22×C4—1221no—
316C22×C4—1222no—
317C22×C4C23×D6111yescharacteristic
318M16—22no—
319M16—22no—
320M16—614no—
321M16—1228no—
322M16—1222no—
323M16—1222no—
324M16—2442no—
325M16Q8×D6114yes—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—21no—
329C2×C8—612no—
330C2×C8—1224no—
331C2×C8—1224no—
332C2×C8—1228no—
333C2×C8—1222no—
334C2×C8—1222no—
335C2×C8—1221no—
336C2×C8—2442no—
337C2×C8D4×D6112yes—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—611no—
341C4⋊C4—611no—
342C4⋊C4—1221no—
343C4⋊C4—1221no—
344C4⋊C4—1222no—
345C4⋊C4—1222no—
346C4⋊C4D4×D6111yescharacteristic
347C4⋊C4D4×D6111yescharacteristic
348C22⋊C4—21no—
349C22⋊C4—1221no—
350C22⋊C4—1221no—
351C22⋊C4—1221no—
352C22⋊C4—2442no—
353C22⋊C4—2442no—
354C42—21no—
355C42—1221no—
356C42—1221no—
357C42—1221no—
358C42—2442no—
359C42—2442no—
360C23—611no—
361C23—611no—
362C23—614no—
363C23—2441no—
364C23—2441no—
365C23#41×D6111yescharacteristic
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—42no—
369C2×C4—82no—
370C2×C4—611no—
371C2×C4—611no—
372C2×C4—614no—
373C2×C4—614no—
374C2×C4—614no—
375C2×C4—612no—
376C2×C4—612no—
377C2×C4—612no—
378C2×C4—612no—
379C2×C4—1221no—
380C2×C4—1221no—
381C2×C4—4882no—
382C2×C4—2442no—
383C2×C4—2442no—
384C2×C4—2442no—
385C2×C4C2×D4×D6111yescharacteristic
386C2×C4C2×D4×D6111yescharacteristic
387C2×C4C2×Q8×D6112yes—
388C2×C4#41×D6112yes—
389C8—24no—
390C8—22no—
391C8—41no—
392C8—1224no—
393C8—1222no—
394C8—2441no—
395C22—21no—
396C22—21no—
397C22—612no—
398C22—612no—
399C22—1221no—
400C22—1221no—
401C22—1221no—
402C22—1221no—
403C22—2444no—
404C22#121×D6111yescharacteristic
405C22#125×D6112yes—
406C4—41no—
407C4—41no—
408C4—82no—
409C4—612no—
410C4—612no—
411C4—4882no—
412C4—2441no—
413C4—2441no—
414C4C22⋊Q8×D6112yes—
415C4#123×D6111yescharacteristic
416C4C4⋊Q8×D6111yescharacteristic
417C2—22no—
418C2—1222no—
419C2—1222no—
420C2#398×D6111yescharacteristic
421C2#2145553711yescharacteristic
422C2#2145554111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2089C2
socleC22×C61C2×#121
prime radicalC23×D61#41
hypercenter#2089D61
hypocenterC61#23674
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2089

Complements

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

NH ≅ G/Ntype
4. #21447333C2semidirect
5. #21447332C2semidirect
6. #21447326C2semidirect
7. #21447325C2semidirect
8. #21447324C2semidirect
11. C61×#2089C2semidirect
214. #2089D61semidirect
254. C61#23674semidirect

Maximal subgroups

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

subgroupindex
#214544062
C2×#168397992
#424×D612
#214473332
#214473322
#214473262
#214473252
#214473242
#214455562
#341×D612
C61×#20892
#2367461