Finite Group Databasebeta

← #29738931

Subgroups — #29738931

Order 15616.

690 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456359C211yesmaximal, characteristic, complemented
2#21455672C212yesmaximal
3#21454416C211yesmaximal, characteristic
4#21447324C212yesmaximal, complemented
5#21447163C212yesmaximal, complemented
6#21446605C211yesmaximal, characteristic, complemented
7#21446046C212yesmaximal
8#21445979C211yesmaximal, characteristic, complemented
9#21445880C211yesmaximal, characteristic
10#21445656C211yesmaximal, characteristic
11C61×#1945C211yesmaximal, characteristic, complemented
12C2×#14519783C2211yescharacteristic
13#16840390C2212yes—
14#16840242C2212yes—
15#16840169—22no—
16#16840166—21no—
17#16840163—21no—
18#16840070—22no—
19#16840063—22no—
20#16840058—22no—
21#16840043—22no—
22#16839805C2212yes—
23#16839805C2212yes—
24#16839804C2212yes—
25#16839801—22no—
26#16839799—21no—
27C2×#14519711C2212yes—
28C2×#14519711C2211yescharacteristic
29C2×#14519711C2211yescharacteristic
30#107×D61—21no—
31C2×#14519705C2211yescharacteristic
32#16839764C2212yes—
33#16839759C2211yescharacteristic
34#16839717C2211yescharacteristic
35#16839632C2212yes—
36#16839530C2212yes—
37#16839530C2212yes—
38#16839527C2211yescharacteristic
39#16839522C2211yescharacteristic
40C2×#108×C61C2211yescharacteristic
41C61×#427C2211yescharacteristic
42C61×#424C2212yes—
43C2×C4×C8×C61C2211yescharacteristic
44C61×#339C2212yes—
45C2×#11758222C2311yescharacteristic
46#14519783—22no—
47#14519783Q814yescomplemented
48C2×#11758194C2311yescharacteristic
49C2×#11758194C2311yescharacteristic
50C2×#11758189—21no—
51C2×C8×D61—21no—
52#14519777—22no—
53#14519776—22no—
54#14519769—22no—
55#14519768—22no—
56#14519754—42no—
57C4×Dic122—22no—
58C4×D244—22no—
59#14519737—42no—
60#14519736C2311yescharacteristic
61#14519735—22no—
62C2×C4⋊Dic61—42no—
63C2×C4⋊Dic61C2312yes—
64C2×C4⋊Dic61C2312yes—
65C4⋊C4×D61—22no—
66#14519711—22no—
67#14519711—21no—
68#14519711—21no—
69#14519711—22no—
70#14519711—21no—
71#14519711—21no—
72#14519711—22no—
73#14519710—22no—
74#14519710—22no—
75#14519709—22no—
76#14519708—22no—
77#14519708—22no—
78#14519705Q814yescomplemented
79#14519702—22no—
80#14519702D411yescharacteristic
81#14519702D411yescharacteristic
82#14519699—22no—
83#14519699D411yescharacteristic
84#14519699D411yescharacteristic
85#14519687—21no—
86C8×Dic61—21no—
87C2×M16×C61C2312yes—
88C22×C8×C61C2311yescharacteristic
89C22×C8×C61C2311yescharacteristic
90#117×C61C2312yes—
91C2×C42×C61C2311yescharacteristic
92#108×C61—22no—
93#107×C61D411yescharacteristic
94#107×C61D411yescharacteristic
95#106×C61D412yes—
96C4⋊C8×C61—22no—
97C4⋊C8×C61—22no—
98C4×C8×C61—22no—
99C22×C4×C61—21no—
100C22×C4×C61C2411yescharacteristic
101C2×C22⋊D61—21no—
102C22×Dic61—22no—
103C22×Dic61—22no—
104#11758222—21no—
105#11758222—21no—
106#11758222C2×Q812yes—
107#11758222C2×Q812yes—
108C2×D244C2×D411yescharacteristic
109C2×C4×D61—21no—
110C2×C4×D61—42no—
111C2×Dic122—42no—
112C2×Dic122C2×D411yescharacteristic
113M16×C61—22no—
114M16×C61—22no—
115C2×C8×C61—21no—
116C2×C8×C61—21no—
117C2×C8×C61—22no—
118C2×C8×C61C2×D411yescharacteristic
119C2×C8×C61C2×D411yescharacteristic
120C2×C8×C61C2×D411yescharacteristic
121C2×C8×C61C2×D411yescharacteristic
122C2×C8×C61C2×Q812yes—
123C2×C8×C61#4112yes—
124C4⋊C4×C61C2×D412yes—
125C4⋊C4×C61C2×D412yes—
126C22⋊C4×C61—22no—
127C42×C61—22no—
128C42×C61—21no—
129C42×C61C2×Q812yes—
130C22⋊Dic61—21no—
131#11758198—42no—
132C4⋊Dic61—22no—
133C4⋊Dic61—22no—
134C4⋊Dic61—22no—
135C4⋊Dic61—22no—
136C4⋊Dic61—42no—
137C4⋊Dic61—84no—
138C4⋊Dic61C2×D411yescharacteristic
139C4⋊Dic61C2×D411yescharacteristic
140#11758196—42no—
141#11758196—42no—
142#11758196—42no—
143C4×Dic61—21no—
144C4×Dic61—42no—
145#11758194—21no—
146#11758194—21no—
147#11758194C2×Q812yes—
148#11758194C2×Q812yes—
149C2×#92418—21no—
150C2×#92418—21no—
151#11758189—24no—
152C8×D61—24no—
153C23×C61H(5,𝔽2)11yescharacteristic
154C22×D61—21no—
155C2×C4×C61—22no—
156C2×C4×C61—22no—
157C2×C4×C61—22no—
158C2×C4×C61—22no—
159C2×C4×C61—22no—
160C2×C4×C61C22×D411yescharacteristic
161C2×C4×C61C22×D411yescharacteristic
162C2×C4×C61C22×Q811yescharacteristic
163C2×C4×C61C22×Q811yescharacteristic
164C2×C4×C61C2×#4111yescharacteristic
165C2×C4×C612−1+411yescharacteristic
166C22⋊D61—22no—
167C22⋊D61—22no—
168C2×Dic61—21no—
169C2×Dic61—21no—
170C2×Dic61—21no—
171C2×Dic61—42no—
172C2×Dic61—42no—
173C2×Dic61—82no—
174C2×Dic61—82no—
175D244—21no—
176D244C4⋊Q812yes—
177C4×D61—22no—
178C4×D61—22no—
179C4×D61—162no—
180Dic122—21no—
181Dic122—162no—
182Dic122C4⋊Q812yes—
183C8×C61—21no—
184C8×C61—21no—
185C8×C61—42no—
186C8×C61C22⋊Q812yes—
187C8×C61C22⋊Q812yes—
188#92418—22no—
189#92418—22no—
190#26083—611nomaximal, Sylow
191C22×C61—21no—
192C22×C61—21no—
193C22×C61#54411yescharacteristic
194C22×C61D4211yescharacteristic
195C22×C61#55411yescharacteristic
196C2×D61—22no—
197C2×D61—41no—
198C2×D61—41no—
199C4×C61—24no—
200C4×C61—42no—
201C4×C61—42no—
202C4×C61C2×C22⋊Q811yescharacteristic
203C4×C61C2×C4⋊Q811yescharacteristic
204C4×C61#54711yescharacteristic
205C4×C61D4×Q811yescharacteristic
206Dic61—22no—
207Dic61—41no—
208Dic61—41no—
209Dic61—82no—
210Dic61—82no—
211#3164—611no—
212#3083—612no—
213#2913—611no—
214#2089—612no—
215#2060—612no—
216#1952—611no—
217#1945D6111yescharacteristic, complemented
218#1859—612no—
219#1845—611no—
220#1816—611no—
221#1764—611no—
222C2×C61—21no—
223C2×C61—21no—
224C2×C61#244011yescharacteristic
225C2×C61#329211yescharacteristic
226C2×C61#329311yescharacteristic
227D61—42no—
228C2×#131—611no—
229#544—612no—
230#520—612no—
231#501—1222no—
232#500—1221no—
233#499—1221no—
234#479—1222no—
235#478—1222no—
236#477—1222no—
237#474—1222no—
238C2×#108C2×D6111yescharacteristic
239#428—612no—
240#428—612no—
241#428—612no—
242#428—1222no—
243#427—1221no—
244#427C2×D6111yescharacteristic
245C2×#107—612no—
246C2×#107—611no—
247C2×#107—611no—
248C2×#107—1221no—
249#424C2×D6112yes—
250C2×#104—611no—
251#417—612no—
252#416—611no—
253#405—611no—
254C2×C4×C8C2×D6111yescharacteristic
255#384—612no—
256#343—612no—
257#343—612no—
258#341—611no—
259#339—611no—
260#339C2×D6112yes—
261C61#2608311yescharacteristic, complemented, Sylow
262C2×#41—611no—
263#131—614no—
264#131—1222no—
265C2×M16—611no—
266C2×M16—611no—
267C2×M16—1221no—
268C2×M16C22×D6112yes—
269C22×C8—1221no—
270C22×C8C22×D6111yescharacteristic
271C22×C8C22×D6111yescharacteristic
272C4⋊Q8—1222no—
273C4⋊Q8—1222no—
274#125—1222no—
275#125—1222no—
276C22⋊Q8—2442no—
277C4×Q8—1222no—
278C4×D4—1222no—
279C4×D4—2442no—
280#117—611no—
281#117—1222no—
282#117C22×D6112yes—
283C2×C4⋊C4—612no—
284C2×C4⋊C4—612no—
285C2×C4⋊C4—1222no—
286C2×C4⋊C4—2442no—
287C2×C42C22×D6111yescharacteristic
288#108—22no—
289#107—1222no—
290#107—1221no—
291#107—1221no—
292#107—1221no—
293#107—1221no—
294#107—1222no—
295#107—1222no—
296#107—1222no—
297#107—1222no—
298#107D24411yescharacteristic
299#107D24411yescharacteristic
300#106—1222no—
301#106—1222no—
302#106—1222no—
303#106D24412yes—
304C4⋊C8—22no—
305C4⋊C8—22no—
306#104—614no—
307#103—611no—
308#103—611no—
309#103—1222no—
310#102—611no—
311#102—611no—
312#102—1222no—
313#97—1221no—
314C4×C8—22no—
315C4×C8—1221no—
316#41—612no—
317#41—612no—
318#41—1221no—
319#41—1221no—
320C2×Q8—611no—
321C2×Q8—2442no—
322C2×D4—611no—
323C2×D4—1221no—
324C22×C4—21no—
325C22×C4—1221no—
326C22×C4—1222no—
327C22×C4—1222no—
328C22×C4—2442no—
329C22×C4C23×D6111yescharacteristic
330M16—22no—
331M16—22no—
332M16—612no—
333M16—612no—
334M16—1221no—
335M16—1221no—
336M16—1224no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—22no—
340C2×C8—1221no—
341C2×C8—1221no—
342C2×C8—1224no—
343C2×C8C2×D24411yescharacteristic
344C2×C8C2×D24411yescharacteristic
345C2×C8D4×D6111yescharacteristic
346C2×C8D4×D6111yescharacteristic
347C2×C8#1175822412yes—
348C2×C8Q8×D6112yes—
349C4⋊C4—611no—
350C4⋊C4—611no—
351C4⋊C4—1222no—
352C4⋊C4—1222no—
353C4⋊C4—1222no—
354C4⋊C4—1222no—
355C4⋊C4—4884no—
356C4⋊C4—2442no—
357C4⋊C4—2442no—
358C4⋊C4—2442no—
359C4⋊C4—2442no—
360C4⋊C4C2×D24412yes—
361C4⋊C4C2×D24412yes—
362C22⋊C4—22no—
363C22⋊C4—1221no—
364C22⋊C4—2442no—
365C42—22no—
366C42—21no—
367C42—1221no—
368C42—2442no—
369C42Q8×D6112yes—
370C23—1221no—
371C23#1451982311yescharacteristic
372Q8—612no—
373Q8—1221no—
374Q8—9762no—
375D4—612no—
376D4—1222no—
377D4—1222no—
378D4—1221no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—1221no—
385C2×C4—1222no—
386C2×C4—1222no—
387C2×C4—1221no—
388C2×C4—1221no—
389C2×C4—9762no—
390C2×C4—4882no—
391C2×C4—4882no—
392C2×C4—2442no—
393C2×C4—2442no—
394C2×C4C2×D4×D6111yescharacteristic
395C2×C4C22×D24411yescharacteristic
396C2×C4C2×Q8×D6111yescharacteristic
397C2×C4C2×Q8×D6111yescharacteristic
398C2×C4C2×#1175822411yescharacteristic
399C2×C4#1451982411yescharacteristic
400C8—21no—
401C8—21no—
402C8—42no—
403C8—1222no—
404C8—1222no—
405C8#1451975712yes—
406C8#1451975712yes—
407C22—21no—
408C22—21no—
409C22—1222no—
410C22—2441no—
411C22—2441no—
412C22#1684039211yescharacteristic
413C22D4×D24411yescharacteristic
414C22#1684047011yescharacteristic
415C4—24no—
416C4—42no—
417C4—42no—
418C4—1222no—
419C4—4882no—
420C4—4882no—
421C4—2441no—
422C4—2441no—
423C4C2×#1451975711yescharacteristic
424C4C4⋊Q8×D6111yescharacteristic
425C4#1684041211yescharacteristic
426C4Q8×D24411yescharacteristic
427C2—21no—
428C2—21no—
429C2—2442no—
430C2#2144951911yescharacteristic
431C2#2145761911yescharacteristic
432C2#2145762411yescharacteristic
subgroupG/H
centerC22D4×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1945C2
socleC22×C61D42
prime radicalC2×#11758222C23
hypercenter#1945D61
hypocenterC61#26083
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1945

Complements

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

NH ≅ G/Ntype
1. #21456359C2semidirect
4. #21447324C2semidirect
5. #21447163C2semidirect
6. #21446605C2semidirect
8. #21445979C2semidirect
11. C61×#1945C2semidirect
47. #14519783Q8semidirect
78. #14519705Q8semidirect
217. #1945D61semidirect
261. C61#26083semidirect

Maximal subgroups

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

subgroupindex
#214563592
#214556722
#214544162
#214473242
#214471632
#214466052
#214460462
#214459792
#214458802
#214456562
C61×#19452
#2608361