Finite Group Databasebeta

← #29738924

Subgroups — #29738924

Order 15616.

674 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456358C211yesmaximal, characteristic, complemented
2#21455680C212yesmaximal
3#21454415C211yesmaximal, characteristic
4#21447333C212yesmaximal, complemented
5#21447167C212yesmaximal, complemented
6#21446565C211yesmaximal, characteristic, complemented
7#21446055C212yesmaximal
8#21445982C211yesmaximal, characteristic, complemented
9#21445880C211yesmaximal, characteristic
10#21445658C211yesmaximal, characteristic
11C61×#1952C211yesmaximal, characteristic, complemented
12C2×#14519783C2211yescharacteristic
13#16840390C2212yes—
14#16840243C2212yes—
15#16840171—22no—
16#16840167—21no—
17#16840164—21no—
18#16840068—22no—
19#16840065—22no—
20#16840051—22no—
21#16840046—22no—
22#16839804C2212yes—
23#16839803—22no—
24#16839802C2212yes—
25#16839800C2211yescharacteristic
26#16839799—21no—
27C2×#14519711C2211yescharacteristic
28#107×D61—21no—
29C2×#14519708C2212yes—
30#16839790C2212yes—
31C2×#14519703C2211yescharacteristic
32#16839764C2212yes—
33#16839756C2211yescharacteristic
34C2×C8×Dic61C2211yescharacteristic
35#16839627C2212yes—
36#16839530C2212yes—
37#16839526C2211yescharacteristic
38#16839523C2212yes—
39#16839521C2211yescharacteristic
40C2×#108×C61C2211yescharacteristic
41C61×#428C2212yes—
42C2×#107×C61C2211yescharacteristic
43C61×#405C2211yescharacteristic
44C61×#343C2212yes—
45C2×#11758222C2311yescharacteristic
46#14519783—22no—
47#14519783Q814yescomplemented
48C2×#11758194C2311yescharacteristic
49C2×#11758189—21no—
50C22×#92418C2311yescharacteristic
51C2×C8×D61—21no—
52#14519777—22no—
53#14519776—22no—
54#14519769—22no—
55#14519768—22no—
56#14519754—42no—
57#14519742—22no—
58#14519739—22no—
59C4×C22⋊D61—42no—
60#14519736C2312yes—
61#14519735—22no—
62C2×C4⋊Dic61C2312yes—
63C4⋊C4×D61—22no—
64C2×C4×Dic61C2311yescharacteristic
65#14519711—21no—
66#14519711—21no—
67#14519711—22no—
68#14519711D411yescharacteristic
69#14519711D411yescharacteristic
70#14519710—22no—
71#14519709—22no—
72#14519709D412yes—
73#14519708—22no—
74#14519708—22no—
75#14519708—22no—
76#14519706—22no—
77#14519706—22no—
78#14519703Q814yescomplemented
79#14519702—22no—
80#14519700D411yescharacteristic
81#14519700D411yescharacteristic
82#14519699—22no—
83#14519697D411yescharacteristic
84#14519697D411yescharacteristic
85C8×Dic61—22no—
86#14519684—42no—
87C2×M16×C61C2312yes—
88C2×M16×C61C2311yescharacteristic
89C22×C8×C61C2311yescharacteristic
90#117×C61C2311yescharacteristic
91C2×C4⋊C4×C61C2312yes—
92#108×C61—22no—
93#107×C61—21no—
94#107×C61—21no—
95#107×C61—22no—
96#106×C61—22no—
97#97×C61—21no—
98C4×C8×C61—21no—
99C22×C4×C61—22no—
100C22×C4×C61C2411yescharacteristic
101C2×C22⋊D61—21no—
102C22×Dic61—21no—
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—21no—
114M16×C61—22no—
115M16×C61—22no—
116M16×C61C2×Q812yes—
117C2×C8×C61—21no—
118C2×C8×C61—22no—
119C2×C8×C61C2×D411yescharacteristic
120C2×C8×C61C2×D411yescharacteristic
121C2×C8×C61C2×Q812yes—
122C2×C8×C61#4112yes—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—22no—
125C4⋊C4×C61C2×D411yescharacteristic
126C4⋊C4×C61C2×D411yescharacteristic
127C22⋊C4×C61—21no—
128C42×C61—21no—
129C22⋊Dic61—22no—
130#11758198—42no—
131C4⋊Dic61—22no—
132C4⋊Dic61—22no—
133C4⋊Dic61—42no—
134C4⋊Dic61C2×D412yes—
135C4⋊Dic61C2×D412yes—
136#11758196—42no—
137#11758196—42no—
138#11758196—42no—
139C4×Dic61—22no—
140C4×Dic61—21no—
141C4×Dic61—42no—
142C4×Dic61C2×Q812yes—
143#11758194—21no—
144#11758194C2×Q812yes—
145C2×#92418—21no—
146C2×#92418—21no—
147C2×#92418—21no—
148C2×#92418C2×D411yescharacteristic
149C2×#92418C2×D411yescharacteristic
150#11758189—24no—
151C8×D61—24no—
152C23×C61H(5,𝔽2)11yescharacteristic
153C22×D61—21no—
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×Q811yescharacteristic
161C2×C4×C61C22×Q811yescharacteristic
162C2×C4×C61C2×#4111yescharacteristic
163C2×C4×C612−1+411yescharacteristic
164C22⋊D61—22no—
165C22⋊D61—22no—
166C2×Dic61—21no—
167C2×Dic61—22no—
168C2×Dic61—22no—
169C2×Dic61—22no—
170C2×Dic61—41no—
171C2×Dic61—41no—
172C2×Dic61—42no—
173C2×Dic61—82no—
174D244—21no—
175D244C4⋊Q812yes—
176C4×D61—22no—
177C4×D61—22no—
178C4×D61—162no—
179Dic122—21no—
180Dic122—162no—
181Dic122C4⋊Q812yes—
182C8×C61—22no—
183C8×C61—42no—
184C8×C61C22⋊Q812yes—
185C8×C61C22⋊Q812yes—
186#92418—22no—
187#92418—21no—
188#92418—21no—
189#26082—611nomaximal, Sylow
190C22×C61—21no—
191C22×C61—21no—
192C22×C61#54411yescharacteristic
193C22×C61D4211yescharacteristic
194C22×C61#55411yescharacteristic
195C2×D61—22no—
196C2×D61—41no—
197C2×D61—41no—
198C4×C61—41no—
199C4×C61—41no—
200C4×C61—82no—
201C4×C61C2×C22⋊Q811yescharacteristic
202C4×C61C2×C4⋊Q811yescharacteristic
203C4×C61#54711yescharacteristic
204C4×C61D4×Q811yescharacteristic
205Dic61—22no—
206Dic61—42no—
207Dic61—42no—
208Dic61—42no—
209Dic61—82no—
210#3164—611no—
211#3084—612no—
212#2913—611no—
213#2090—612no—
214#2061—612no—
215#1952D6111yescharacteristic, complemented
216#1945—611no—
217#1860—612no—
218#1846—611no—
219#1816—611no—
220#1764—611no—
221C2×C61—21no—
222C2×C61—21no—
223C2×C61#244011yescharacteristic
224C2×C61#329011yescharacteristic
225C2×C61#329011yescharacteristic
226D61—42no—
227C2×#131—611no—
228#544—612no—
229#520—612no—
230#501—1222no—
231#500—1221no—
232#499—1221no—
233#479—1222no—
234#478—1222no—
235#476—1222no—
236#475—1222no—
237C2×#108C2×D6111yescharacteristic
238#428—612no—
239#428—612no—
240#428—1222no—
241#428C2×D6112yes—
242#427—611no—
243#427—1221no—
244C2×#107—611no—
245C2×#107—1221no—
246C2×#107C2×D6111yescharacteristic
247C2×#106—612no—
248#424—612no—
249C2×#104—611no—
250#417—612no—
251#416—611no—
252#405C2×D6111yescharacteristic
253C2×C4×C8—611no—
254#382—612no—
255#343—612no—
256#343C2×D6112yes—
257#341—611no—
258#339—612no—
259#339—611no—
260C61#2608211yescharacteristic, complemented, Sylow
261C2×#41—611no—
262#131—614no—
263#131—1222no—
264C2×M16—611no—
265C2×M16—1221no—
266C2×M16C22×D6112yes—
267C2×M16C22×D6111yescharacteristic
268C22×C8—611no—
269C22×C8—1221no—
270C22×C8C22×D6111yescharacteristic
271C4⋊Q8—1222no—
272C4⋊Q8—1222no—
273#125—1222no—
274#125—1222no—
275C22⋊Q8—2442no—
276C4×Q8—1222no—
277C4×D4—1222no—
278C4×D4—2442no—
279#117—612no—
280#117—1222no—
281#117C22×D6111yescharacteristic
282C2×C4⋊C4—612no—
283C2×C4⋊C4—1222no—
284C2×C4⋊C4C22×D6112yes—
285C2×C42—611no—
286#108—22no—
287#107—21no—
288#107—21no—
289#107—22no—
290#107—611no—
291#107—611no—
292#107—1222no—
293#107—1222no—
294#107—1221no—
295#107—1221no—
296#106—22no—
297#106—612no—
298#106—1222no—
299#106—1222no—
300#106—1222no—
301#106—1222no—
302C4⋊C8—1222no—
303C4⋊C8—1222no—
304#104—614no—
305#103—611no—
306#103—611no—
307#103—1222no—
308#102—611no—
309#102—611no—
310#102—1222no—
311#97—21no—
312C4×C8—21no—
313C4×C8—1222no—
314#95—2442no—
315#41—612no—
316#41—612no—
317#41—1221no—
318#41—1221no—
319C2×Q8—611no—
320C2×Q8—2442no—
321C2×D4—611no—
322C2×D4—1221no—
323C22×C4—22no—
324C22×C4—1221no—
325C22×C4—1221no—
326C22×C4—1222no—
327C22×C4—2442no—
328C22×C4C23×D6111yescharacteristic
329M16—21no—
330M16—22no—
331M16—22no—
332M16—612no—
333M16—1221no—
334M16—1224no—
335M16Q8×D6112yes—
336C2×C8—21no—
337C2×C8—22no—
338C2×C8—611no—
339C2×C8—611no—
340C2×C8—1221no—
341C2×C8—1221no—
342C2×C8—1221no—
343C2×C8—1224no—
344C2×C8#1175822212yes—
345C2×C8D4×D6111yescharacteristic
346C2×C8D4×D6111yescharacteristic
347C2×C8Q8×D6112yes—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—612no—
351C4⋊C4—612no—
352C4⋊C4—1222no—
353C4⋊C4—1222no—
354C4⋊C4—2442no—
355C4⋊C4—2442no—
356C4⋊C4—2442no—
357C4⋊C4—2442no—
358C4⋊C4D4×D6111yescharacteristic
359C4⋊C4D4×D6111yescharacteristic
360C22⋊C4—21no—
361C22⋊C4—1222no—
362C22⋊C4—2442no—
363C42—21no—
364C42—612no—
365C42—1222no—
366C42—1221no—
367C42—2442no—
368C23—1221no—
369C23#1451982311yescharacteristic
370Q8—612no—
371Q8—1221no—
372Q8—9762no—
373D4—612no—
374D4—1222no—
375D4—1222no—
376D4—1221no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—42no—
380C2×C4—82no—
381C2×C4—1221no—
382C2×C4—1222no—
383C2×C4—1222no—
384C2×C4—1222no—
385C2×C4—1222no—
386C2×C4—1222no—
387C2×C4—9762no—
388C2×C4—4882no—
389C2×C4—2442no—
390C2×C4—2441no—
391C2×C4—2441no—
392C2×C4C2×D4×D6111yescharacteristic
393C2×C4C2×D4×D6111yescharacteristic
394C2×C4C2×Q8×D6111yescharacteristic
395C2×C4C2×Q8×D6111yescharacteristic
396C2×C4C2×#1175822211yescharacteristic
397C2×C4#1451982411yescharacteristic
398C8—22no—
399C8—42no—
400C8—1222no—
401C8—1221no—
402C8—1221no—
403C8#1451975512yes—
404C8#1451975512yes—
405C22—21no—
406C22—21no—
407C22—1222no—
408C22—2441no—
409C22—2441no—
410C22#1684039211yescharacteristic
411C22#1684039611yescharacteristic
412C22#1684047211yescharacteristic
413C4—41no—
414C4—41no—
415C4—82no—
416C4—1222no—
417C4—4882no—
418C4—2442no—
419C4—2442no—
420C4—2442no—
421C4C2×#1451975511yescharacteristic
422C4C4⋊Q8×D6111yescharacteristic
423C4#1684041711yescharacteristic
424C4#1684042511yescharacteristic
425C2—21no—
426C2—21no—
427C2—2442no—
428C2#2144952311yescharacteristic
429C2#2145760011yescharacteristic
430C2#2145760811yescharacteristic
subgroupG/H
centerC22#16840396
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1952C2
socleC22×C61D42
prime radicalC2×#11758222C23
hypercenter#1952D61
hypocenterC61#26082
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1952

Complements

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

NH ≅ G/Ntype
1. #21456358C2semidirect
4. #21447333C2semidirect
5. #21447167C2semidirect
6. #21446565C2semidirect
8. #21445982C2semidirect
11. C61×#1952C2semidirect
47. #14519783Q8semidirect
78. #14519703Q8semidirect
215. #1952D61semidirect
260. C61#26082semidirect

Maximal subgroups

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

subgroupindex
#214563582
#214556802
#214544152
#214473332
#214471672
#214465652
#214460552
#214459822
#214458802
#214456582
C61×#19522
#2608261