Finite Group Databasebeta

← #29663941

Subgroups — #29663941

Order 15616.

702 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840165C211yesmaximal, characteristic, complemented
2#21455307C211yesmaximal, characteristic
3#21455302C211yesmaximal, characteristic
4#21454775C211yesmaximal, characteristic
5#21445212C214yesmaximal, complemented
6#21445207C214yesmaximal, complemented
7#21444577C211yesmaximal, characteristic, complemented
8#21444490C211yesmaximal, characteristic, complemented
9C61×#1546C211yesmaximal, characteristic, complemented
10C2×#14519810—21no—
11C2×#14519808—21no—
12C2×#14519777C2211yescharacteristic
13C4×#11758222C2211yescharacteristic
14#16840165—21no—
15#16840165—21no—
16#16840165C2214yes—
17#16839973—21no—
18#16839973C2212yes—
19#16839969—21no—
20#16839969C2212yes—
21#16839968—21no—
22#16839968C2212yes—
23#16839964—21no—
24#16839964C2212yes—
25#16839886C2212yes—
26#16839827C2212yes—
27#16839825—22no—
28#16839813—21no—
29C2×#14519710—22no—
30#16839786C2211yescharacteristic
31#16839772—21no—
32#16839771—21no—
33C2×C4×#92418C2211yescharacteristic
34C4×C8×D61—21no—
35#16839514C2214yes—
36#16839507C2214yes—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C61×#431C2211yescharacteristic
39C61×#423C2211yescharacteristic
40C61×#332C2214yes—
41C2×#11758222D411yescharacteristic
42C22×Dic122—21no—
43#14519810—44no—
44#14519808—44no—
45C2×Dic244—22no—
46C2×#11758202—22no—
47C2×#11758190—22no—
48C2×#11758201—22no—
49C2×#11758194—21no—
50C22×#92418D412yes—
51C2×C8×D61—22no—
52#14519777—21no—
53#14519777—21no—
54#14519777—21no—
55#14519777C2312yes—
56C4×Dic122D411yescharacteristic
57C4×Dic122C2311yescharacteristic
58C4×D244D411yescharacteristic
59C4×D244C2311yescharacteristic
60C4×C22⋊D61—21no—
61C4×C22⋊D61—21no—
62#14519736D411yescharacteristic
63#14519734—21no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—21no—
66C42×D61—21no—
67#14519710—24no—
68#14519710—24no—
69C4⋊#92418—21no—
70C4⋊#92418C2312yes—
71#14519706—22no—
72#14519702—22no—
73#14519702—22no—
74#14519701—22no—
75#14519701—22no—
76#14519689—22no—
77C4×#92418D411yescharacteristic
78C4×#92418D411yescharacteristic
79C4×#92418C2312yes—
80C8×Dic61—22no—
81C22×Q8×C61—21no—
82C2×M16×C61—21no—
83C4⋊Q8×C61—21no—
84C4⋊Q8×C61—21no—
85C4⋊Q8×C61—21no—
86C4⋊Q8×C61C2312yes—
87C2×C4⋊C4×C61—21no—
88C2×C4⋊C4×C61—21no—
89C2×C42×C61C2311yescharacteristic
90C4⋊C8×C61—21no—
91C4⋊C8×C61C2312yes—
92C4⋊C8×C61C2312yes—
93C22⋊C8×C61—21no—
94C4×C8×C61—21no—
95C2×Q8×C61—22no—
96C2×Q8×C61—41no—
97C2×Q8×C61—42no—
98C2×Q8×C61—42no—
99C22×C4×C61—41no—
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61#4111yescharacteristic
103C2×C22⋊D61—21no—
104C22×Dic61—41no—
105#11758222—22no—
106#11758222D814yes—
107C2×D244C2×D411yescharacteristic
108C2×C4×D61—21no—
109C2×C4×D61—21no—
110C2×C4×D61—21no—
111C2×Dic122—22no—
112C2×Dic122—42no—
113C2×Dic122—42no—
114C2×Dic122—41no—
115C2×Dic122C2×D411yescharacteristic
116M16×C61—42no—
117C2×C8×C61—22no—
118C2×C8×C61—22no—
119C4⋊C4×C61—22no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—41no—
122C4⋊C4×C61—41no—
123C42×C61C2×D411yescharacteristic
124C42×C61C2×D411yescharacteristic
125C42×C61#4111yescharacteristic
126C42×C61C2411yescharacteristic
127C22⋊Dic61—21no—
128#11758202—44no—
129#11758201—44no—
130#11758198—21no—
131#11758198—21no—
132C4⋊Dic61—22no—
133C4⋊Dic61—22no—
134C4⋊Dic61—41no—
135C4⋊Dic61—41no—
136C4⋊Dic61C2×D411yescharacteristic
137C4⋊Dic61C2×D411yescharacteristic
138#11758196—21no—
139#11758196—21no—
140C4×Dic61—21no—
141C4×Dic61—21no—
142C4×Dic61—21no—
143#11758194—42no—
144C2×#92418—24no—
145C2×#92418—22no—
146C2×#92418C2×D412yes—
147C2×#92418C2×D412yes—
148Dic244—44no—
149#11758190—44no—
150C8×D61—44no—
151C23×C61#12111yescharacteristic
152C22×D61—21no—
153Q8×C61—42no—
154Q8×C61—42no—
155Q8×C61—82no—
156C2×C4×C61—22no—
157C2×C4×C61—22no—
158C2×C4×C61—42no—
159C2×C4×C61—82no—
160C2×C4×C61#12111yescharacteristic
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61#12111yescharacteristic
163C2×C4×C61C2×D812yes—
164C2×C4×C61#13512yes—
165C2×C4×C61C22×D411yescharacteristic
166C2×C4×C61C22×D411yescharacteristic
167C2×C4×C61C2×#4111yescharacteristic
168C22⋊D61—24no—
169C2×Dic61—21no—
170C2×Dic61—21no—
171C2×Dic61—21no—
172C2×Dic61—42no—
173C2×Dic61—82no—
174D244—21no—
175D244C2×D812yes—
176C4×D61—24no—
177C4×D61—42no—
178C4×D61—42no—
179Dic122—21no—
180Dic122—42no—
181Dic122—42no—
182Dic122—82no—
183Dic122C2×D812yes—
184C8×C61—42no—
185C8×C61—42no—
186#92418—22no—
187#92418—42no—
188#92418#13714yescomplemented
189#19271—611nomaximal, Sylow
190C22×C61—21no—
191C22×C61—21no—
192C22×C61#46612yes—
193C22×C61C2×#12111yescharacteristic
194C2×D61—22no—
195C2×D61—41no—
196C2×D61—41no—
197C4×C61—21no—
198C4×C61—21no—
199C4×C61—82no—
200C4×C61#46612yes—
201C4×C61#47312yes—
202C4×C61C22×D811yescharacteristic
203C4×C61C2×#13511yescharacteristic
204C4×C61C2×#13712yes—
205Dic61—22no—
206Dic61—41no—
207Dic61—41no—
208Dic61—82no—
209C2×#500—611no—
210#3039—611no—
211#3039—611no—
212C8×#41—611no—
213#1674—614no—
214#1673—614no—
215#1567—611no—
216#1546—611no—
217#1546D6111yescharacteristic, complemented
218C2×C61—22no—
219C2×C61#303911yescharacteristic
220C2×C61C2×#46611yescharacteristic
221C2×C61C2×#47311yescharacteristic
222D61—42no—
223C2×#137—1221no—
224C2×#137—1221no—
225C2×C4⋊Q8—611no—
226C2×C4⋊Q8C2×D6111yescharacteristic
227C4×#41—611no—
228#500—614no—
229#500—1221no—
230#500—1221no—
231#462—612no—
232#462—612no—
233#462—1221no—
234#462—1221no—
235#461—612no—
236#461—612no—
237#461—1221no—
238#461—1221no—
239C8×Q8—612no—
240C8×D4—612no—
241C8×D4—1222no—
242#431—1221no—
243#431C2×D6111yescharacteristic
244C2×#107—1222no—
245#423—611no—
246#423C2×D6111yescharacteristic
247#419—1221no—
248#419—1221no—
249C2×C4×C8—611no—
250C2×C4×C8—1221no—
251#335—614no—
252#332—614no—
253#332C2×D6114yes—
254C61#1927111yescharacteristic, complemented, Sylow
255C2×#41—611no—
256C22×Q8—21no—
257C22×Q8—1221no—
258#137—2444no—
259#137—2444no—
260C2×Q16—1222no—
261C2×Q16—1222no—
262C2×SD16—1222no—
263C2×SD16—1222no—
264C2×M16—21no—
265C2×M16—1221no—
266C22×C8—612no—
267C22×C8—1222no—
268C4⋊Q8—21no—
269C4⋊Q8—21no—
270C4⋊Q8—21no—
271C4⋊Q8—612no—
272C4⋊Q8—1221no—
273C4⋊Q8—1221no—
274C4⋊Q8—1221no—
275C4⋊Q8C22×D6112yes—
276C4×Q8—611no—
277C4×Q8—611no—
278C4×D4—611no—
279C4×D4—611no—
280C4×D4—1221no—
281C4×D4—1221no—
282#117—611no—
283#117—1221no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—1221no—
287C2×C4⋊C4—1221no—
288C2×C42—1221no—
289C2×C42C22×D6111yescharacteristic
290#107—1224no—
291#107—1224no—
292C4⋊C8—21no—
293C4⋊C8—612no—
294C4⋊C8—1222no—
295C4⋊C8—1221no—
296C4⋊C8C22×D6112yes—
297C4⋊C8C22×D6112yes—
298#103—1222no—
299#103—1222no—
300#103—1222no—
301#103—1222no—
302C22⋊C8—21no—
303C22⋊C8—1222no—
304C4×C8—21no—
305C4×C8—611no—
306C4×C8—611no—
307C4×C8—612no—
308C4×C8—1222no—
309#41—614no—
310#41—1222no—
311C2×Q8—22no—
312C2×Q8—41no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—611no—
316C2×Q8—1222no—
317C2×Q8—2442no—
318C2×Q8—2442no—
319C2×Q8—2441no—
320C2×D4—611no—
321C2×D4—1221no—
322C22×C4—41no—
323C22×C4—1221no—
324C22×C4—1221no—
325C22×C4—1221no—
326C22×C4—2441no—
327C22×C4#1175822211yescharacteristic
328C22×C4D4×D6111yescharacteristic
329C22×C4D4×D6111yescharacteristic
330Q16—2444no—
331Q16—2444no—
332SD16—2444no—
333SD16—2444no—
334M16—42no—
335M16—2442no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—612no—
339C2×C8—612no—
340C2×C8—1224no—
341C2×C8—1222no—
342C2×C8—2444no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—611no—
348C4⋊C4—611no—
349C4⋊C4—1221no—
350C4⋊C4—1221no—
351C4⋊C4—1222no—
352C4⋊C4—1222no—
353C4⋊C4—2441no—
354C4⋊C4—2441no—
355C22⋊C4—1221no—
356C22⋊C4—1221no—
357C22⋊C4—1221no—
358C42—1221no—
359C42—1221no—
360C42—1221no—
361C42#1175822211yescharacteristic
362C42D4×D6111yescharacteristic
363C42D4×D6111yescharacteristic
364C42C23×D6111yescharacteristic
365C23—1221no—
366C23#1451975011yescharacteristic
367Q8—42no—
368Q8—42no—
369Q8—82no—
370Q8—612no—
371Q8—1221no—
372Q8—4882no—
373Q8—2442no—
374Q8—2442no—
375D4—612no—
376D4—1224no—
377D4—1221no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—42no—
381C2×C4—82no—
382C2×C4—1221no—
383C2×C4—1224no—
384C2×C4—1221no—
385C2×C4—1221no—
386C2×C4—4882no—
387C2×C4—2442no—
388C2×C4—2442no—
389C2×C4—2442no—
390C2×C4#1451975011yescharacteristic
391C2×C4#1451975011yescharacteristic
392C2×C4#1451975011yescharacteristic
393C2×C4D8×D6112yes—
394C2×C4#1451979812yes—
395C2×C4C2×D4×D6111yescharacteristic
396C2×C4C2×D4×D6111yescharacteristic
397C2×C4C2×#1175822211yescharacteristic
398C8—42no—
399C8—42no—
400C8—614no—
401C8—1222no—
402C8—2442no—
403C22—21no—
404C22—21no—
405C22—1222no—
406C22—2441no—
407C22—2441no—
408C22#1683999712yes—
409C22C2×#1451975011yescharacteristic
410C4—21no—
411C4—21no—
412C4—82no—
413C4—1222no—
414C4—4882no—
415C4—2441no—
416C4—2441no—
417C4#1683999712yes—
418C4#1684003612yes—
419C4C2×D8×D6111yescharacteristic
420C4C2×#1451979811yescharacteristic
421C4#137×D6112yes—
422C2—22no—
423C2—2442no—
424C2#2145530511yescharacteristic
425C2C2×#1683999711yescharacteristic
426C2C2×#1684003611yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1546C2
socleC22×C61C2×#121
prime radicalC2×#11758222D4
hypercenter#1546D61
hypocenterC61#19271
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
1. C2×#16840165C2semidirect
5. #21445212C2semidirect
6. #21445207C2semidirect
7. #21444577C2semidirect
8. #21444490C2semidirect
9. C61×#1546C2semidirect
188. #92418#137semidirect
217. #1546D61semidirect
254. C61#19271semidirect

Maximal subgroups

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

subgroupindex
C2×#168401652
#214553072
#214553022
#214547752
#214452122
#214452072
#214445772
#214444902
C61×#15462
#1927161