Finite Group Databasebeta

← #29667903

Subgroups — #29667903

Order 15616.

836 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456739C211yesmaximal, characteristic
2#21456379C211yesmaximal, characteristic, complemented
3C2×#16840136C211yesmaximal, characteristic, complemented
4#21456215C211yesmaximal, characteristic, complemented
5#21456195C211yesmaximal, characteristic, complemented
6#21445475C212yesmaximal, complemented
7#21445461C212yesmaximal, complemented
8#21445422C212yesmaximal, complemented
9#21445412C212yesmaximal, complemented
10#21444100C211yesmaximal, characteristic, complemented
11C61×#3230C211yesmaximal, characteristic, complemented
12C22×#11758201—22no—
13#16840312C2211yescharacteristic
14C2×#14519774C2211yescharacteristic
15#16840158C2211yescharacteristic, complemented
16#16840139C2212yes—
17#16840136—22no—
18#16840136C2212yes—
19#16840136C2212yes—
20#16840136C2211yescharacteristic, complemented
21#16840128—22no—
22#16840124—22no—
23#16840114C2212yescomplemented
24#16840112C2211yescharacteristic, complemented
25#16840110C2212yes—
26#16840110C2212yes—
27#16840110C2212yescomplemented
28#16840110C2211yescharacteristic, complemented
29#16840109C2212yes—
30#16839993—22no—
31#16839958—22no—
32#16839898—22no—
33C4×#11758201—22no—
34#16839766—22no—
35#16839761—22no—
36C4×#11758194C2211yescharacteristic
37C2×C4×#92418C2211yescharacteristic
38#16839484C2212yescomplemented
39C8×#92418C2212yescomplemented
40C61×#533C2211yescharacteristic
41C2×C4⋊Q8×C61C2211yescharacteristic
42C61×#500C2212yescomplemented
43C61×#487C2212yescomplemented
44C61×#484—22no—
45C61×#477—22no—
46C61×#431C2211yescharacteristic
47C2×#11758222—21no—
48C22×D244—21no—
49C22×D244—21no—
50C22×D244—21no—
51C2×#11758201—22no—
52C2×#11758201—24no—
53C2×#11758201—24no—
54C2×#11758201—48no—
55C2×D488—24no—
56C2×#11758194D412yes—
57C22×#92418D412yes—
58#14519777C2311yescharacteristic
59#14519774—21no—
60#14519774—21no—
61#14519774—21no—
62#14519774C2312yes—
63#14519774C2311yescharacteristic
64#14519765—21no—
65#14519751—21no—
66#14519751—21no—
67C4×D244—21no—
68C4×D244—21no—
69#14519708—22no—
70#14519701—22no—
71#14519700—24no—
72#14519699—24no—
73#14519697—22no—
74#14519697—24no—
75#14519697—24no—
76#14519697—24no—
77#14519688D412yes—
78C4×#92418D412yes—
79C4×#92418C2311yescharacteristic
80C4×#92418C2311yescharacteristic
81C4×#92418C2312yes—
82C22×Q8×C61—21no—
83C4⋊Q8×C61—21no—
84C4⋊Q8×C61—21no—
85C4⋊Q8×C61—21no—
86C4⋊Q8×C61C2312yes—
87C4⋊Q8×C61C2311yescharacteristic
88C4⋊Q8×C61C2311yescharacteristic
89#125×C61—21no—
90C22⋊Q8×C61—21no—
91C22⋊Q8×C61—21no—
92C4×Q8×C61—21no—
93C4×Q8×C61—21no—
94#117×C61—21no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—21no—
97C2×C42×C61C2311yescharacteristic
98#107×C61—24no—
99C4⋊C8×C61D412yes—
100#103×C61—24no—
101C22⋊C8×C61D412yes—
102C4×C8×C61C2312yes—
103C23×D61—41no—
104C2×Q8×C61—21no—
105C2×Q8×C61—21no—
106C2×Q8×C61—22no—
107C2×Q8×C61—44no—
108C2×Q8×C61—41no—
109C22×C4×C61—41no—
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C2×C22⋊D61—41no—
114#11758222—44no—
115C2×D244—22no—
116C2×D244—22no—
117C2×D244—22no—
118C2×D244—21no—
119C2×D244—21no—
120C2×D244—21no—
121C2×D244—44no—
122C2×D244—44no—
123C2×D244—41no—
124C2×D244—41no—
125C2×D244—41no—
126C2×D244—82no—
127C2×C4×D61—41no—
128C2×Dic122—21no—
129C2×C8×C61C2×D414yes—
130C4⋊C4×C61—22no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—41no—
137C4⋊C4×C61—41no—
138C4⋊C4×C61—41no—
139C4⋊C4×C61—41no—
140C22⋊C4×C61—41no—
141C42×C61—41no—
142C42×C61C2×D411yescharacteristic
143C42×C61C2×D411yescharacteristic
144C42×C61C2×D411yescharacteristic
145C42×C61C2411yescharacteristic
146#11758201—48no—
147#11758201—44no—
148#11758201—48no—
149#11758198—41no—
150#11758198—41no—
151C4⋊Dic61—21no—
152C4⋊Dic61—21no—
153#11758194—24no—
154C2×#92418—24no—
155C2×#92418C2×D412yes—
156C2×#92418C2×D414yes—
157C2×#92418C2×D412yes—
158D488—84no—
159C23×C61#12711yescharacteristic
160C22×D61—41no—
161C22×D61—42no—
162C22×D61—82no—
163C22×D61—82no—
164C22×D61—82no—
165Q8×C61—44no—
166Q8×C61—42no—
167Q8×C61—42no—
168Q8×C61—82no—
169C2×C4×C61—22no—
170C2×C4×C61—22no—
171C2×C4×C61—41no—
172C2×C4×C61—41no—
173C2×C4×C61—42no—
174C2×C4×C61—82no—
175C2×C4×C61#12711yescharacteristic
176C2×C4×C61#12711yescharacteristic
177C2×C4×C61#12711yescharacteristic
178C2×C4×C61#13514yes—
179C2×C4×C61C22×D411yescharacteristic
180C2×C4×C61C22×D411yescharacteristic
181C2×C4×C61C22×D411yescharacteristic
182C22⋊D61—82no—
183C2×Dic61—41no—
184D244—44no—
185D244—42no—
186D244—44no—
187D244—42no—
188D244—42no—
189D244—44no—
190D244—82no—
191D244—82no—
192D244—82no—
193C4×D61—82no—
194Dic122—42no—
195C8×C61—24no—
196#92418—24no—
197#92418—22no—
198#92418AGL(1,ℤ/8)14yes—
199#19625—611nomaximal, Sylow
200C22×C61—21no—
201C22×C61—21no—
202C22×C61#49512yes—
203C22×C61C2×#12711yescharacteristic
204C2×D61—81no—
205C2×D61—82no—
206C2×D61—82no—
207C2×D61—81no—
208C2×D61—82no—
209C2×D61—81no—
210C2×D61—164no—
211C4×C61—21no—
212C4×C61—21no—
213C4×C61—82no—
214C4×C61—81no—
215C4×C61—81no—
216C4×C61#49512yes—
217C4×C61C2×#13512yes—
218C4×C61C2×AGL(1,ℤ/8)12yes—
219C4×C61#57712yes—
220Dic61—81no—
221#3230D6111yescharacteristic, complemented
222#3205—611no—
223#3167—611no—
224C2×#492—611no—
225#3141—611no—
226#3138—611no—
227#1719—612no—
228#1717—612no—
229#1708—612no—
230#1706—612no—
231C8×M16—611no—
232C2×C61—22no—
233C2×C61C2×#49511yescharacteristic
234C2×C61#315111yescharacteristic
235C2×C61#315411yescharacteristic
236D61—161no—
237D61—162no—
238C22×SD16—1222no—
239#533C2×D6111yescharacteristic
240#532—611no—
241C2×C4⋊Q8C2×D6111yescharacteristic
242C2×#127—611no—
243#500C2×D6112yescomplemented
244#498—611no—
245#493—612no—
246#492—612no—
247#492—612no—
248#492—611no—
249#492—1222no—
250#490—1222no—
251#489—1222no—
252#487—612no—
253#487—611no—
254#487C2×D6112yescomplemented
255#486—612no—
256#486—612no—
257#486—612no—
258#486—611no—
259#486—612no—
260#484—22no—
261#477—22no—
262#465—1222no—
263#459—1222no—
264#447—1222no—
265C4×SD16—1222no—
266#431C2×D6111yescharacteristic
267#418—1222no—
268#417—1222no—
269C4×M16—611no—
270C2×C4×C8—611no—
271#322—612no—
272C82—612no—
273C61#1962511yescharacteristic, complemented, Sylow
274C2×#41—1221no—
275C22×Q8—21no—
276C22×D4—1221no—
277C22×D4—1221no—
278C22×D4—1221no—
279C2×SD16—1224no—
280C2×SD16—1224no—
281C2×SD16—1222no—
282C2×SD16—2448no—
283C2×D8—1224no—
284C2×M16—612no—
285C22×C8—612no—
286C4⋊Q8—21no—
287C4⋊Q8—21no—
288C4⋊Q8—21no—
289C4⋊Q8—611no—
290C4⋊Q8C22×D6112yes—
291C4⋊Q8C22×D6111yescharacteristic
292C4⋊Q8C22×D6111yescharacteristic
293#127—612no—
294#127—611no—
295#127—1221no—
296#127—1221no—
297#127—1221no—
298#125—21no—
299#124—1221no—
300C22⋊Q8—21no—
301C22⋊Q8—21no—
302#121—1221no—
303#121—1221no—
304C4×Q8—21no—
305C4×Q8—21no—
306C4×D4—1221no—
307C4×D4—1221no—
308#117—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C42C22×D6111yescharacteristic
312#107—24no—
313#106—1222no—
314C4⋊C8C22⋊D6112yes—
315#103—24no—
316#103—1224no—
317#103—1222no—
318#102—1224no—
319#102—1224no—
320#102—1224no—
321#102—1222no—
322#102—1224no—
323C22⋊C8C22⋊D6112yes—
324#97—612no—
325C4×C8—612no—
326C4×C8—611no—
327C4×C8—612no—
328C4×C8—611no—
329C4×C8C22×D6112yes—
330C24—2441no—
331#41—2444no—
332C2×Q8—21no—
333C2×Q8—22no—
334C2×Q8—21no—
335C2×Q8—44no—
336C2×Q8—41no—
337C2×Q8—1221no—
338C2×D4—1221no—
339C2×D4—1222no—
340C2×D4—1221no—
341C2×D4—1222no—
342C2×D4—1222no—
343C2×D4—1221no—
344C2×D4—4882no—
345C2×D4—2444no—
346C2×D4—2441no—
347C2×D4—2441no—
348C2×D4—2441no—
349C2×D4—2441no—
350C2×D4—2444no—
351C22×C4—41no—
352C22×C4—2441no—
353C22×C4D4×D6111yescharacteristic
354C22×C4D4×D6111yescharacteristic
355C22×C4C2×C22⋊D6111yescharacteristic
356SD16—2444no—
357SD16—2448no—
358SD16—2448no—
359D8—4884no—
360M16—1224no—
361C2×C8—612no—
362C2×C8—612no—
363C2×C8—614no—
364C2×C8—1224no—
365C2×C8C2×C22⋊D6114yes—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—22no—
371C4⋊C4—21no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—1221no—
377C4⋊C4—1221no—
378C22⋊C4—41no—
379C22⋊C4—2441no—
380C22⋊C4—2441no—
381C42—41no—
382C42D4×D6111yescharacteristic
383C42D4×D6111yescharacteristic
384C42C2×C22⋊D6111yescharacteristic
385C42C23×D6111yescharacteristic
386C23—4882no—
387C23—4882no—
388C23—4882no—
389C23—2441no—
390C23—2442no—
391C23#1451977311yescharacteristic
392Q8—44no—
393Q8—42no—
394Q8—42no—
395Q8—82no—
396Q8—2442no—
397D4—4882no—
398D4—4882no—
399D4—4882no—
400D4—4882no—
401D4—2444no—
402D4—2442no—
403D4—2444no—
404D4—2442no—
405D4—2442no—
406D4—2444no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—82no—
413C2×C4—4882no—
414C2×C4—2441no—
415C2×C4#1451977311yescharacteristic
416C2×C4#1451977311yescharacteristic
417C2×C4#1451977311yescharacteristic
418C2×C4#1451980014yes—
419C2×C4C2×D4×D6111yescharacteristic
420C2×C4C2×D4×D6111yescharacteristic
421C2×C4C22×C22⋊D6111yescharacteristic
422C8—24no—
423C8—614no—
424C8—1224no—
425C8—1222no—
426C22—21no—
427C22—21no—
428C22—9764no—
429C22—4881no—
430C22—4882no—
431C22—4882no—
432C22—4881no—
433C22—4882no—
434C22—4881no—
435C22#1684014712yes—
436C22C2×#1451977311yescharacteristic
437C4—21no—
438C4—21no—
439C4—82no—
440C4—81no—
441C4—81no—
442C4—4881no—
443C4#1684014712yes—
444C4C2×#1451980012yes—
445C4AGL(1,ℤ/8)×D6112yes—
446C4#1684057012yes—
447C2—22no—
448C2—9762no—
449C2—9761no—
450C2C2×#1684014711yescharacteristic
451C2#2145626711yescharacteristic
452C2#2145629711yescharacteristic
subgroupG/H
centerC22C2×#14519773
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3230C2
socleC22×C61C2×#127
prime radical#21456739C2
hypercenter#3230D61
hypocenterC61#19625
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3230

Complements

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

NH ≅ G/Ntype
2. #21456379C2semidirect
3. C2×#16840136C2semidirect
4. #21456215C2semidirect
5. #21456195C2semidirect
6. #21445475C2semidirect
7. #21445461C2semidirect
8. #21445422C2semidirect
9. #21445412C2semidirect
10. #21444100C2semidirect
11. C61×#3230C2semidirect
15. #16840158C22semidirect
20. #16840136C22semidirect
23. #16840114C22semidirect
24. #16840112C22semidirect
27. #16840110C22semidirect
28. #16840110C22semidirect
38. #16839484C22semidirect
39. C8×#92418C22semidirect
42. C61×#500C22semidirect
43. C61×#487C22semidirect
221. #3230D61semidirect
243. #500C2×D61semidirect
254. #487C2×D61semidirect
273. C61#19625semidirect

Maximal subgroups

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

subgroupindex
#214567392
#214563792
C2×#168401362
#214562152
#214561952
#214454752
#214454612
#214454222
#214454122
#214441002
C61×#32302
#1962561