Finite Group Databasebeta

← #29698485

Subgroups — #29698485

Order 15616.

862 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839877C211yesmaximal, characteristic
2#21454700C212yesmaximal
3#21454411C211yesmaximal, characteristic, complemented
4#21446596C212yesmaximal, complemented
5#21446045C212yesmaximal, complemented
6#21446042C212yesmaximal, complemented
7#21445974C211yesmaximal, characteristic, complemented
8#21445879C211yesmaximal, characteristic, complemented
9#21445878C211yesmaximal, characteristic, complemented
10#21445700C211yesmaximal, characteristic, complemented
11C61×#2912C211yesmaximal, characteristic, complemented
12C2×#14519797C2211yescharacteristic
13C2×#14519783C2211yescharacteristic
14#16840242C2212yes—
15#16839877C418yescomplemented
16#16839872—22no—
17#16839867—22no—
18C4×#11758201—22no—
19C4×#11758200—22no—
20C2×#14519713C2211yescharacteristic
21C2×#14519712C2211yescharacteristic
22#16839805C2212yes—
23#16839803—22no—
24#16839797—21no—
25C2×#14519709C2211yescharacteristic
26C2×#14519708C2212yes—
27#106×D61—21no—
28C2×#14519704C2211yescharacteristic
29C2×#14519703C2211yescharacteristic
30#16839763C2212yes—
31#16839761C2212yes—
32#16839718C2212yes—
33C2×C8×Dic61C2211yescharacteristic
34#16839631C2212yes—
35#16839531C2212yes—
36#16839530C2212yes—
37#16839521C2212yes—
38C2×#131×C61C2211yescharacteristic
39C61×#520C2212yes—
40C61×#428—22no—
41C61×#428C2212yes—
42C61×#427—21no—
43C61×#427C2211yescharacteristic
44C2×#106×C61—21no—
45C2×#106×C61C2211yescharacteristic
46C2×#11758222C2311yescharacteristic
47#14519797C2×C414yes—
48#14519797Q814yes—
49C2×#11758202D411yescharacteristic
50C2×#11758200D411yescharacteristic
51C2×#11758201D411yescharacteristic
52C2×#11758199D411yescharacteristic
53#14519783—22no—
54#14519783C2×C414yescomplemented
55C2×#11758194C2312yes—
56C2×#11758194C2311yescharacteristic
57C2×#11758189—21no—
58C22×#92418C2311yescharacteristic
59C2×C8×D61—21no—
60C4×Dic122—22no—
61C4×D244—22no—
62#14519737—42no—
63#14519735—22no—
64C2×C4⋊Dic61C2312yes—
65C2×#11758196—42no—
66C4⋊C4×D61—22no—
67C2×C4×Dic61C2311yescharacteristic
68#14519713—22no—
69#14519712C2×C414yescomplemented
70#14519710—22no—
71#14519709—21no—
72#14519709—21no—
73#14519708—22no—
74#14519708—22no—
75#14519708—22no—
76#14519704C2×C414yescomplemented
77#14519703C2×C414yescomplemented
78#14519701—22no—
79#14519700—22no—
80#14519698—22no—
81#14519697—22no—
82#14519688—22no—
83C4×#92418—22no—
84C8×Dic61C2×C414yescomplemented
85C2×#41×C61C2311yescharacteristic
86#131×C61—22no—
87#131×C61C2×C414yescomplemented
88C2×M16×C61—21no—
89C2×M16×C61C2311yescharacteristic
90C22×C8×C61—21no—
91C22×C8×C61C2311yescharacteristic
92C4×Q8×C61—22no—
93C4×D4×C61—22no—
94C4×D4×C61—42no—
95#117×C61—22no—
96#117×C61C2312yes—
97C2×C4⋊C4×C61—22no—
98C2×C4⋊C4×C61C2312yes—
99#107×C61—22no—
100#107×C61D412yes—
101#106×C61—21no—
102#106×C61—21no—
103#106×C61—22no—
104#106×C61D411yescharacteristic
105#106×C61D411yescharacteristic
106#41×C61—21no—
107#41×C61—21no—
108#41×C61C22×C412yes—
109#41×C61C2×Q812yes—
110C2×Q8×C61C2×D411yescharacteristic
111C2×D4×C61—21no—
112C2×D4×C61C2×D411yescharacteristic
113C22×C4×C61—21no—
114C22×C4×C61—22no—
115C22×C4×C61—42no—
116C22×C4×C61C2411yescharacteristic
117C2×C22⋊D61—21no—
118C22×Dic61—21no—
119C22×Dic61—22no—
120#11758222—21no—
121#11758222—21no—
122#11758222C22×C412yes—
123#11758222C2×Q812yes—
124C2×D244C2×D411yescharacteristic
125C2×C4×D61—21no—
126C2×C4×D61—42no—
127C2×Dic122C2×D411yescharacteristic
128M16×C61—21no—
129M16×C61—24no—
130M16×C61C22×C412yes—
131C2×C8×C61—21no—
132C2×C8×C61—24no—
133C2×C8×C61C22×C412yes—
134C2×C8×C61C2×D411yescharacteristic
135C2×C8×C61C2×D411yescharacteristic
136C2×C8×C61#4112yes—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—42no—
140C4⋊C4×C61C2×D412yes—
141C4⋊C4×C61C2×D412yes—
142C22⋊C4×C61—22no—
143C22⋊C4×C61—42no—
144C42×C61—22no—
145C42×C61—42no—
146#11758202C4⋊C414yescomplemented
147#11758201C4⋊C414yescomplemented
148#11758200C4⋊C414yescomplemented
149#11758199C4⋊C414yescomplemented
150#11758198—42no—
151C4⋊Dic61—22no—
152C4⋊Dic61—22no—
153#11758196—42no—
154#11758196—84no—
155C4×Dic61—21no—
156C4×Dic61—42no—
157C4×Dic61C22×C412yes—
158#11758194—21no—
159#11758194—22no—
160#11758194—22no—
161#11758194C22×C412yes—
162C2×#92418—21no—
163C2×#92418—22no—
164C2×#92418C22×C412yes—
165C2×#92418C2×D411yescharacteristic
166C2×#92418C2×D411yescharacteristic
167C2×#92418C2×Q812yes—
168#11758189—24no—
169C8×D61—24no—
170C23×C61—21no—
171C23×C61H(5,𝔽2)11yescharacteristic
172C22×D61—21no—
173Q8×C61—21no—
174Q8×C61C2×C4⋊C412yes—
175D4×C61—22no—
176D4×C61—22no—
177D4×C61—21no—
178D4×C61C2×C4⋊C412yes—
179C2×C4×C61—21no—
180C2×C4×C61—22no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—42no—
185C2×C4×C61—82no—
186C2×C4×C61—162no—
187C2×C4×C61C23×C411yescharacteristic
188C2×C4×C61C22×D411yescharacteristic
189C2×C4×C61C22×D411yescharacteristic
190C2×C4×C61C22×Q811yescharacteristic
191C2×C4×C61C2×#4111yescharacteristic
192C2×C4×C612−1+411yescharacteristic
193C22⋊D61—22no—
194C22⋊D61—22no—
195C2×Dic61—24no—
196C2×Dic61—22no—
197C2×Dic61—21no—
198C2×Dic61—42no—
199C2×Dic61—82no—
200D244—21no—
201D244C2×C4⋊C412yes—
202C4×D61—22no—
203C4×D61—22no—
204C4×D61—162no—
205Dic122—21no—
206Dic122C2×C4⋊C412yes—
207C8×C61—22no—
208C8×C61C4×D412yes—
209C8×C61C4×D412yes—
210#92418—22no—
211#92418—42no—
212#92418C2×C4⋊C412yes—
213#92418C2×C4⋊C412yes—
214#22689—611nomaximal, Sylow
215C22×C61—22no—
216C22×C61—22no—
217C22×C61—42no—
218C22×C61#52011yescharacteristic
219C22×C61D4211yescharacteristic
220C22×C61#55411yescharacteristic
221C2×D61—22no—
222C2×D61—42no—
223C4×C61—22no—
224C4×C61—42no—
225C4×C61—42no—
226C4×C61—82no—
227C4×C61C22×C4⋊C411yescharacteristic
228C4×C61C2×C4×D411yescharacteristic
229C4×C61#54711yescharacteristic
230C4×C61D4×Q811yescharacteristic
231Dic61—24no—
232Dic61—22no—
233Dic61—82no—
234C2×#443—611no—
235#2952—612no—
236#2912—611no—
237#2912D6111yescharacteristic, complemented
238#1951—612no—
239#1859—612no—
240#1859—612no—
241#1844—611no—
242#1815—611no—
243#1815—611no—
244#1772—611no—
245C2×C61—21no—
246C2×C61—21no—
247C2×C61—42no—
248C2×C61D4×C4⋊C411yescharacteristic
249C2×C61#329112yes—
250D61—42no—
251C2×#135—611no—
252C2×#131—611no—
253C2×#131C2×D6111yescharacteristic
254#520—612no—
255#520C2×D6112yes—
256#443—618no—
257#442—1222no—
258#441—1222no—
259C4×SD16—1222no—
260C4×SD16—1222no—
261C2×#108—611no—
262C2×#108—611no—
263#428—22no—
264#428—612no—
265#428—1222no—
266#428C2×D6112yes—
267#427—21no—
268#427—1221no—
269#427C2×D6111yescharacteristic
270C2×#106—21no—
271C2×#106—611no—
272C2×#106—612no—
273C2×#106—1221no—
274C2×#106C2×D6111yescharacteristic
275C2×#104—611no—
276C2×#104—611no—
277#417—612no—
278#417—612no—
279#405—612no—
280C2×C4×C8—611no—
281#384—612no—
282#343—612no—
283#343—612no—
284#339—612no—
285C61#2268911yescharacteristic, complemented, Sylow
286C2×#41—611no—
287C2×#41C22×D6111yescharacteristic
288#135—614no—
289#135—614no—
290C2×Q16—611no—
291C2×SD16—611no—
292C2×SD16—611no—
293C2×D8—611no—
294#131—22no—
295#131—614no—
296#131—1222no—
297#131C4×D6114yescomplemented
298C2×M16—21no—
299C2×M16—611no—
300C2×M16—612no—
301C2×M16—1221no—
302C2×M16C22×D6111yescharacteristic
303C22×C8—21no—
304C22×C8—611no—
305C22×C8—1221no—
306C22×C8C22×D6111yescharacteristic
307C4×Q8—22no—
308C4×Q8—1222no—
309C4×D4—22no—
310C4×D4—42no—
311C4×D4—1222no—
312C4×D4—2442no—
313#117—22no—
314#117—1222no—
315#117C22×D6112yes—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—612no—
318C2×C4⋊C4—1222no—
319C2×C4⋊C4—2442no—
320C2×C4⋊C4C22×D6112yes—
321C2×C42—611no—
322#108—614no—
323#108—1222no—
324#107—22no—
325#107—1222no—
326#107C22⋊D6112yes—
327#106—21no—
328#106—21no—
329#106—22no—
330#106—1222no—
331#106—1222no—
332#106—1221no—
333#106—1221no—
334#106—1222no—
335#106C22⋊D6111yescharacteristic
336#106C22⋊D6111yescharacteristic
337#104—614no—
338#104—614no—
339#103—1222no—
340#103—1222no—
341#102—1222no—
342#102—1222no—
343#97—1222no—
344C4×C8—614no—
345C4×C8—1222no—
346#41—21no—
347#41—21no—
348#41—612no—
349#41—612no—
350#41—1221no—
351#41—1221no—
352#41C2×C4×D6112yes—
353#41Q8×D6112yes—
354C2×Q8—611no—
355C2×Q8D4×D6111yescharacteristic
356C2×D4—21no—
357C2×D4—611no—
358C2×D4—1221no—
359C2×D4D4×D6111yescharacteristic
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—42no—
363C22×C4—1221no—
364C22×C4—1221no—
365C22×C4—1222no—
366C22×C4—2442no—
367C22×C4C23×D6111yescharacteristic
368Q16—614no—
369SD16—614no—
370SD16—614no—
371D8—614no—
372M16—21no—
373M16—24no—
374M16—612no—
375M16—1221no—
376M16—1224no—
377M16—1222no—
378M16—1222no—
379M16C2×C4×D6112yes—
380C2×C8—21no—
381C2×C8—24no—
382C2×C8—612no—
383C2×C8—611no—
384C2×C8—611no—
385C2×C8—612no—
386C2×C8—1221no—
387C2×C8—1224no—
388C2×C8—1222no—
389C2×C8C2×C4×D6112yes—
390C2×C8#1175822212yes—
391C2×C8C2×C22⋊D6111yescharacteristic
392C2×C8C2×C22⋊D6111yescharacteristic
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—42no—
396C4⋊C4—1222no—
397C4⋊C4—1222no—
398C4⋊C4—4884no—
399C4⋊C4—2442no—
400C4⋊C4C2×C22⋊D6112yes—
401C4⋊C4C2×C22⋊D6112yes—
402C22⋊C4—22no—
403C22⋊C4—42no—
404C22⋊C4—2442no—
405C42—22no—
406C42—42no—
407C42—612no—
408C42—1221no—
409C42—2442no—
410C23—21no—
411C23—1221no—
412C23#1451982211yescharacteristic
413Q8—21no—
414Q8—612no—
415Q8—1221no—
416Q8C4⋊C4×D6112yes—
417D4—22no—
418D4—22no—
419D4—21no—
420D4—612no—
421D4—1222no—
422D4—1222no—
423D4—1221no—
424D4C4⋊C4×D6112yes—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—42no—
431C2×C4—82no—
432C2×C4—162no—
433C2×C4—1224no—
434C2×C4—1222no—
435C2×C4—1221no—
436C2×C4—1222no—
437C2×C4—1222no—
438C2×C4—9762no—
439C2×C4—4882no—
440C2×C4—2442no—
441C2×C4C22×C4×D6111yescharacteristic
442C2×C4C2×D4×D6111yescharacteristic
443C2×C4C22×C22⋊D6111yescharacteristic
444C2×C4C2×Q8×D6111yescharacteristic
445C2×C4C2×#1175822211yescharacteristic
446C2×C4#1451982511yescharacteristic
447C8—22no—
448C8—612no—
449C8—612no—
450C8—1222no—
451C8—2442no—
452C8C4×C22⋊D6112yes—
453C8C4×C22⋊D6112yes—
454C22—22no—
455C22—22no—
456C22—42no—
457C22—1222no—
458C22—2442no—
459C22#1684024311yescharacteristic
460C22D4×C22⋊D6111yescharacteristic
461C22#1684046711yescharacteristic
462C4—22no—
463C4—42no—
464C4—42no—
465C4—82no—
466C4—1224no—
467C4—1222no—
468C4—4882no—
469C4C2×C4⋊C4×D6111yescharacteristic
470C4C2×C4×C22⋊D6111yescharacteristic
471C4#1684041311yescharacteristic
472C4Q8×C22⋊D6111yescharacteristic
473C2—21no—
474C2—21no—
475C2—42no—
476C2—2442no—
477C2C4⋊C4×C22⋊D6111yescharacteristic
478C2#2145761012yes—
subgroupG/H
centerC22D4×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2912C2
socleC22×C61D42
prime radicalC2×#14519797C22
cosocleC22×C4×C61C24
hypercenter#2912D61
hypocenterC61#22689
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2912

Complements

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

NH ≅ G/Ntype
3. #21454411C2semidirect
4. #21446596C2semidirect
5. #21446045C2semidirect
6. #21446042C2semidirect
7. #21445974C2semidirect
8. #21445879C2semidirect
9. #21445878C2semidirect
10. #21445700C2semidirect
11. C61×#2912C2semidirect
15. #16839877C4semidirect
54. #14519783C2×C4semidirect
69. #14519712C2×C4semidirect
76. #14519704C2×C4semidirect
77. #14519703C2×C4semidirect
84. C8×Dic61C2×C4semidirect
87. #131×C61C2×C4semidirect
146. #11758202C4⋊C4semidirect
147. #11758201C4⋊C4semidirect
148. #11758200C4⋊C4semidirect
149. #11758199C4⋊C4semidirect
237. #2912D61semidirect
285. C61#22689semidirect
297. #131C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398772
#214547002
#214544112
#214465962
#214460452
#214460422
#214459742
#214458792
#214458782
#214457002
C61×#29122
#2268961