Finite Group Databasebeta

← #29667927

Subgroups — #29667927

Order 15616.

708 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456967C211yesmaximal, characteristic, complemented
2#21456940C211yesmaximal, characteristic
3#21456379C211yesmaximal, characteristic
4#21456213C211yesmaximal, characteristic
5C2×#16840114C211yesmaximal, characteristic, complemented
6#21445473C212yesmaximal, complemented
7#21445466C212yesmaximal, complemented
8#21445432C212yesmaximal, complemented
9#21445427C212yesmaximal, complemented
10M16×#92418C211yesmaximal, characteristic, complemented
11C61×#3139C211yesmaximal, characteristic, complemented
12#16840312C2211yescharacteristic
13C2×#14519777C2211yescharacteristic
14#16840165C2212yes—
15#16840165C2212yes—
16#16840158C2211yescharacteristic
17#16840136C2211yescharacteristic
18#16840122—22no—
19#16840114—22no—
20#16840114C2212yes—
21#16840114C2212yes—
22#16840113C2212yes—
23#16840112C2211yescharacteristic
24#16840112C2212yes—
25#16840110C2212yes—
26#16840110C2211yescharacteristic
27#16840109C2212yes—
28#16840095—22no—
29#16840075—22no—
30#16840057—22no—
31#16840039—22no—
32#16839993—22no—
33C4×#11758201—22no—
34#16839815C2211yescharacteristic
35#16839764—22no—
36C2×#14519700—22no—
37C2×C4×#92418C2211yescharacteristic
38#16839483C2212yes—
39C8×#92418C2212yes—
40C61×#533C2211yescharacteristic
41C2×C4⋊Q8×C61C2211yescharacteristic
42C61×#491—22no—
43C61×#487C2212yes—
44C61×#487C2212yes—
45C61×#419—22no—
46C4×M16×C61C2211yescharacteristic
47C2×#11758222—21no—
48C22×Dic122—21no—
49C2×#11758201—22no—
50C22×#92418D412yes—
51#14519777—21no—
52#14519777—21no—
53#14519777—21no—
54#14519777C2312yes—
55#14519777C2311yescharacteristic
56#14519774C2311yescharacteristic
57#14519765—21no—
58#14519751—21no—
59#14519751—21no—
60C4×D244—21no—
61C4×D244—21no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61—21no—
64#14519710—24no—
65#14519710—24no—
66#14519708—22no—
67C4⋊#92418D412yes—
68#14519702—24no—
69#14519701—22no—
70#14519701—24no—
71#14519700—24no—
72#14519700—24no—
73#14519699—24no—
74#14519697—22no—
75#14519697—24no—
76C22⋊#92418D412yes—
77C4×#92418D412yes—
78C4×#92418C2311yescharacteristic
79C4×#92418C2311yescharacteristic
80C4×#92418C2312yes—
81C22×Q8×C61—21no—
82C2×M16×C61D412yes—
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#103×C61—24no—
99#103×C61—24no—
100#97×C61D412yes—
101C4×C8×C61C2312yes—
102C2×Q8×C61—22no—
103C2×Q8×C61—21no—
104C2×Q8×C61—21no—
105C2×Q8×C61—41no—
106C2×Q8×C61—44no—
107C22×C4×C61—41no—
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C2×C22⋊D61—41no—
112C22×Dic61—41no—
113#11758222—44no—
114C2×D244—21no—
115C2×D244—21no—
116C2×D244—21no—
117C2×C4×D61—41no—
118C2×Dic122—22no—
119C2×Dic122—21no—
120C2×Dic122—44no—
121C2×Dic122—41no—
122M16×C61—24no—
123C2×C8×C61C2×D414yes—
124C4⋊C4×C61—21no—
125C4⋊C4×C61—21no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—21no—
129C4⋊C4×C61—21no—
130C4⋊C4×C61—41no—
131C4⋊C4×C61—41no—
132C4⋊C4×C61—41no—
133C4⋊C4×C61—41no—
134C22⋊C4×C61—41no—
135C42×C61—41no—
136C42×C61C2×D411yescharacteristic
137C42×C61C2×D411yescharacteristic
138C42×C61C2×D411yescharacteristic
139C42×C61C2411yescharacteristic
140#11758201—44no—
141#11758198—41no—
142#11758198—41no—
143C4⋊Dic61—22no—
144C4⋊Dic61—21no—
145C4⋊Dic61—22no—
146C4⋊Dic61—21no—
147C4⋊Dic61—41no—
148C4⋊Dic61—41no—
149C2×#92418—42no—
150C2×#92418C2×D412yes—
151C2×#92418C2×D412yes—
152C2×#92418C2×D414yes—
153C23×C61#12711yescharacteristic
154C22×D61—41no—
155Q8×C61—44no—
156Q8×C61—42no—
157Q8×C61—42no—
158Q8×C61—82no—
159C2×C4×C61—22no—
160C2×C4×C61—22no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—42no—
164C2×C4×C61—82no—
165C2×C4×C61#12711yescharacteristic
166C2×C4×C61#12711yescharacteristic
167C2×C4×C61#12711yescharacteristic
168C2×C4×C61#13514yes—
169C2×C4×C61C22×D411yescharacteristic
170C2×C4×C61C22×D411yescharacteristic
171C2×C4×C61C22×D411yescharacteristic
172C22⋊D61—82no—
173C2×Dic61—42no—
174C2×Dic61—41no—
175C2×Dic61—82no—
176D244—42no—
177D244—42no—
178D244—42no—
179C4×D61—82no—
180Dic122—42no—
181Dic122—44no—
182Dic122—82no—
183C8×C61—24no—
184#92418—22no—
185#92418—22no—
186#92418—24no—
187#19628—611nomaximal, Sylow
188C22×C61—21no—
189C22×C61—21no—
190C22×C61#49512yes—
191C22×C61C2×#12711yescharacteristic
192C2×D61—81no—
193C2×D61—81no—
194C2×D61—81no—
195C4×C61—21no—
196C4×C61—21no—
197C4×C61—82no—
198C4×C61—81no—
199C4×C61—81no—
200C4×C61#49512yes—
201C4×C61C2×#13512yes—
202C4×C61#57512yes—
203C4×C61#57812yes—
204Dic61—81no—
205Dic61—82no—
206#3230—611no—
207#3226—611no—
208#3167—611no—
209#3141—611no—
210#3139D6111yescharacteristic, complemented
211C2×#487—611no—
212#1719—612no—
213#1718—612no—
214#1711—612no—
215#1710—612no—
216C8×M16—611no—
217C2×C61—22no—
218C2×C61C2×#49511yescharacteristic
219C2×C61#315311yescharacteristic
220C2×C61#315611yescharacteristic
221D61—161no—
222#533C2×D6111yescharacteristic
223#532—611no—
224C2×C4⋊Q8—611no—
225C2×C4⋊Q8C2×D6111yescharacteristic
226#500—612no—
227#500—612no—
228#498—611no—
229#492—611no—
230#491—22no—
231#489—1222no—
232#487—612no—
233#487—612no—
234#487—611no—
235#487—612no—
236#487—612no—
237#487—1222no—
238#487C2×D6112yes—
239#487C2×D6112yes—
240#486—611no—
241#486—612no—
242#486—612no—
243#484—1222no—
244#480—1222no—
245#477—1222no—
246#474—1222no—
247#465—1222no—
248C4×SD16—1222no—
249#431—611no—
250#419—22no—
251#417—1222no—
252C2×#103—1222no—
253C4×M16C2×D6111yescharacteristic
254C2×C4×C8—611no—
255#322—612no—
256C82—612no—
257C61#1962811yescharacteristic, complemented, Sylow
258C2×#41—1221no—
259C22×Q8—21no—
260C22×Q8—1221no—
261C2×SD16—1222no—
262C2×M16C22⋊D6112yes—
263C22×C8—612no—
264C4⋊Q8—21no—
265C4⋊Q8—21no—
266C4⋊Q8—21no—
267C4⋊Q8—612no—
268C4⋊Q8—611no—
269C4⋊Q8—1221no—
270C4⋊Q8—1221no—
271C4⋊Q8—1221no—
272C4⋊Q8C22×D6112yes—
273C4⋊Q8C22×D6111yescharacteristic
274C4⋊Q8C22×D6111yescharacteristic
275#127—611no—
276#125—21no—
277#124—1221no—
278C22⋊Q8—21no—
279C22⋊Q8—21no—
280#121—1221no—
281#121—1221no—
282C4×Q8—21no—
283C4×Q8—21no—
284C4×D4—1221no—
285C4×D4—1221no—
286#117—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—1221no—
290C2×C4⋊C4—1221no—
291C2×C42C22×D6111yescharacteristic
292#107—1224no—
293#107—1224no—
294#106—1222no—
295C4⋊C8—612no—
296#103—24no—
297#103—24no—
298#103—1224no—
299#103—1224no—
300#103—1222no—
301#103—1224no—
302#103—1224no—
303#102—1222no—
304#102—1224no—
305#102—1224no—
306C22⋊C8—612no—
307#97C22⋊D6112yes—
308C4×C8—612no—
309C4×C8—611no—
310C4×C8—612no—
311C4×C8—611no—
312C4×C8C22×D6112yes—
313#41—2444no—
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—22no—
317C2×Q8—41no—
318C2×Q8—44no—
319C2×Q8—1222no—
320C2×Q8—1221no—
321C2×Q8—2444no—
322C2×Q8—2441no—
323C2×D4—1221no—
324C2×D4—1221no—
325C2×D4—1221no—
326C2×D4—2441no—
327C22×C4—41no—
328C22×C4—2441no—
329C22×C4—2441no—
330C22×C4D4×D6111yescharacteristic
331C22×C4D4×D6111yescharacteristic
332C22×C4C2×C22⋊D6111yescharacteristic
333SD16—2444no—
334M16—24no—
335C2×C8—612no—
336C2×C8—612no—
337C2×C8—614no—
338C2×C8—2442no—
339C2×C8C2×C22⋊D6114yes—
340C4⋊C4—21no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—1221no—
351C4⋊C4—1222no—
352C4⋊C4—1221no—
353C4⋊C4—1222no—
354C4⋊C4—2441no—
355C4⋊C4—2441no—
356C22⋊C4—41no—
357C22⋊C4—2441no—
358C22⋊C4—2441no—
359C42—41no—
360C42D4×D6111yescharacteristic
361C42D4×D6111yescharacteristic
362C42C2×C22⋊D6111yescharacteristic
363C42C23×D6111yescharacteristic
364C23—2441no—
365C23#1451977311yescharacteristic
366Q8—44no—
367Q8—42no—
368Q8—42no—
369Q8—82no—
370Q8—4882no—
371Q8—2442no—
372Q8—2444no—
373D4—4882no—
374D4—2442no—
375D4—2442no—
376D4—2442no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—82no—
383C2×C4—4882no—
384C2×C4—4882no—
385C2×C4—2441no—
386C2×C4—2442no—
387C2×C4#1451977311yescharacteristic
388C2×C4#1451977311yescharacteristic
389C2×C4#1451977311yescharacteristic
390C2×C4#1451979714yes—
391C2×C4C2×D4×D6111yescharacteristic
392C2×C4C2×D4×D6111yescharacteristic
393C2×C4C22×C22⋊D6111yescharacteristic
394C8—24no—
395C8—1222no—
396C8—1222no—
397C8—1224no—
398C22—21no—
399C22—21no—
400C22—4881no—
401C22—4881no—
402C22—4881no—
403C22#1684014412yes—
404C22C2×#1451977311yescharacteristic
405C4—21no—
406C4—21no—
407C4—82no—
408C4—81no—
409C4—81no—
410C4—4881no—
411C4—4882no—
412C4#1684014412yes—
413C4C2×#1451979712yes—
414C4#1684056112yes—
415C4#1684057812yes—
416C2—22no—
417C2—9761no—
418C2C2×#1684014411yescharacteristic
419C2#2145628611yescharacteristic
420C2#2145631211yescharacteristic
subgroupG/H
centerC22C2×#14519773
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3139C2
socleC22×C61C2×#127
prime radical#16840312C22
hypercenter#3139D61
hypocenterC61#19628
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3139

Complements

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

NH ≅ G/Ntype
1. #21456967C2semidirect
5. C2×#16840114C2semidirect
6. #21445473C2semidirect
7. #21445466C2semidirect
8. #21445432C2semidirect
9. #21445427C2semidirect
10. M16×#92418C2semidirect
11. C61×#3139C2semidirect
210. #3139D61semidirect
257. C61#19628semidirect

Maximal subgroups

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

subgroupindex
#214569672
#214569402
#214563792
#214562132
C2×#168401142
#214454732
#214454662
#214454322
#214454272
M16×#924182
C61×#31392
#1962861