Finite Group Databasebeta

← #29663914

Subgroups — #29663914

Order 15616.

702 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840141C211yesmaximal, characteristic, complemented
2#21455635C211yesmaximal, characteristic
3#21455628C211yesmaximal, characteristic
4#21454775C211yesmaximal, characteristic
5#21445188C214yesmaximal, complemented
6#21445183C214yesmaximal, complemented
7#21444577C211yesmaximal, characteristic, complemented
8#21444503C211yesmaximal, characteristic, complemented
9C61×#1549C211yesmaximal, characteristic, complemented
10C22×#11758202—22no—
11C2×#14519777C2211yescharacteristic
12C4×#11758222C2211yescharacteristic
13#16840141—21no—
14#16840141—21no—
15#16840141C2214yes—
16#16840058—21no—
17#16840054—21no—
18#16840053—21no—
19#16840049—21no—
20#16840048C2212yes—
21#16840044C2212yes—
22#16840043C2212yes—
23#16840039C2212yes—
24#16839886C2212yes—
25#16839827C2212yes—
26#16839825—22no—
27#16839813—21no—
28#16839804—21no—
29#16839802—21no—
30#16839786C2211yescharacteristic
31#16839764—21no—
32#16839761—21no—
33C2×C4×#92418C2211yescharacteristic
34C4×C8×D61—21no—
35#16839514C2214yes—
36#16839509C2214yes—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C61×#431C2211yescharacteristic
39C61×#423C2211yescharacteristic
40C61×#332C2214yes—
41C2×#11758222D411yescharacteristic
42C22×Dic122—21no—
43C2×#11758202—24no—
44C2×#11758202—24no—
45C2×#11758202—48no—
46C2×#11758194—21no—
47C22×#92418D412yes—
48C2×C8×D61—22no—
49#14519777—21no—
50#14519777—21no—
51#14519777—21no—
52#14519777C2312yes—
53C4×Dic122D411yescharacteristic
54C4×Dic122C2311yescharacteristic
55C4×D244D411yescharacteristic
56C4×D244C2311yescharacteristic
57C4×C22⋊D61—21no—
58C4×C22⋊D61—21no—
59#14519736D411yescharacteristic
60#14519734—21no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—21no—
63C42×D61—21no—
64#14519711—22no—
65#14519710—22no—
66#14519709—22no—
67#14519708—22no—
68C4⋊#92418—21no—
69C4⋊#92418C2312yes—
70#14519706—22no—
71#14519702—22no—
72#14519700—22no—
73#14519699—22no—
74#14519697—22no—
75#14519689—22no—
76C4×#92418D411yescharacteristic
77C4×#92418D411yescharacteristic
78C4×#92418C2312yes—
79C8×Dic61—22no—
80C22×Q8×C61—21no—
81C2×M16×C61—21no—
82C4⋊Q8×C61—21no—
83C4⋊Q8×C61—21no—
84C4⋊Q8×C61—21no—
85C4⋊Q8×C61C2312yes—
86C2×C4⋊C4×C61—21no—
87C2×C4⋊C4×C61—21no—
88C2×C42×C61C2311yescharacteristic
89C4⋊C8×C61—21no—
90C4⋊C8×C61C2312yes—
91C4⋊C8×C61C2312yes—
92C22⋊C8×C61—21no—
93C4×C8×C61—21no—
94C2×Q8×C61—22no—
95C2×Q8×C61—44no—
96C2×Q8×C61—41no—
97C22×C4×C61—41no—
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61#4111yescharacteristic
101C2×C22⋊D61—21no—
102C22×Dic61—41no—
103#11758222—22no—
104#11758222Q1614yescomplemented
105C2×D244C2×D411yescharacteristic
106C2×C4×D61—21no—
107C2×C4×D61—21no—
108C2×C4×D61—21no—
109C2×Dic122—22no—
110C2×Dic122—44no—
111C2×Dic122—41no—
112C2×Dic122C2×D411yescharacteristic
113M16×C61—42no—
114C2×C8×C61—22no—
115C2×C8×C61—22no—
116C4⋊C4×C61—22no—
117C4⋊C4×C61—22no—
118C4⋊C4×C61—41no—
119C4⋊C4×C61—41no—
120C42×C61C2×D411yescharacteristic
121C42×C61C2×D411yescharacteristic
122C42×C61#4111yescharacteristic
123C42×C61C2411yescharacteristic
124C22⋊Dic61—21no—
125#11758202—48no—
126#11758202—48no—
127#11758198—21no—
128#11758198—21no—
129C4⋊Dic61—22no—
130C4⋊Dic61—22no—
131C4⋊Dic61—41no—
132C4⋊Dic61—41no—
133C4⋊Dic61C2×D411yescharacteristic
134C4⋊Dic61C2×D411yescharacteristic
135#11758196—21no—
136#11758196—21no—
137C4×Dic61—21no—
138C4×Dic61—21no—
139C4×Dic61—21no—
140#11758194—42no—
141C2×#92418—24no—
142C2×#92418—22no—
143C2×#92418C2×D412yes—
144C2×#92418C2×D412yes—
145C8×D61—44no—
146C23×C61#12111yescharacteristic
147C22×D61—21no—
148Q8×C61—44no—
149Q8×C61—82no—
150C2×C4×C61—22no—
151C2×C4×C61—22no—
152C2×C4×C61—42no—
153C2×C4×C61—82no—
154C2×C4×C61#12111yescharacteristic
155C2×C4×C61#12111yescharacteristic
156C2×C4×C61#12111yescharacteristic
157C2×C4×C61C2×Q1612yes—
158C2×C4×C61#13512yes—
159C2×C4×C61C22×D411yescharacteristic
160C2×C4×C61C22×D411yescharacteristic
161C2×C4×C61C2×#4111yescharacteristic
162C22⋊D61—24no—
163C2×Dic61—21no—
164C2×Dic61—21no—
165C2×Dic61—21no—
166C2×Dic61—42no—
167C2×Dic61—82no—
168D244—21no—
169D244C2×Q1612yescomplemented
170C4×D61—24no—
171C4×D61—42no—
172C4×D61—42no—
173Dic122—21no—
174Dic122—44no—
175Dic122—82no—
176Dic122C2×Q1612yescomplemented
177C8×C61—42no—
178C8×C61—42no—
179#92418—22no—
180#92418—42no—
181#92418AGL(1,ℤ/8)14yes—
182#19268—611nomaximal, Sylow
183C22×C61—21no—
184C22×C61—21no—
185C22×C61#46712yes—
186C22×C61C2×#12111yescharacteristic
187C2×D61—22no—
188C2×D61—41no—
189C2×D61—41no—
190C4×C61—21no—
191C4×C61—21no—
192C4×C61—82no—
193C4×C61#46712yes—
194C4×C61#47312yes—
195C4×C61C22×Q1611yescharacteristic
196C4×C61C2×#13511yescharacteristic
197C4×C61C2×AGL(1,ℤ/8)12yes—
198Dic61—22no—
199Dic61—41no—
200Dic61—41no—
201Dic61—82no—
202C2×#494—611no—
203#3079—611no—
204#3079—611no—
205C8×#41—611no—
206#1670—614no—
207#1669—614no—
208#1567—611no—
209#1549—611no—
210#1549D6111yescharacteristic, complemented
211C2×C61—22no—
212C2×C61#303811yescharacteristic
213C2×C61C2×#46711yescharacteristic
214C2×C61C2×#47311yescharacteristic
215D61—42no—
216C22×Q16—1222no—
217C2×C4⋊Q8—611no—
218C2×C4⋊Q8C2×D6111yescharacteristic
219C4×#41—611no—
220#494—614no—
221#494—1221no—
222#494—1221no—
223#477—1221no—
224#477—1221no—
225#476—1221no—
226#476—1221no—
227#475—612no—
228#475—612no—
229#474—612no—
230#474—612no—
231C8×Q8—612no—
232C8×D4—612no—
233C8×D4—1222no—
234#431—1221no—
235#431C2×D6111yescharacteristic
236#428—1221no—
237#428—1221no—
238#423—611no—
239#423C2×D6111yescharacteristic
240#417—1221no—
241#417—1221no—
242C2×C4×C8—611no—
243C2×C4×C8—1221no—
244#335—614no—
245#332—614no—
246#332C2×D6114yes—
247C61#1926811yescharacteristic, complemented, Sylow
248C2×#41—611no—
249C22×Q8—21no—
250C22×Q8—1221no—
251C2×Q16—1224no—
252C2×Q16—1224no—
253C2×Q16—2448no—
254C2×M16—21no—
255C2×M16—1221no—
256C22×C8—612no—
257C22×C8—1222no—
258C4⋊Q8—21no—
259C4⋊Q8—21no—
260C4⋊Q8—21no—
261C4⋊Q8—612no—
262C4⋊Q8—1221no—
263C4⋊Q8—1221no—
264C4⋊Q8—1221no—
265C4⋊Q8C22×D6112yes—
266C4×Q8—611no—
267C4×Q8—611no—
268C4×D4—611no—
269C4×D4—611no—
270C4×D4—1221no—
271C4×D4—1221no—
272#117—611no—
273#117—1221no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—1221no—
277C2×C4⋊C4—1221no—
278C2×C42—1221no—
279C2×C42C22×D6111yescharacteristic
280#107—1222no—
281#107—1222no—
282#106—1222no—
283#106—1222no—
284C4⋊C8—21no—
285C4⋊C8—612no—
286C4⋊C8—1222no—
287C4⋊C8—1221no—
288C4⋊C8C22×D6112yes—
289C4⋊C8C22×D6112yes—
290#103—1222no—
291#103—1222no—
292#102—1222no—
293#102—1222no—
294C22⋊C8—21no—
295C22⋊C8—1222no—
296C4×C8—21no—
297C4×C8—611no—
298C4×C8—611no—
299C4×C8—612no—
300C4×C8—1222no—
301#41—614no—
302#41—1222no—
303C2×Q8—22no—
304C2×Q8—44no—
305C2×Q8—41no—
306C2×Q8—611no—
307C2×Q8—1222no—
308C2×Q8—2444no—
309C2×Q8—2441no—
310C2×D4—611no—
311C2×D4—1221no—
312C22×C4—41no—
313C22×C4—1221no—
314C22×C4—1221no—
315C22×C4—1221no—
316C22×C4—2441no—
317C22×C4#1175822211yescharacteristic
318C22×C4D4×D6111yescharacteristic
319C22×C4D4×D6111yescharacteristic
320Q16—2448no—
321Q16—2448no—
322M16—42no—
323M16—2442no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—612no—
327C2×C8—612no—
328C2×C8—1224no—
329C2×C8—1222no—
330C2×C8—2444no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—611no—
336C4⋊C4—611no—
337C4⋊C4—1221no—
338C4⋊C4—1221no—
339C4⋊C4—1222no—
340C4⋊C4—1222no—
341C4⋊C4—2441no—
342C4⋊C4—2441no—
343C22⋊C4—1221no—
344C22⋊C4—1221no—
345C22⋊C4—1221no—
346C42—1221no—
347C42—1221no—
348C42—1221no—
349C42#1175822211yescharacteristic
350C42D4×D6111yescharacteristic
351C42D4×D6111yescharacteristic
352C42C23×D6111yescharacteristic
353C23—1221no—
354C23#1451975011yescharacteristic
355Q8—44no—
356Q8—82no—
357Q8—612no—
358Q8—1221no—
359Q8—4882no—
360Q8—2444no—
361D4—612no—
362D4—1224no—
363D4—1221no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—42no—
367C2×C4—82no—
368C2×C4—1221no—
369C2×C4—1224no—
370C2×C4—1221no—
371C2×C4—1221no—
372C2×C4—4882no—
373C2×C4—2442no—
374C2×C4—2442no—
375C2×C4—2442no—
376C2×C4#1451975011yescharacteristic
377C2×C4#1451975011yescharacteristic
378C2×C4#1451975011yescharacteristic
379C2×C4Q16×D6112yes—
380C2×C4#1451979812yes—
381C2×C4C2×D4×D6111yescharacteristic
382C2×C4C2×D4×D6111yescharacteristic
383C2×C4C2×#1175822211yescharacteristic
384C8—42no—
385C8—42no—
386C8—614no—
387C8—1222no—
388C8—2442no—
389C22—21no—
390C22—21no—
391C22—1222no—
392C22—2441no—
393C22—2441no—
394C22#1684000212yes—
395C22C2×#1451975011yescharacteristic
396C4—21no—
397C4—21no—
398C4—82no—
399C4—1222no—
400C4—4882no—
401C4—2441no—
402C4—2441no—
403C4#1684000212yes—
404C4#1684003612yes—
405C4C2×Q16×D6111yescharacteristic
406C4C2×#1451979811yescharacteristic
407C4AGL(1,ℤ/8)×D6112yes—
408C2—22no—
409C2—2442no—
410C2#2145529811yescharacteristic
411C2C2×#1684000211yescharacteristic
412C2C2×#1684003611yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1549C2
socleC22×C61C2×#121
prime radicalC2×#11758222D4
hypercenter#1549D61
hypocenterC61#19268
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
1. C2×#16840141C2semidirect
5. #21445188C2semidirect
6. #21445183C2semidirect
7. #21444577C2semidirect
8. #21444503C2semidirect
9. C61×#1549C2semidirect
104. #11758222Q16semidirect
169. D244C2×Q16semidirect
176. Dic122C2×Q16semidirect
210. #1549D61semidirect
247. C61#19268semidirect

Maximal subgroups

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

subgroupindex
C2×#168401412
#214556352
#214556282
#214547752
#214451882
#214451832
#214445772
#214445032
C61×#15492
#1926861