Finite Group Databasebeta

← #29582584

Subgroups — #29582584

Order 15616.

594 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840113C211yesmaximal, characteristic, complemented
2C2×#16840043C211yesmaximal, characteristic
3C2×#16839968C211yesmaximal, characteristic
4#21446613C211yesmaximal, characteristic
5C2×#16839509C211yesmaximal, characteristic, complemented
6C2×#16839507C211yesmaximal, characteristic, complemented
7C61×#1277C211yesmaximal, characteristic, complemented
8C22×#11758190—21no—
9C2×#14519777C2211yescharacteristic
10C2×#14519777C2211yescharacteristic
11C2×C4×D244C2211yescharacteristic
12#16840113—22no—
13#16840113—22no—
14#16840043—22no—
15#16840043—22no—
16#16839968—22no—
17#16839968—22no—
18C2×#14519711—21no—
19C2×C4⋊#92418C2211yescharacteristic
20C2×#14519702—21no—
21C2×#14519702—22no—
22C2×#14519699—21no—
23C2×#14519689—22no—
24#16839509—22no—
25#16839507—22no—
26C2×C4⋊C8×C61C2211yescharacteristic
27C2×C4⋊C8×C61C2211yescharacteristic
28C2×C4×C8×C61C2211yescharacteristic
29C22×Dic122—21no—
30C22×Dic122—21no—
31C22×D244D411yescharacteristic
32C22×C4×D61—21no—
33C2×#11758190—24no—
34C2×#11758190—42no—
35C2×#11758190—42no—
36C22×#92418—21no—
37#14519777—21no—
38#14519777—21no—
39#14519777—21no—
40#14519777—21no—
41#14519777—21no—
42#14519777—21no—
43#14519777D412yes—
44#14519777D412yes—
45C4×D244—21no—
46C4×D244—21no—
47C4×D244—21no—
48C4×D244—21no—
49C2×C4⋊Dic61—21no—
50C2×C4⋊Dic61—21no—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61D411yescharacteristic
54C2×#11758198—21no—
55#14519711—24no—
56C4⋊#92418—21no—
57C4⋊#92418—21no—
58#14519702—24no—
59#14519702—28no—
60#14519699—24no—
61#14519689—44no—
62C22×C8×C61—21no—
63C22×C8×C61—21no—
64C22×C8×C61D412yes—
65C2×C42×C61C2311yescharacteristic
66C4⋊C8×C61—21no—
67C4⋊C8×C61—21no—
68C4⋊C8×C61D412yes—
69C4⋊C8×C61D412yes—
70C4×C8×C61—21no—
71C4×C8×C61—21no—
72C23×D61—21no—
73C22×C4×C61C2×D411yescharacteristic
74C22×C4×C61C2×D411yescharacteristic
75C22×C4×C61#4111yescharacteristic
76C22×Dic61—21no—
77C22×Dic61—41no—
78C22×Dic61—41no—
79C2×D244—21no—
80C2×D244—21no—
81C2×D244—21no—
82C2×D244—21no—
83C2×D244—21no—
84C2×D244SD1612yes—
85C2×C4×D61—41no—
86C2×C4×D61—41no—
87C2×C4×D61—42no—
88C2×C4×D61—42no—
89C2×Dic122—22no—
90C2×Dic122—22no—
91C2×Dic122—41no—
92C2×Dic122—42no—
93C2×Dic122—41no—
94C2×Dic122—42no—
95C2×Dic122—41no—
96C2×Dic122—41no—
97C2×Dic122—41no—
98C2×Dic122—41no—
99C2×C8×C61—22no—
100C2×C8×C61—22no—
101C2×C8×C61—22no—
102C2×C8×C61—22no—
103C2×C8×C61—22no—
104C2×C8×C61—41no—
105C2×C8×C61—41no—
106C2×C8×C61—41no—
107C2×C8×C61—41no—
108C42×C61C2×D411yescharacteristic
109C42×C61C2×D411yescharacteristic
110C42×C61C2×D411yescharacteristic
111C42×C61C2×D411yescharacteristic
112#11758198—41no—
113#11758198—41no—
114C4⋊Dic61—22no—
115C4⋊Dic61—22no—
116C4⋊Dic61—21no—
117C4⋊Dic61—21no—
118C4⋊Dic61—22no—
119C4⋊Dic61—22no—
120C4⋊Dic61—41no—
121C4⋊Dic61—41no—
122C4⋊Dic61—41no—
123C4⋊Dic61—41no—
124C2×#92418—41no—
125C2×#92418—42no—
126#11758190—48no—
127C23×C61#12111yescharacteristic
128C22×D61—22no—
129C22×D61—41no—
130C22×D61—42no—
131C22×D61—41no—
132C22×D61—42no—
133C2×C4×C61—22no—
134C2×C4×C61#12011yescharacteristic
135C2×C4×C61#12011yescharacteristic
136C2×C4×C61#12011yescharacteristic
137C2×C4×C61#12111yescharacteristic
138C2×C4×C61#12111yescharacteristic
139C2×C4×C61#12711yescharacteristic
140C2×C4×C61C2×SD1611yescharacteristic
141C2×C4×C61#13511yescharacteristic
142C2×C4×C61#13512yes—
143C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C61AGL(1,ℤ/8)12yes—
145C2×C4×C61#13711yescharacteristic
146C2×Dic61—42no—
147C2×Dic61—42no—
148C2×Dic61—41no—
149C2×Dic61—42no—
150C2×Dic61—82no—
151C2×Dic61—81no—
152C2×Dic61—81no—
153D244—22no—
154D244—22no—
155D244—41no—
156D244—41no—
157C4×D61—82no—
158C4×D61—81no—
159C4×D61—81no—
160Dic122—44no—
161Dic122—42no—
162Dic122—42no—
163Dic122—81no—
164Dic122—81no—
165Dic122—81no—
166Dic122—81no—
167C8×C61—42no—
168C8×C61—42no—
169C8×C61—42no—
170#92418—81no—
171#10270—611nomaximal, Sylow
172C22×C61#39211yescharacteristic
173C22×C61#46312yes—
174C22×C61#46511yescharacteristic
175C22×C61#46811yescharacteristic
176C22×C61#47312yes—
177C2×D61—44no—
178C2×D61—42no—
179C2×D61—82no—
180C2×D61—81no—
181C2×D61—81no—
182C4×C61—21no—
183C4×C61—21no—
184C4×C61#44811yescharacteristic
185C4×C61#44911yescharacteristic
186C4×C61#44912yes—
187C4×C61#45011yescharacteristic
188C4×C61#45211yescharacteristic
189C4×C61#49511yescharacteristic
190C4×C61#49611yescharacteristic
191Dic61—82no—
192Dic61—82no—
193Dic61—81no—
194C2×#487—611no—
195C2×#474—611no—
196C2×#461—611no—
197#1954—611no—
198C2×#332—611no—
199C2×#332—611no—
200#1277D6111yescharacteristic, complemented
201C2×C61#167912yes—
202C2×C61#169912yes—
203C2×C61#200011yescharacteristic
204C2×C61#201211yescharacteristic
205C2×C61#202011yescharacteristic
206D61—82no—
207C22×SD16—1221no—
208C2×C4⋊Q8—611no—
209C2×C4⋊Q8—611no—
210C2×C4×D4—611no—
211#487—1222no—
212#487—1222no—
213#474—1222no—
214#474—1222no—
215#461—1222no—
216#461—1222no—
217C2×#107—1221no—
218C2×C4⋊C8—611no—
219C2×C4⋊C8C2×D6111yescharacteristic
220C2×C4⋊C8C2×D6111yescharacteristic
221C2×#103—1221no—
222C2×#103—1222no—
223C2×#102—1221no—
224C2×C22⋊C8—1222no—
225C2×C4×C8C2×D6111yescharacteristic
226#332—1222no—
227#332—1222no—
228C61#1027011yescharacteristic, complemented, Sylow
229C22×Q8—1221no—
230C22×Q8—1221no—
231C22×D4—611no—
232C23×C4—1221no—
233C2×SD16—1224no—
234C2×SD16—2442no—
235C2×SD16—2442no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—1221no—
239C22×C8D24412yes—
240C4⋊Q8—612no—
241C4⋊Q8—612no—
242C4⋊Q8—1221no—
243C4⋊Q8—1221no—
244C4⋊Q8—1221no—
245C4⋊Q8—1221no—
246C4⋊Q8—1221no—
247C4⋊Q8—1221no—
248C4×D4—1221no—
249C4×D4—1221no—
250C4×D4—1221no—
251C4×D4—1221no—
252C2×C4⋊C4—611no—
253C2×C4⋊C4—1221no—
254C2×C4⋊C4—1221no—
255C2×C4⋊C4—1221no—
256C2×C4⋊C4—1221no—
257C2×C22⋊C4—1221no—
258C2×C42C22×D6111yescharacteristic
259#107—1224no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—1221no—
263C4⋊C8—1221no—
264C4⋊C8D24412yes—
265C4⋊C8D24412yes—
266#103—1224no—
267#103—1228no—
268#102—1224no—
269C22⋊C8—2444no—
270C4×C8—21no—
271C4×C8—21no—
272C24—1221no—
273C2×Q8—1222no—
274C2×Q8—1222no—
275C2×Q8—2441no—
276C2×Q8—2441no—
277C2×Q8—2441no—
278C2×Q8—2442no—
279C2×Q8—2441no—
280C2×Q8—2442no—
281C2×Q8—2441no—
282C2×Q8—2441no—
283C2×D4—612no—
284C2×D4—1221no—
285C2×D4—1221no—
286C2×D4—1221no—
287C2×D4—1221no—
288C2×D4—1221no—
289C22×C4—1221no—
290C22×C4—2441no—
291C22×C4—2441no—
292C22×C4—2441no—
293C22×C4—2441no—
294C22×C4—2442no—
295C22×C4—2442no—
296C22×C4C2×D24411yescharacteristic
297C22×C4D4×D6111yescharacteristic
298C22×C4#1175822611yescharacteristic
299SD16—2448no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—2441no—
310C2×C8—2442no—
311C4⋊C4—1222no—
312C4⋊C4—1221no—
313C4⋊C4—1222no—
314C4⋊C4—1221no—
315C4⋊C4—1222no—
316C4⋊C4—1222no—
317C4⋊C4—2441no—
318C4⋊C4—2441no—
319C4⋊C4—2441no—
320C4⋊C4—2441no—
321C22⋊C4—2441no—
322C22⋊C4—2441no—
323C42C2×D24411yescharacteristic
324C42C2×D24411yescharacteristic
325C42D4×D6111yescharacteristic
326C42D4×D6111yescharacteristic
327C23—1222no—
328C23—2442no—
329C23—2441no—
330C23—2442no—
331C23—2441no—
332C23#1451975211yescharacteristic
333Q8—4881no—
334Q8—4881no—
335Q8—4881no—
336Q8—4881no—
337Q8—2444no—
338Q8—2442no—
339Q8—2442no—
340D4—1222no—
341D4—1222no—
342D4—2441no—
343D4—2441no—
344C2×C4—22no—
345C2×C4—4882no—
346C2×C4—4881no—
347C2×C4—4881no—
348C2×C4—4881no—
349C2×C4—4881no—
350C2×C4—4882no—
351C2×C4—2442no—
352C2×C4—2441no—
353C2×C4—2442no—
354C2×C4—2442no—
355C2×C4#1451974711yescharacteristic
356C2×C4#1451974711yescharacteristic
357C2×C4#1451974711yescharacteristic
358C2×C4#1451975211yescharacteristic
359C2×C4#1451975211yescharacteristic
360C2×C4#1451977411yescharacteristic
361C2×C4SD16×D6111yescharacteristic
362C2×C4#1451979811yescharacteristic
363C2×C4#1451979912yes—
364C2×C4#1451980111yescharacteristic
365C2×C4#1451980512yes—
366C2×C4#1451980911yescharacteristic
367C8—42no—
368C8—42no—
369C8—42no—
370C8—4881no—
371C22—4881no—
372C22—4881no—
373C22—4882no—
374C22—2444no—
375C22—2442no—
376C22#1683967211yescharacteristic
377C22#1683997812yes—
378C22#1683999211yescharacteristic
379C22#1684000911yescharacteristic
380C22#1684003812yes—
381C4—21no—
382C4—21no—
383C4—4882no—
384C4—4881no—
385C4—4882no—
386C4#1683990211yescharacteristic
387C4#1683990811yescharacteristic
388C4#1683990912yes—
389C4#1683991411yescharacteristic
390C4#1683992711yescharacteristic
391C4#1684014611yescharacteristic
392C4#1684015111yescharacteristic
393C2—4882no—
394C2#2144524312yes—
395C2#2144536712yes—
396C2#2144682511yescharacteristic
397C2#2144689911yescharacteristic
398C2#2144694111yescharacteristic
subgroupG/H
centerC23#14519752
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1277C2
socleC23×C61#121
prime radicalC22×D244D4
hypercenter#1277D61
hypocenterC61#10270
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1277

Complements

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

NH ≅ G/Ntype
1. C2×#16840113C2semidirect
5. C2×#16839509C2semidirect
6. C2×#16839507C2semidirect
7. C61×#1277C2semidirect
200. #1277D61semidirect
228. C61#10270semidirect

Maximal subgroups

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

subgroupindex
C2×#168401132
C2×#168400432
C2×#168399682
#214466132
C2×#168395092
C2×#168395072
C61×#12772
#1027061