Finite Group Databasebeta

← #10773113

Subgroups — #10773113

Order 768.

946 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 946 conjugacy classes of subgroups — too many to draw. Only its 179 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3548C211yesmaximal, characteristic, complemented
2#83334C211yesmaximal, characteristic, complemented
3#83313C212yesmaximal, complemented
4#71911C214yesmaximal, complemented
5#71910C214yesmaximal, complemented
6#71834C211yesmaximal, characteristic, complemented
7#71802C212yesmaximal, complemented
8#18696—31nomaximal, Sylow
9C3×#562—22no—
10C3×#554—22no—
11C3×#550—22no—
12C3×#547C2214yes—
13C3×#540—22no—
14C3×#537—24no—
15C3×#533C2211yescharacteristic
16C3×#532C2212yes—
17#5477C2211yescharacteristic
18#5093C2212yes—
19#5088C2212yes—
20#5084—22no—
21#5072—22no—
22#5068—22no—
23#5039—24no—
24#5015C2214yes—
25#5008C2214yes—
26#5002—22no—
27#4924C2212yes—
28C2×C4×C3⋊C8C2211yescharacteristic
29#4547C2214yes—
30D4⋊(C3⋊C8)C2218yes—
31#3548D311yescharacteristic, complemented
32#3253—32no—
33#3131—31no—
34#1635—34no—
35#1625—34no—
36#1561—31no—
37#1492—32no—
38C2×C3×#41—24no—
39C2×C3×#41—22no—
40C2×#235—21no—
41C3×C4⋊Q8D412yes—
42C3×C4⋊Q8C2312yes—
43C3×#127—22no—
44C3×#127D412yes—
45C3×#125—22no—
46C3×#125D412yes—
47C3×#124—22no—
48C3×#123—28no—
49C3×C22⋊Q8—28no—
50C3×C22⋊Q8—22no—
51C3×C22⋊Q8—24no—
52C3×#121—28no—
53C3×#121—28no—
54C3×#121—24no—
55C3×#121—22no—
56C3×#121D414yes—
57C3×C4×Q8—22no—
58C3×C4×D4—24no—
59C3×C4×D4—22no—
60C3×C4×D4C2314yes—
61C3×#117D412yes—
62C2×C3×C4⋊C4—24no—
63C2×C3×C42C2311yescharacteristic
64C2×#213—24no—
65#930—22no—
66C2×#208—22no—
67C22×C3⋊C8—21no—
68#875—21no—
69#874C2311yescharacteristic
70C4×D12—22no—
71#869C2311yescharacteristic
72#832—24no—
73#810—24no—
74#809—24no—
75#809—24no—
76#808—24no—
77C4⋊(C3⋊C8)—22no—
78C4⋊(C3⋊C8)C2314yes—
79C4×C3⋊C8—21no—
80C4×C3⋊C8C2312yes—
81#562—22no—
82#554—22no—
83#550—22no—
84#547C2×D314yes—
85#540—22no—
86#537—24no—
87#533C2×D311yescharacteristic
88#532—31no—
89#532C2×D312yes—
90#488—62no—
91#487—32no—
92#486—32no—
93#476—34no—
94#476—62no—
95#469—64no—
96#459—34no—
97#459—62no—
98#423—32no—
99#416—62no—
100C2×C4×C8—31no—
101#332—34no—
102D4⋊C8—38no—
103C3×#41—82no—
104C3×#41—82no—
105C3×#41—84no—
106C3×#41—84no—
107C2×C3×Q8—22no—
108C2×C3×Q8—22no—
109C2×C3×D4—28no—
110C2×C3×D4—24no—
111C2×C3×D4—22no—
112C2×C3×D4—22no—
113C2×C3×D4C2×D414yes—
114C22×C3×C4—28no—
115C22×C3×C4—24no—
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D412yes—
118C2×#67—41no—
119#235—44no—
120C2×D12—21no—
121C2×D12—22no—
122C2×C4×D3—41no—
123C2×Dic6—21no—
124C3×C4⋊C4—28no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—24no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4C2×D414yes—
129C3×C4⋊C4C2×D414yes—
130C3×C22⋊C4—28no—
131C3×C22⋊C4—22no—
132C3×C22⋊C4—24no—
133C3×C22⋊C4—24no—
134C3×C42—22no—
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D412yes—
137C3×C42C2411yescharacteristic
138#213—84no—
139(C2×D3)⋊C4—42no—
140C4⋊Dic3—22no—
141#208—82no—
142C2×C3⋊C8—22no—
143C2×C3⋊C8—24no—
144C2×C3⋊C8—42no—
145C2×#41—24no—
146C2×#41—22no—
147C2×#41—61no—
148C2×D8—64no—
149C2×M16—62no—
150C22×C8—61no—
151C4⋊Q8—31no—
152C4⋊Q8#6712yes—
153C4⋊Q8C22×D312yes—
154#127—22no—
155#127—31no—
156#127#6712yes—
157#125—22no—
158#125#6712yes—
159#124—22no—
160#124—61no—
161#123—28no—
162C22⋊Q8—28no—
163C22⋊Q8—22no—
164C22⋊Q8—24no—
165#121—28no—
166#121—28no—
167#121—24no—
168#121—22no—
169#121—62no—
170#121#6714yes—
171C4×Q8—22no—
172C4×D4—24no—
173C4×D4—22no—
174C4×D4—62no—
175C4×D4C22×D314yes—
176#117#6712yes—
177C2×C4⋊C4—24no—
178C2×C42C22×D311yescharacteristic
179#106—64no—
180C4⋊C8—34no—
181C4⋊C8—62no—
182#103—64no—
183#102—64no—
184#102—64no—
185#102—64no—
186C4×C8—32no—
187C4×C8—61no—
188C23×C3—24no—
189C23×C3#12011yescharacteristic
190C22×D3—41no—
191C3×Q8—42no—
192C3×Q8—42no—
193C3×Q8—82no—
194C3×D4—24no—
195C3×D4—24no—
196C3×D4—44no—
197C3×D4—82no—
198C3×D4—84no—
199C3×D4—88no—
200C2×C3×C4—22no—
201C2×C3×C4—22no—
202C2×C3×C4—22no—
203C2×C3×C4—24no—
204C2×C3×C4—24no—
205C2×C3×C4—44no—
206C2×C3×C4—44no—
207C2×C3×C4—88no—
208C2×C3×C4#12012yes—
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4C22×D411yescharacteristic
211C2×C3×C4C22×D412yes—
212#67—82no—
213C2×Dic3—41no—
214D12—42no—
215D12—42no—
216D12—42no—
217C4×D3—82no—
218Dic6—42no—
219C3⋊C8—42no—
220C3⋊C8—44no—
221#41—82no—
222#41—82no—
223#41—84no—
224#41—84no—
225#41—124no—
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—61no—
229C2×D4—28no—
230C2×D4—24no—
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—61no—
234C2×D4—62no—
235C2×D4—121no—
236C2×D4C2×#6714yes—
237C22×C4—28no—
238C22×C4—24no—
239C22×C4—121no—
240C22×C4C2×#6711yescharacteristic
241C22×C4C2×#6712yes—
242D8—244no—
243M16—242no—
244C2×C8—62no—
245C2×C8—64no—
246C2×C8—122no—
247C4⋊C4—28no—
248C4⋊C4—22no—
249C4⋊C4—24no—
250C4⋊C4—22no—
251C4⋊C4—62no—
252C4⋊C4C2×#6714yes—
253C4⋊C4C2×#6714yes—
254C22⋊C4—28no—
255C22⋊C4—22no—
256C22⋊C4—24no—
257C22⋊C4—24no—
258C22⋊C4—122no—
259C42—22no—
260C42C2×#6711yescharacteristic
261C42C2×#6712yes—
262C42C23×D311yescharacteristic
263C22×C3—22no—
264C22×C3—21no—
265C22×C3—44no—
266C22×C3—44no—
267C22×C3—44no—
268C22×C3C2×#12011yescharacteristic
269C2×D3—82no—
270C2×D3—81no—
271C3×C4—22no—
272C3×C4—21no—
273C3×C4—44no—
274C3×C4—44no—
275C3×C4#45412yes—
276C3×C4#57612yes—
277C3×C4#57712yes—
278Dic3—81no—
279C23—24no—
280C23—121no—
281C23#95311yescharacteristic
282Q8—42no—
283Q8—42no—
284Q8—82no—
285Q8—122no—
286D4—24no—
287D4—24no—
288D4—44no—
289D4—82no—
290D4—84no—
291D4—88no—
292D4—122no—
293D4—122no—
294D4—122no—
295D4—242no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—88no—
304C2×C4—121no—
305C2×C4—242no—
306C2×C4#95312yes—
307C2×C4#95311yescharacteristic
308C2×C4C22×#6711yescharacteristic
309C2×C4C22×#6712yes—
310C8—122no—
311C8—124no—
312C2×C3—41no—
313C2×C3—84no—
314C2×C3#301212yes—
315C2×C3C2×#45411yescharacteristic
316D3—161no—
317C22—22no—
318C22—21no—
319C22—44no—
320C22—44no—
321C22—44no—
322C22—242no—
323C22—241no—
324C22C2×#95311yescharacteristic
325C4—22no—
326C4—21no—
327C4—44no—
328C4—44no—
329C4—241no—
330C4#524512yes—
331C4#583512yes—
332C4#583612yes—
333C3#1869611yescharacteristic, complemented, Sylow
334C2—41no—
335C2—84no—
336C2—481no—
337C2#8410212yes—
338C2C2×#524511yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3548C2
socleC22×C3C2×#120
hypercenter#3548D3
hypocenterC3#18696
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3548

Complements

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

NH ≅ G/Ntype
1. C3×#3548C2semidirect
2. #83334C2semidirect
3. #83313C2semidirect
4. #71911C2semidirect
5. #71910C2semidirect
6. #71834C2semidirect
7. #71802C2semidirect
31. #3548D3semidirect
333. C3#18696semidirect

Maximal subgroups

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

subgroupindex
C3×#35482
#833342
#833132
#719112
#719102
#718342
#718022
#186963