Finite Group Databasebeta

← #10694210

Subgroups — #10694210

Order 768.

596 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2858C211yesmaximal, characteristic
2C2×#4540C214yesmaximal, complemented
3#69391C211yesmaximal, characteristic, complemented
4#69389C211yesmaximal, characteristic, complemented
5#9689—31nomaximal, Sylow
6C2×C3×C4⋊Q8C411yescharacteristic
7C2×C3×#127C411yescharacteristic
8C2×C3×#124C412yes—
9C2×C3×#124C2212yes—
10C3×#390—21no—
11C3×#390—21no—
12C3×#390—21no—
13C3×#390—21no—
14C3×C43C2211yescharacteristic
15C2×C4⋊(C3⋊C8)—21no—
16C2×#803C2212yes—
17C2×#803C2212yes—
18#4540—28no—
19#4524—21no—
20#4524—21no—
21#4524—21no—
22#2858D311yescharacteristic
23#1767—31no—
24#1745—31no—
25C2×#329—34no—
26C22×C3×Q8C2×C412yes—
27C22×C3×D4—21no—
28C22×C3×D4—21no—
29C22×C3×D4C2×C412yes—
30C3×C4⋊Q8—42no—
31C3×#127—42no—
32C3×#124—24no—
33C3×#124—24no—
34C3×#124—44no—
35C2×C3×C4⋊C4—21no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C22⋊C4—22no—
38C2×C3×C22⋊C4—22no—
39C2×C3×C22⋊C4—22no—
40C2×C3×C22⋊C4—22no—
41C2×C3×C42C2×C411yescharacteristic
42C2×C3×C42C2×C411yescharacteristic
43C2×C3×C42D411yescharacteristic
44C2×C3×C42D411yescharacteristic
45C2×C3×C42D411yescharacteristic
46C2×C3×C42D411yescharacteristic
47C2×C3×C42C2311yescharacteristic
48C22×C3⋊C8—22no—
49C22×C3⋊C8—22no—
50C4⋊(C3⋊C8)—24no—
51#803—22no—
52#803—24no—
53#803D414yescomplemented
54C2×C4⋊Q8Dic311yescharacteristic
55C2×#127Dic311yescharacteristic
56C2×#124Dic312yes—
57C2×#124C2×D312yes—
58C2×C4⋊C8—61no—
59C2×#97—32no—
60C2×#97—32no—
61#390—21no—
62#390—21no—
63#390—21no—
64#390—21no—
65C43C2×D311yescharacteristic
66#336—61no—
67#336—61no—
68#336—61no—
69#329—68no—
70C24×C3—22no—
71C2×C3×Q8—24no—
72C2×C3×Q8—42no—
73C2×C3×Q8—42no—
74C2×C3×D4—24no—
75C2×C3×D4—42no—
76C2×C3×D4—42no—
77C2×C3×D4—42no—
78C2×C3×D4—42no—
79C2×C3×D4—42no—
80C2×C3×D4—42no—
81C2×C3×D4—42no—
82C2×C3×D4—42no—
83C22×C3×C4—22no—
84C22×C3×C4C22⋊C411yescharacteristic
85C22×C3×C4C22⋊C411yescharacteristic
86C22×C3×C4C22⋊C411yescharacteristic
87C22×C3×C4C22⋊C411yescharacteristic
88C22×C3×C4C22×C411yescharacteristic
89C22×C3×C4C2×D411yescharacteristic
90C22×C3×C4C2×D411yescharacteristic
91C3×C4⋊C4—42no—
92C3×C4⋊C4—42no—
93C3×C22⋊C4—44no—
94C3×C22⋊C4—44no—
95C3×C22⋊C4—44no—
96C3×C22⋊C4—42no—
97C3×C22⋊C4—42no—
98C3×C42—22no—
99C3×C42—22no—
100C3×C42—22no—
101C3×C42—41no—
102C3×C42—41no—
103C3×C42—41no—
104C3×C42C2×D412yes—
105C3×C42#4112yes—
106C2×C3⋊C8—24no—
107C2×C3⋊C8—24no—
108C2×C3⋊C8—42no—
109C2×C3⋊C8—42no—
110C2×C3⋊C8—42no—
111C2×C3⋊C8—42no—
112C22×Q8C2×Dic312yes—
113C22×D4—21no—
114C22×D4—21no—
115C22×D4C2×Dic312yes—
116C22×C8—62no—
117C22×C8—62no—
118C4⋊Q8—42no—
119#127—42no—
120#124—24no—
121#124—24no—
122#124—44no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—22no—
129C2×C42C2×Dic311yescharacteristic
130C2×C42C2×Dic311yescharacteristic
131C2×C42#6711yescharacteristic
132C2×C42#6711yescharacteristic
133C2×C42#6711yescharacteristic
134C2×C42#6711yescharacteristic
135C2×C42C22×D311yescharacteristic
136C4⋊C8—64no—
137#97—34no—
138#97—64no—
139#97—62no—
140C23×C3—42no—
141C23×C3—42no—
142C23×C3—42no—
143C23×C3—42no—
144C23×C3—42no—
145C23×C3—42no—
146C23×C3—42no—
147C23×C3C2×C22⋊C411yescharacteristic
148C3×Q8—84no—
149C3×D4—84no—
150C3×D4—84no—
151C3×D4—84no—
152C2×C3×C4—21no—
153C2×C3×C4—21no—
154C2×C3×C4—21no—
155C2×C3×C4—21no—
156C2×C3×C4—21no—
157C2×C3×C4—21no—
158C2×C3×C4—21no—
159C2×C3×C4—21no—
160C2×C3×C4—21no—
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4—42no—
168C2×C3×C4—42no—
169C2×C3×C4—42no—
170C2×C3×C4#10012yes—
171C2×C3×C4#10012yes—
172C2×C3×C4#10412yes—
173C2×C3×C4C4×D412yes—
174C2×C3×C4#12112yes—
175C2×C3×C4#12411yescharacteristic
176C2×C3×C4#12711yescharacteristic
177C3⋊C8—44no—
178C3⋊C8—44no—
179C24—22no—
180C2×Q8—24no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×D4—24no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—42no—
192C22×C4—22no—
193C22×C4#21711yescharacteristic
194C22×C4#21711yescharacteristic
195C22×C4#21711yescharacteristic
196C22×C4#21711yescharacteristic
197C22×C4C22×Dic311yescharacteristic
198C22×C4C2×#6711yescharacteristic
199C22×C4C2×#6711yescharacteristic
200C2×C8—64no—
201C2×C8—64no—
202C2×C8—122no—
203C2×C8—122no—
204C2×C8—122no—
205C2×C8—122no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C22⋊C4—44no—
209C22⋊C4—44no—
210C22⋊C4—42no—
211C22⋊C4—44no—
212C22⋊C4—42no—
213C42—22no—
214C42—22no—
215C42—22no—
216C42—41no—
217C42—41no—
218C42—41no—
219C42D3×D412yes—
220C42#23712yes—
221C22×C3—82no—
222C22×C3—82no—
223C22×C3—82no—
224C22×C3—82no—
225C22×C3—82no—
226C22×C3—82no—
227C22×C3—82no—
228C22×C3#39011yescharacteristic
229C22×C3C2×#10011yescharacteristic
230C22×C3C2×#10011yescharacteristic
231C22×C3C2×#10411yescharacteristic
232C22×C3#42111yescharacteristic
233C22×C3#42111yescharacteristic
234C22×C3#42111yescharacteristic
235C3×C4—22no—
236C3×C4—22no—
237C3×C4—21no—
238C3×C4—21no—
239C3×C4—22no—
240C3×C4—22no—
241C3×C4—22no—
242C3×C4—41no—
243C3×C4—82no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23C2×#21711yescharacteristic
252Q8—84no—
253D4—84no—
254D4—84no—
255D4—84no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4#83312yes—
275C2×C4#83312yes—
276C2×C4#83712yes—
277C2×C4D4×Dic312yes—
278C2×C4#93611yescharacteristic
279C2×C4#93812yes—
280C2×C4#94011yescharacteristic
281C8—124no—
282C8—124no—
283C2×C3—162no—
284C2×C3#152711yescharacteristic
285C2×C3#153011yescharacteristic
286C2×C3#153311yescharacteristic
287C2×C3#153611yescharacteristic
288C2×C3#196311yescharacteristic
289C2×C3#196511yescharacteristic
290C2×C3#198011yescharacteristic
291C22—82no—
292C22—82no—
293C22—82no—
294C22—82no—
295C22—82no—
296C22—82no—
297C22—82no—
298C22C2×#83311yescharacteristic
299C22C2×#83311yescharacteristic
300C22#522111yescharacteristic
301C22C2×#83711yescharacteristic
302C22#523611yescharacteristic
303C22#523611yescharacteristic
304C22#523611yescharacteristic
305C4—22no—
306C4—22no—
307C4—22no—
308C4—22no—
309C4—22no—
310C4—21no—
311C4—21no—
312C4—41no—
313C4—82no—
314C3#968911yescharacteristic, complemented, Sylow
315C2—162no—
316C2#7171811yescharacteristic
317C2#7171911yescharacteristic
318C2#7176611yescharacteristic
319C2#7179811yescharacteristic
320C2#7179911yescharacteristic
321C2#7181611yescharacteristic
322C2#7181811yescharacteristic
subgroupG/H
centerC23C2×#217
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2858C2
socleC23×C3C2×C22⋊C4
prime radicalC2×C3×#127C4
cosocleC2×C3×C42C23
hypercenter#2858D3
hypocenterC3#9689
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C43 ◁ #2858

Complements

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

NH ≅ G/Ntype
2. C2×#4540C2semidirect
3. #69391C2semidirect
4. #69389C2semidirect
53. #803D4semidirect
314. C3#9689semidirect

Maximal subgroups

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

subgroupindex
C3×#28582
C2×#45402
#693912
#693892
#96893