Finite Group Databasebeta

← #10691210

Subgroups — #10691210

Order 768.

524 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2905C211yesmaximal, characteristic
2#82262C211yesmaximal, characteristic
3#71571C211yesmaximal, characteristic
4#70506C212yesmaximal, complemented
5#67713C212yesmaximal, complemented
6#9195—31nomaximal, Sylow
7C22×C3×M16—21no—
8C2×C3×#117C2211yescharacteristic
9C2×#924—21no—
10C2×#869C2211yescharacteristic
11C3×#424—24no—
12C3×#423—21no—
13C3×#423C2212yes—
14#4981—21no—
15#4981C412yes—
16#4953—21no—
17#4953C412yes—
18#4936C2211yescharacteristic
19#4929—21no—
20#4925C2212yes—
21#4528—21no—
22#4525—21no—
23#4482—24no—
24#2905D311yescharacteristic
25#2621—31no—
26#1546—32no—
27#1458—31no—
28#1286—32no—
29C23×C3×C4D411yescharacteristic
30C22×Dic6—21no—
31C2×C3×M16—22no—
32C2×C3×M16—22no—
33C2×C3×M16—42no—
34C2×C3×M16—42no—
35C22×C3×C8—22no—
36C3×#117—22no—
37C3×#117—22no—
38C2×C3×C4⋊C4C2×C411yescharacteristic
39C2×C3×C4⋊C4D411yescharacteristic
40C2×C3×C22⋊C4C2×C411yescharacteristic
41C2×C3×C22⋊C4D411yescharacteristic
42C2×C3×C42D411yescharacteristic
43C2×C3×C42C2311yescharacteristic
44C2×#217—22no—
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3C2×C412yes—
48#924—42no—
49#924—42no—
50C2×Dic3⋊C4—22no—
51#869—22no—
52#869—22no—
53C3×C4⋊C8—24no—
54C3×C4⋊C8—22no—
55C3×C4⋊C8D414yes—
56C3⋊#95—22no—
57C3⋊#95—22no—
58C3⋊#95—21no—
59C3⋊#95C2×C412yes—
60#830—22no—
61C4⋊(C3⋊C8)—22no—
62C4×C3⋊C8—22no—
63C22×M16—21no—
64C2×C4⋊Q8—31no—
65C2×C22⋊Q8—61no—
66C2×#117C2×D311yescharacteristic
67#431—32no—
68#424—24no—
69#423—21no—
70#423C2×D312yes—
71#398—32no—
72#398—61no—
73#397—32no—
74#397—61no—
75#391—61no—
76#382—31no—
77#332—64no—
78#324—61no—
79#323—61no—
80C24×C3C22⋊C411yescharacteristic
81C22×C3×C4—21no—
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—21no—
85C22×C3×C4C22⋊C411yescharacteristic
86C22×C3×C4C22⋊C411yescharacteristic
87C22×C3×C4C22⋊C411yescharacteristic
88C22×C3×C4D812yes—
89C22×C3×C4SD1612yes—
90C22×C3×C4C22×C411yescharacteristic
91C22×C3×C4C2×D411yescharacteristic
92C22×C3×C4C2×D411yescharacteristic
93C22×Dic3—22no—
94C22×Dic3—22no—
95C2×Dic6—41no—
96C2×Dic6—41no—
97C2×Dic6—41no—
98C2×Dic6—41no—
99C2×Dic6—41no—
100C2×Dic6—41no—
101C3×M16—44no—
102C3×M16—42no—
103C3×M16—42no—
104C2×C3×C8—24no—
105C2×C3×C8—44no—
106C3×C4⋊C4—22no—
107C3×C4⋊C4—22no—
108C3×C22⋊C4—22no—
109C3×C22⋊C4—22no—
110C3×C42—21no—
111C3×C42—21no—
112C3×C42C2×D412yes—
113C3×C42#4112yes—
114#217—44no—
115C4⋊Dic3—24no—
116C4⋊Dic3—42no—
117C4⋊Dic3—41no—
118C4⋊Dic3—41no—
119Dic3⋊C4—44no—
120C2×C3⋊C8—42no—
121C22×Q8—61no—
122C23×C4#6711yescharacteristic
123C2×M16—22no—
124C2×M16—22no—
125C2×M16—42no—
126C2×M16—42no—
127C22×C8—22no—
128C4⋊Q8—62no—
129C4⋊Q8—62no—
130C22⋊Q8—122no—
131C22⋊Q8—122no—
132#117—22no—
133#117—22no—
134C2×C4⋊C4—32no—
135C2×C4⋊C4—62no—
136C2×C4⋊C4—61no—
137C2×C4⋊C4—61no—
138C2×C4⋊C4C4×D311yescharacteristic
139C2×C4⋊C4D1211yescharacteristic
140C2×C22⋊C4—62no—
141C2×C22⋊C4C4×D311yescharacteristic
142C2×C22⋊C4D1211yescharacteristic
143C2×C42#6711yescharacteristic
144C2×C42C22×D311yescharacteristic
145C4⋊C8—24no—
146C4⋊C8—22no—
147C4⋊C8—62no—
148C4⋊C8#6714yes—
149C22⋊C8—62no—
150C4×C8—62no—
151#95—32no—
152#95—62no—
153#95—62no—
154#95—61no—
155C23×C3—21no—
156C23×C3—21no—
157C23×C3—41no—
158C23×C3—41no—
159C23×C3#10212yes—
160C23×C3C2×C22⋊C411yescharacteristic
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4—22no—
164C2×C3×C4—22no—
165C2×C3×C4—22no—
166C2×C3×C4—22no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—41no—
170C2×C3×C4—41no—
171C2×C3×C4—41no—
172C2×C3×C4—41no—
173C2×C3×C4#10212yes—
174C2×C3×C4C4×D412yes—
175C2×C3×C4#12011yescharacteristic
176C2×C3×C4#12112yes—
177C2×C3×C4#12311yescharacteristic
178C2×C3×C4C2×D811yescharacteristic
179C2×C3×C4C2×SD1611yescharacteristic
180C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
181C2×C3×C4#13711yescharacteristic
182C2×Dic3—42no—
183C2×Dic3—42no—
184C2×Dic3—42no—
185C2×Dic3—42no—
186C2×Dic3—42no—
187C2×Dic3—42no—
188Dic6—81no—
189Dic6—81no—
190Dic6—81no—
191Dic6—81no—
192C3×C8—44no—
193C3⋊C8—82no—
194C24(C2×D3)⋊C411yescharacteristic
195C2×Q8—121no—
196C2×Q8—121no—
197C2×Q8—121no—
198C2×Q8—121no—
199C2×Q8—121no—
200C2×Q8—121no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—62no—
206C22×C4—62no—
207C22×C4(C2×D3)⋊C411yescharacteristic
208C22×C4(C2×D3)⋊C411yescharacteristic
209C22×C4(C2×D3)⋊C411yescharacteristic
210C22×C4#21312yes—
211C22×C4#21512yes—
212C22×C4C2×C4×D311yescharacteristic
213C22×C4C2×D1211yescharacteristic
214C22×C4C2×#6711yescharacteristic
215M16—44no—
216M16—42no—
217M16—42no—
218C2×C8—24no—
219C2×C8—44no—
220C2×C8—122no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—64no—
224C4⋊C4—124no—
225C4⋊C4—122no—
226C4⋊C4—121no—
227C4⋊C4—121no—
228C22⋊C4—22no—
229C22⋊C4—22no—
230C22⋊C4—124no—
231C42—21no—
232C42—21no—
233C42#23512yes—
234C42C2×#6712yes—
235C22×C3—21no—
236C22×C3—21no—
237C22×C3—22no—
238C22×C3—42no—
239C22×C3—41no—
240C22×C3—41no—
241C22×C3—81no—
242C22×C3#38611yescharacteristic
243C22×C3C2×#10211yescharacteristic
244C22×C3#41711yescharacteristic
245C3×C4—22no—
246C3×C4—22no—
247C3×C4—22no—
248C3×C4—41no—
249C3×C4—41no—
250C3×C4#44711yescharacteristic
251C3×C4#44811yescharacteristic
252C3×C4#48011yescharacteristic
253C3×C4#48111yescharacteristic
254Dic3—82no—
255Dic3—82no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23#80912yes—
261C23C2×(C2×D3)⋊C411yescharacteristic
262Q8—241no—
263Q8—241no—
264Q8—241no—
265Q8—241no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—122no—
279C2×C4—122no—
280C2×C4—122no—
281C2×C4—122no—
282C2×C4—122no—
283C2×C4—122no—
284C2×C4#80912yes—
285C2×C4C4×#6712yes—
286C2×C4#92911yescharacteristic
287C2×C4#93012yes—
288C2×C4C2×#21311yescharacteristic
289C2×C4#93211yescharacteristic
290C2×C4C2×#21511yescharacteristic
291C2×C4#94211yescharacteristic
292C2×C4#95311yescharacteristic
293C8—44no—
294C8—242no—
295C2×C3—21no—
296C2×C3—21no—
297C2×C3—42no—
298C2×C3#187411yescharacteristic
299C2×C3#190911yescharacteristic
300C2×C3#191511yescharacteristic
301C22—21no—
302C22—21no—
303C22—22no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22—81no—
308C22C2×#80911yescharacteristic
309C22#496611yescharacteristic
310C22#521311yescharacteristic
311C4—22no—
312C4—22no—
313C4—22no—
314C4—41no—
315C4—41no—
316C4—242no—
317C4—242no—
318C4#496711yescharacteristic
319C4#496811yescharacteristic
320C4#521711yescharacteristic
321C4#522711yescharacteristic
322C3#919511yescharacteristic, complemented, Sylow
323C2—21no—
324C2—21no—
325C2—42no—
326C2#7156011yescharacteristic
327C2#7202411yescharacteristic
328C2#7203111yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2905C2
socleC22×C3#386
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#2905D3
hypocenterC3#9195
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2905

Complements

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

NH ≅ G/Ntype
4. #70506C2semidirect
5. #67713C2semidirect
322. C3#9195semidirect

Maximal subgroups

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

subgroupindex
C3×#29052
#822622
#715712
#705062
#677132
#91953