Finite Group Databasebeta

← #10693037

Subgroups — #10693037

Order 768.

626 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 135 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×#2535C211yesmaximal, characteristic
2#82301C211yesmaximal, characteristic, complemented
3#71617C211yesmaximal, characteristic, complemented
4#71598C212yesmaximal, complemented
5#67631C212yesmaximal, complemented
6#9485—31nomaximal, Sylow
7C2×C3×#121—21no—
8C2×C3×C4×D4C2211yescharacteristic
9C2×C3×C4×D4C2211yescharacteristic
10C3×#397C412yes—
11C3×#392C412yes—
12C3×#386—22no—
13C3×#382C2211yescharacteristic
14C3×#380—21no—
15C2×#833—21no—
16C2×#830—22no—
17D4×C3⋊C8—24no—
18#4925C2212yes—
19#4924C2212yes—
20D4⋊(C3⋊C8)—24no—
21#4530—21no—
22#2972—31no—
23#2535D311yescharacteristic
24#1800—31no—
25#1486—32no—
26#1286—32no—
27C22×C3×D4—22no—
28C22×C3×D4C2×C411yescharacteristic
29C22×C3×D4D411yescharacteristic
30C23×C3×C4—21no—
31C23×C3×C4D412yes—
32C3×#121—42no—
33C3×#121—42no—
34C3×C4×D4—22no—
35C3×C4×D4—22no—
36C3×C4×D4—22no—
37C3×C4×D4—22no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4C2×C411yescharacteristic
40C2×C3×C4⋊C4D411yescharacteristic
41C2×C3×C22⋊C4—22no—
42C2×C3×C22⋊C4—21no—
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4C2×C412yes—
45C2×C3×C42C2311yescharacteristic
46C2×#208—22no—
47C22×C3⋊C8—24no—
48C3×#95—22no—
49C3×#95—21no—
50C3×#95C2×C412yes—
51#833—42no—
52#830—24no—
53#830—22no—
54C4⋊(C3⋊C8)—22no—
55C4⋊(C3⋊C8)—22no—
56C4⋊(C3⋊C8)—22no—
57C4×C3⋊C8D414yescomplemented
58C2×#121—21no—
59C2×C4×D4C2×D311yescharacteristic
60C2×C4×D4C2×D311yescharacteristic
61C8×D4—64no—
62#431—32no—
63#423—32no—
64C2×#100—61no—
65C2×C22⋊C8—62no—
66#397Dic312yes—
67#392Dic312yes—
68#386—22no—
69#382C2×D311yescharacteristic
70#380—21no—
71#343—61no—
72D4⋊C8—64no—
73C24×C3—21no—
74C24×C3C22⋊C412yes—
75C2×C3×D4—22no—
76C2×C3×D4—22no—
77C2×C3×D4—22no—
78C2×C3×D4—42no—
79C2×C3×D4—44no—
80C2×C3×D4—44no—
81C2×C3×D4—82no—
82C2×C3×D4D812yes—
83C2×C3×D4SD1612yes—
84C22×C3×C4—22no—
85C22×C3×C4—22no—
86C22×C3×C4—22no—
87C22×C3×C4—21no—
88C22×C3×C4—42no—
89C22×C3×C4—41no—
90C22×C3×C4—41no—
91C22×C3×C4—42no—
92C22×C3×C4—82no—
93C22×C3×C4C22⋊C412yes—
94C22×C3×C4C22×C411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C3×C4⋊C4—22no—
98C3×C4⋊C4—22no—
99C3×C4⋊C4—41no—
100C3×C4⋊C4—41no—
101C3×C22⋊C4—24no—
102C3×C22⋊C4—44no—
103C3×C22⋊C4—41no—
104C3×C22⋊C4—41no—
105C3×C22⋊C4—82no—
106C3×C42C2×D412yes—
107C3×C42#4112yes—
108#208—42no—
109#208—42no—
110C2×C3⋊C8—24no—
111C2×C3⋊C8—42no—
112C2×C3⋊C8—84no—
113C22×D4—22no—
114C22×D4C2×Dic311yescharacteristic
115C22×D4#6711yescharacteristic
116C23×C4—21no—
117C23×C4#6712yes—
118C2×M16—62no—
119C22×C8—64no—
120#121—42no—
121#121—42no—
122C4×D4—22no—
123C4×D4—22no—
124C4×D4—22no—
125C4×D4—22no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4C2×Dic311yescharacteristic
128C2×C4⋊C4#6711yescharacteristic
129C2×C22⋊C4—22no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4C2×Dic312yes—
133C2×C42C22×D311yescharacteristic
134C4⋊C8—62no—
135C4⋊C8—62no—
136C4⋊C8—62no—
137#100—122no—
138C22⋊C8—64no—
139C22⋊C8—62no—
140C4×C8—34no—
141#95—22no—
142#95—21no—
143#95C2×Dic312yes—
144C23×C3—22no—
145C23×C3—22no—
146C23×C3—22no—
147C23×C3—42no—
148C23×C3—42no—
149C23×C3—41no—
150C23×C3—41no—
151C23×C3—41no—
152C23×C3—41no—
153C23×C3—41no—
154C23×C3—41no—
155C23×C3#10214yes—
156C23×C3C2×C22⋊C411yescharacteristic
157C3×D4—24no—
158C3×D4—24no—
159C3×D4—24no—
160C3×D4—42no—
161C3×D4—88no—
162C2×C3×C4—22no—
163C2×C3×C4—41no—
164C2×C3×C4—41no—
165C2×C3×C4—44no—
166C2×C3×C4—42no—
167C2×C3×C4—42no—
168C2×C3×C4—42no—
169C2×C3×C4—41no—
170C2×C3×C4—41no—
171C2×C3×C4—42no—
172C2×C3×C4—41no—
173C2×C3×C4—82no—
174C2×C3×C4—82no—
175C2×C3×C4—82no—
176C2×C3×C4C4×D412yes—
177C2×C3×C4#12011yescharacteristic
178C2×C3×C4#12112yes—
179C2×C3×C4#12311yescharacteristic
180C2×C3×C4C2×D811yescharacteristic
181C2×C3×C4C2×SD1611yescharacteristic
182C2×C3×C4#13512yes—
183C3⋊C8—44no—
184C3⋊C8—82no—
185C24—21no—
186C24#21712yes—
187C2×D4—22no—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—42no—
191C2×D4—44no—
192C2×D4—44no—
193C2×D4—82no—
194C2×D4#21312yes—
195C2×D4#21412yes—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—82no—
205C22×C4#21712yes—
206C22×C4C22×Dic311yescharacteristic
207C22×C4C2×#6711yescharacteristic
208C22×C4C2×#6711yescharacteristic
209M16—122no—
210M16—122no—
211C2×C8—64no—
212C2×C8—122no—
213C2×C8—244no—
214C4⋊C4—22no—
215C4⋊C4—22no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C22⋊C4—24no—
219C22⋊C4—44no—
220C22⋊C4—41no—
221C22⋊C4—41no—
222C22⋊C4—82no—
223C42D3×D412yes—
224C42#23712yes—
225C22×C3—21no—
226C22×C3—21no—
227C22×C3—24no—
228C22×C3—44no—
229C22×C3—42no—
230C22×C3—42no—
231C22×C3—42no—
232C22×C3—42no—
233C22×C3—42no—
234C22×C3—82no—
235C22×C3—81no—
236C22×C3—81no—
237C22×C3—81no—
238C22×C3—81no—
239C22×C3#38611yescharacteristic
240C22×C3C2×#10211yescharacteristic
241C22×C3#41611yescharacteristic
242C3×C4—22no—
243C3×C4—21no—
244C3×C4—21no—
245C3×C4—42no—
246C3×C4—82no—
247C3×C4—81no—
248C3×C4#46512yes—
249C3×C4#46612yes—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—42no—
254C23—42no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23#83214yes—
262C23C2×#21711yescharacteristic
263D4—24no—
264D4—24no—
265D4—24no—
266D4—42no—
267D4—88no—
268C2×C4—22no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—44no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4C2×#21311yescharacteristic
283C2×C4C2×#21411yescharacteristic
284C2×C4D4×Dic312yes—
285C2×C4#93511yescharacteristic
286C2×C4#93711yescharacteristic
287C2×C4#93912yes—
288C2×C4#95012yes—
289C8—124no—
290C8—242no—
291C2×C3—21no—
292C2×C3—21no—
293C2×C3—44no—
294C2×C3—82no—
295C2×C3#189111yescharacteristic
296C2×C3#189311yescharacteristic
297C2×C3#191111yescharacteristic
298C22—21no—
299C22—21no—
300C22—24no—
301C22—44no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—42no—
306C22—42no—
307C22—82no—
308C22—81no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22C2×#83211yescharacteristic
313C22#522011yescharacteristic
314C22#523311yescharacteristic
315C4—22no—
316C4—21no—
317C4—21no—
318C4—42no—
319C4—82no—
320C4—81no—
321C4#503812yes—
322C4#504112yes—
323C3#948511yescharacteristic, complemented, Sylow
324C2—21no—
325C2—21no—
326C2—44no—
327C2—82no—
328C2#7168111yescharacteristic
329C2#7171411yescharacteristic
330C2#7185311yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#2535C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#2535D3
hypocenterC3#9485
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2535

Complements

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

NH ≅ G/Ntype
2. #82301C2semidirect
3. #71617C2semidirect
4. #71598C2semidirect
5. #67631C2semidirect
57. C4×C3⋊C8D4semidirect
323. C3#9485semidirect

Maximal subgroups

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

subgroupindex
C3×#25352
#823012
#716172
#715982
#676312
#94853