Finite Group Databasebeta

← #10698047

Subgroups — #10698047

Order 768.

726 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 103 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×#3186C211yesmaximal, characteristic
2#82699C212yesmaximal, complemented
3#71675C212yesmaximal, complemented
4#69910C211yesmaximal, characteristic, complemented
5#67676C211yesmaximal, characteristic, complemented
6#10321—31nomaximal, Sylow
7C2×C3×#121C2212yes—
8C2×#935—22no—
9C22×C4×Dic3C2211yescharacteristic
10C3×#480—22no—
11C3×#466—22no—
12C3×#447—22no—
13C2×C3×C22⋊C8C2211yescharacteristic
14#5221—22no—
15#5220—22no—
16#5212C2212yes—
17C2×#830C2211yescharacteristic
18#4972—22no—
19#4944—24no—
20#4550—22no—
21#4537—24no—
22#3186D311yescharacteristic
23#2854—32no—
24#1817—31no—
25#1519—32no—
26#1289—31no—
27C22×C3×D4—22no—
28C22×C3×D4D412yes—
29C23×C3×C4C2311yescharacteristic
30C23×Dic3D411yescharacteristic
31C23×Dic3D411yescharacteristic
32C2×C3×D8—22no—
33C22×C3×C8D412yes—
34C3×#121—22no—
35C3×#121—22no—
36C3×#121—22no—
37C3×#121—22no—
38C2×C3×C4⋊C4D412yes—
39C2×C3×C22⋊C4—22no—
40C2×#217—24no—
41C2×#217—24no—
42C2×#217—22no—
43#935—44no—
44#935—44no—
45C2×C4⋊Dic3—22no—
46C2×Dic3⋊C4—24no—
47C2×C4×Dic3—22no—
48C2×C4×Dic3—22no—
49C2×C4×Dic3—21no—
50C2×C4×Dic3—21no—
51C22×C3⋊C8—21no—
52C3×#107—22no—
53C3×#102—24no—
54C3×C22⋊C8—22no—
55C3⋊#95—24no—
56C3⋊#95D414yes—
57#830—22no—
58C2×#123—62no—
59C2×#121C2×D312yes—
60C22×C42—31no—
61#480—22no—
62#466—22no—
63#447—22no—
64C2×C22⋊C8—31no—
65C2×C22⋊C8C2×D311yescharacteristic
66#390—62no—
67#389—62no—
68#388—64no—
69#386—62no—
70#380—32no—
71#336—62no—
72#327—64no—
73C24×C3—22no—
74C24×C3C2×D411yescharacteristic
75C2×C3×D4—22no—
76C2×C3×D4—22no—
77C2×C3×D4—44no—
78C2×C3×D4—42no—
79C2×C3×D4—42no—
80C2×C3×D4—42no—
81C2×C3×D4—42no—
82C2×C3×D4—42no—
83C2×C3×D4—42no—
84C22×C3×C4—22no—
85C22×C3×C4—42no—
86C22×C3×C4C2×D412yes—
87C22×C3×C4C2×D412yes—
88C22×C3×C4C2×D411yescharacteristic
89C22×C3×C4#4111yescharacteristic
90C22×Dic3—22no—
91C22×Dic3—22no—
92C22×Dic3—21no—
93C22×Dic3—21no—
94C22×Dic3—21no—
95C22×Dic3—22no—
96C22×Dic3—24no—
97C22×Dic3—42no—
98C22×Dic3D812yescomplemented
99C3×D8—82no—
100C2×C3×C8—22no—
101C2×C3×C8—42no—
102C3×C4⋊C4—22no—
103C3×C4⋊C4—22no—
104C3×C22⋊C4—42no—
105C3×C22⋊C4—42no—
106#217—48no—
107#217—48no—
108#217—42no—
109#217—42no—
110C4⋊Dic3—44no—
111Dic3⋊C4—48no—
112C4×Dic3—41no—
113C4×Dic3—42no—
114C4×Dic3—41no—
115C2×C3⋊C8—42no—
116C2×C3⋊C8—41no—
117C22×D4—22no—
118C22×D4#6712yes—
119C23×C4—31no—
120C23×C4—31no—
121C23×C4C22×D311yescharacteristic
122C2×D8—22no—
123C22×C8—61no—
124C22×C8#6712yes—
125#123—124no—
126#123—124no—
127#121—22no—
128#121—22no—
129#121—22no—
130#121—22no—
131C2×C4⋊C4—64no—
132C2×C4⋊C4—62no—
133C2×C4⋊C4#6712yes—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—64no—
136C2×C22⋊C4—64no—
137C2×C22⋊C4—62no—
138C2×C42—62no—
139C2×C42—61no—
140C2×C42—61no—
141C2×C42—62no—
142#107—22no—
143#102—24no—
144C22⋊C8—22no—
145C22⋊C8—62no—
146#95—34no—
147#95—64no—
148C23×C3—41no—
149C23×C3—41no—
150C23×C3—44no—
151C23×C3—44no—
152C23×C3—42no—
153C23×C3—42no—
154C23×C3—42no—
155C23×C3#12012yes—
156C23×C3#12111yescharacteristic
157C23×C3C2×D811yescharacteristic
158C23×C3#13511yescharacteristic
159C23×C3AGL(1,ℤ/8)12yes—
160C3×D4—44no—
161C3×D4—82no—
162C3×D4—82no—
163C3×D4—82no—
164C3×D4—82no—
165C3×D4—82no—
166C2×C3×C4—22no—
167C2×C3×C4—24no—
168C2×C3×C4—42no—
169C2×C3×C4—41no—
170C2×C3×C4—42no—
171C2×C3×C4—42no—
172C2×C3×C4—42no—
173C2×C3×C4#12011yescharacteristic
174C2×C3×C4#12112yes—
175C2×C3×C4#12711yescharacteristic
176C2×Dic3—24no—
177C2×Dic3—22no—
178C2×Dic3—24no—
179C2×Dic3—22no—
180C2×Dic3—42no—
181C2×Dic3—41no—
182C2×Dic3—41no—
183C2×Dic3—41no—
184C2×Dic3—42no—
185C2×Dic3—41no—
186C2×Dic3—44no—
187C2×Dic3—44no—
188C2×Dic3—44no—
189C3×C8—42no—
190C3⋊C8—81no—
191C24—22no—
192C24D3×D411yescharacteristic
193C2×D4—22no—
194C2×D4—22no—
195C2×D4—44no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—42no—
202C22×C4—22no—
203C22×C4—32no—
204C22×C4—42no—
205C22×C4—62no—
206C22×C4—61no—
207C22×C4—62no—
208C22×C4—62no—
209C22×C4—61no—
210C22×C4—61no—
211C22×C4—64no—
212C22×C4—122no—
213C22×C4D3×D412yes—
214C22×C4#23711yescharacteristic
215C22×C4C2×#6712yes—
216C22×C4C2×#6711yescharacteristic
217D8—82no—
218C2×C8—22no—
219C2×C8—42no—
220C2×C8—122no—
221C2×C8—121no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—124no—
225C4⋊C4—128no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—128no—
229C22⋊C4—128no—
230C22⋊C4—122no—
231C22⋊C4—122no—
232C42—122no—
233C42—121no—
234C42—121no—
235C22×C3—22no—
236C22×C3—21no—
237C22×C3—22no—
238C22×C3—21no—
239C22×C3—41no—
240C22×C3—41no—
241C22×C3—42no—
242C22×C3—84no—
243C22×C3—84no—
244C22×C3—82no—
245C22×C3—82no—
246C22×C3—82no—
247C22×C3#39211yescharacteristic
248C22×C3#44712yes—
249C22×C3#44912yes—
250C22×C3#46611yescharacteristic
251C22×C3#46911yescharacteristic
252C3×C4—21no—
253C3×C4—22no—
254C3×C4—82no—
255C3×C4#45812yes—
256Dic3—42no—
257Dic3—42no—
258Dic3—84no—
259C23—41no—
260C23—41no—
261C23—44no—
262C23—44no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23D3×D811yescharacteristic
267C23#91112yes—
268C23#91211yescharacteristic
269C23#93712yes—
270C23#93811yescharacteristic
271D4—44no—
272D4—82no—
273D4—82no—
274D4—82no—
275D4—82no—
276D4—82no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—64no—
285C2×C4—62no—
286C2×C4—64no—
287C2×C4—62no—
288C2×C4—122no—
289C2×C4—121no—
290C2×C4—122no—
291C2×C4—121no—
292C2×C4—121no—
293C2×C4—121no—
294C2×C4—124no—
295C2×C4—124no—
296C2×C4—124no—
297C2×C4#93912yes—
298C2×C4#94011yescharacteristic
299C2×C4#95311yescharacteristic
300C8—42no—
301C8—241no—
302C2×C3—22no—
303C2×C3—22no—
304C2×C3—21no—
305C2×C3—21no—
306C2×C3—162no—
307C2×C3#199911yescharacteristic
308C2×C3#201011yescharacteristic
309C2×C3#202211yescharacteristic
310C22—22no—
311C22—21no—
312C22—22no—
313C22—21no—
314C22—41no—
315C22—42no—
316C22—41no—
317C22—84no—
318C22—84no—
319C22—82no—
320C22—82no—
321C22—82no—
322C22#515612yes—
323C22#515711yescharacteristic
324C22#515812yes—
325C22#515911yescharacteristic
326C22#522311yescharacteristic
327C4—21no—
328C4—22no—
329C4—82no—
330C4—122no—
331C4—122no—
332C4—244no—
333C4#524312yes—
334C3#1032111yescharacteristic, complemented, Sylow
335C2—22no—
336C2—22no—
337C2—21no—
338C2—21no—
339C2—162no—
340C2#7184911yescharacteristic
341C2#7204311yescharacteristic
342C2#7211811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#3186C2
socleC22×C3#392
hypercenter#3186D3
hypocenterC3#10321
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3186

Complements

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

NH ≅ G/Ntype
2. #82699C2semidirect
3. #71675C2semidirect
4. #69910C2semidirect
5. #67676C2semidirect
98. C22×Dic3D8semidirect
334. C3#10321semidirect

Maximal subgroups

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

subgroupindex
C3×#31862
#826992
#716752
#699102
#676762
#103213