Finite Group Databasebeta

← #10697529

Subgroups — #10697529

Order 768.

712 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 137 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×#2537C211yesmaximal, characteristic
2C2×#5008C212yesmaximal, complemented
3C2×D4⋊(C3⋊C8)C212yesmaximal, complemented
4#69579C211yesmaximal, characteristic, complemented
5#67634C211yesmaximal, characteristic, complemented
6#10235—31nomaximal, Sylow
7C2×C3×#121—22no—
8C2×C3×C4×D4C2212yes—
9C2×#869C2211yescharacteristic
10C3×#394—22no—
11C3×#384C2211yescharacteristic
12C3×#380—22no—
13C2×#832—22no—
14#5008—24no—
15#5008—24no—
16C2×#810—22no—
17C2×#808—22no—
18C2×C4⋊(C3⋊C8)C2212yes—
19C2×C4×C3⋊C8C2211yescharacteristic
20D4⋊(C3⋊C8)—24no—
21C2×#476—32no—
22#2537D311yescharacteristic
23#1956—31no—
24C2×D4⋊C8—32no—
25#1296—31no—
26C22×C3×D4—22no—
27C22×C3×D4D412yes—
28C23×C3×C4—22no—
29C22×Dic6—21no—
30C3×#121—44no—
31C3×#121—44no—
32C3×C4×D4—22no—
33C3×C4×D4—22no—
34C3×C4×D4—22no—
35C3×C4×D4—22no—
36C2×C3×C4⋊C4—22no—
37C2×C3×C4⋊C4D412yes—
38C2×C3×C4⋊C4D412yes—
39C2×C3×C22⋊C4—24no—
40C2×C3×C22⋊C4—22no—
41C2×C3×C42C2311yescharacteristic
42C2×C4⋊Dic3—22no—
43C22×C3⋊C8—21no—
44C22×C3⋊C8—22no—
45#869—22no—
46#869—21no—
47#869D412yescomplemented
48C3×#95—22no—
49#832—28no—
50#810—44no—
51#808—28no—
52C4⋊(C3⋊C8)—22no—
53C4⋊(C3⋊C8)D414yescomplemented
54C4×C3⋊C8—22no—
55C2×C4⋊Q8—31no—
56C2×#121—22no—
57C2×C4×D4C2×D312yes—
58#476—64no—
59#476—64no—
60C2×#106—62no—
61C2×C4⋊C8—32no—
62C2×#103—62no—
63C2×#102—62no—
64C2×C4×C8—31no—
65#394—22no—
66#384C2×D311yescharacteristic
67#380—22no—
68D4⋊C8—64no—
69C24×C3—22no—
70C2×C3×D4—22no—
71C2×C3×D4—22no—
72C2×C3×D4—22no—
73C2×C3×D4—22no—
74C2×C3×D4—22no—
75C2×C3×D4—48no—
76C2×C3×D4—82no—
77C2×C3×D4SD1614yes—
78C22×C3×C4—22no—
79C22×C3×C4—22no—
80C22×C3×C4—42no—
81C22×C3×C4—42no—
82C22×C3×C4—42no—
83C22×C3×C4—42no—
84C22×C3×C4—42no—
85C22×C3×C4—42no—
86C22×C3×C4C2×D411yescharacteristic
87C22×C3×C4C2×D412yes—
88C22×Dic3—41no—
89C2×Dic6—22no—
90C2×Dic6—42no—
91C2×Dic6—42no—
92C2×Dic6—41no—
93C3×C4⋊C4—22no—
94C3×C4⋊C4—22no—
95C3×C4⋊C4—42no—
96C3×C4⋊C4—42no—
97C3×C4⋊C4—42no—
98C3×C22⋊C4—48no—
99C3×C22⋊C4—42no—
100C3×C22⋊C4—42no—
101C3×C42C2×D411yescharacteristic
102C3×C42C2×D412yes—
103C3×C42#4111yescharacteristic
104C4⋊Dic3—24no—
105C4⋊Dic3—42no—
106C2×C3⋊C8—22no—
107C2×C3⋊C8—24no—
108C2×C3⋊C8—41no—
109C2×C3⋊C8—41no—
110C2×C3⋊C8—42no—
111C2×C3⋊C8—42no—
112C22×Q8—61no—
113C22×D4—22no—
114C22×D4#6712yes—
115C23×C4—22no—
116C22×C8—61no—
117C22×C8—62no—
118C4⋊Q8—32no—
119C4⋊Q8—62no—
120C4⋊Q8—61no—
121#121—44no—
122#121—44no—
123C4×D4—22no—
124C4×D4—22no—
125C4×D4—22no—
126C4×D4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—62no—
129C2×C4⋊C4#6712yes—
130C2×C4⋊C4#6712yes—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—22no—
133C2×C42C22×D311yescharacteristic
134#106—68no—
135C4⋊C8—34no—
136C4⋊C8—62no—
137#103—124no—
138#102—68no—
139C4×C8—62no—
140#95—22no—
141C23×C3—24no—
142C23×C3—42no—
143C23×C3—42no—
144C23×C3—42no—
145C23×C3—42no—
146C23×C3—42no—
147C23×C3—42no—
148C23×C3#12011yescharacteristic
149C3×D4—24no—
150C3×D4—24no—
151C3×D4—42no—
152C3×D4—42no—
153C3×D4—88no—
154C2×C3×C4—21no—
155C2×C3×C4—21no—
156C2×C3×C4—42no—
157C2×C3×C4—42no—
158C2×C3×C4—44no—
159C2×C3×C4—42no—
160C2×C3×C4—42no—
161C2×C3×C4—82no—
162C2×C3×C4—82no—
163C2×C3×C4—82no—
164C2×C3×C4—82no—
165C2×C3×C4#12012yes—
166C2×C3×C4#12111yescharacteristic
167C2×C3×C4#12112yes—
168C2×C3×C4#12711yescharacteristic
169C2×C3×C4C2×SD1612yes—
170C2×C3×C4#13512yes—
171C2×C3×C4AGL(1,ℤ/8)12yes—
172C2×C3×C4#13712yes—
173C2×Dic3—42no—
174C2×Dic3—81no—
175C2×Dic3—81no—
176Dic6—44no—
177Dic6—82no—
178C3⋊C8—42no—
179C3⋊C8—44no—
180C24—22no—
181C2×Q8—62no—
182C2×Q8—122no—
183C2×Q8—122no—
184C2×Q8—121no—
185C2×D4—22no—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—48no—
191C2×D4—82no—
192C2×D4#21414yes—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—121no—
202C22×C4C2×#6711yescharacteristic
203C22×C4C2×#6712yes—
204C2×C8—62no—
205C2×C8—64no—
206C2×C8—121no—
207C2×C8—122no—
208C2×C8—121no—
209C2×C8—122no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—64no—
216C4⋊C4—122no—
217C22⋊C4—48no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C42D3×D411yescharacteristic
221C42D3×D412yes—
222C42#23711yescharacteristic
223C22×C3—44no—
224C22×C3—44no—
225C22×C3—44no—
226C22×C3—82no—
227C22×C3—82no—
228C22×C3—82no—
229C22×C3—82no—
230C22×C3#39211yescharacteristic
231C22×C3#45012yes—
232C22×C3#45212yes—
233C22×C3#45311yescharacteristic
234C22×C3#45411yescharacteristic
235C3×C4—21no—
236C3×C4—21no—
237C3×C4—82no—
238C3×C4—82no—
239C3×C4#44812yes—
240C3×C4#44912yes—
241C3×C4#46512yes—
242C3×C4#46812yes—
243Dic3—82no—
244C23—24no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23#95311yescharacteristic
252Q8—124no—
253Q8—242no—
254D4—24no—
255D4—24no—
256D4—42no—
257D4—42no—
258D4—88no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—44no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—122no—
271C2×C4—241no—
272C2×C4—241no—
273C2×C4#93212yes—
274C2×C4C2×#21412yes—
275C2×C4#93712yes—
276C2×C4#93811yescharacteristic
277C2×C4#93912yes—
278C2×C4#94011yescharacteristic
279C2×C4#95012yes—
280C2×C4#95112yes—
281C8—122no—
282C8—124no—
283C2×C3—84no—
284C2×C3#162911yescharacteristic
285C2×C3#163411yescharacteristic
286C2×C3#163512yes—
287C2×C3#200012yes—
288C2×C3#200611yescharacteristic
289C22—44no—
290C22—44no—
291C22—44no—
292C22—82no—
293C22—82no—
294C22—82no—
295C22—82no—
296C22#521812yes—
297C22#522311yescharacteristic
298C22#523912yes—
299C22#524111yescharacteristic
300C22#524511yescharacteristic
301C4—21no—
302C4—21no—
303C4—82no—
304C4—82no—
305C4—242no—
306C4#503712yes—
307C4#504012yes—
308C4#504112yes—
309C4#504212yes—
310C3#1023511yescharacteristic, complemented, Sylow
311C2—84no—
312C2#7168512yes—
313C2#7187911yescharacteristic
314C2#7190711yescharacteristic
315C2#7191112yes—
316C2#7192811yescharacteristic
subgroupG/H
centerC23#953
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#2537C2
socleC23×C3#120
hypercenter#2537D3
hypocenterC3#10235
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2537

Complements

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

NH ≅ G/Ntype
2. C2×#5008C2semidirect
3. C2×D4⋊(C3⋊C8)C2semidirect
4. #69579C2semidirect
5. #67634C2semidirect
47. #869D4semidirect
53. C4⋊(C3⋊C8)D4semidirect
310. C3#10235semidirect

Maximal subgroups

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

subgroupindex
C3×#25372
C2×#50082
C2×D4⋊(C3⋊C8)2
#695792
#676342
#102353