Finite Group Databasebeta

← #10697484

Subgroups — #10697484

Order 768.

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

normal subgroups only

This group has 828 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, complemented
2C2×#5015C212yesmaximal, complemented
3C2×D4⋊(C3⋊C8)C212yesmaximal, complemented
4#69580C211yesmaximal, characteristic, complemented
5#67634C211yesmaximal, characteristic, complemented
6#10228—31nomaximal, Sylow
7C2×C3×#121—22no—
8C2×C3×C4×D4C2212yes—
9C22×#213—22no—
10C2×#874C2211yescharacteristic
11C3×#394—22no—
12C3×#384C2211yescharacteristic
13C3×#380—22no—
14#5015—24no—
15#5015—24no—
16C2×#809—22no—
17C2×#809—22no—
18C2×C4⋊(C3⋊C8)C2212yes—
19C2×C4×C3⋊C8C2211yescharacteristic
20D4⋊(C3⋊C8)—24no—
21C2×#459—32no—
22#2537D311yescharacteristic, complemented
23#1957—31no—
24C2×D4⋊C8—32no—
25#1296—31no—
26C22×C3×D4—22no—
27C22×C3×D4D412yes—
28C23×C3×C4—22no—
29C22×D12—21no—
30C22×D12—22no—
31C3×#121—44no—
32C3×#121—44no—
33C3×C4×D4—22no—
34C3×C4×D4—22no—
35C3×C4×D4—22no—
36C3×C4×D4—22no—
37C2×C3×C4⋊C4—22no—
38C2×C3×C4⋊C4D412yes—
39C2×C3×C4⋊C4D412yes—
40C2×C3×C22⋊C4—24no—
41C2×C3×C22⋊C4—22no—
42C2×C3×C42C2311yescharacteristic
43C2×#213—28no—
44C2×#213—44no—
45C2×#213—44no—
46C22×C3⋊C8—21no—
47C22×C3⋊C8—22no—
48#874—22no—
49#874—21no—
50#874D412yescomplemented
51C3×#95—22no—
52#809—28no—
53#809—44no—
54C4⋊(C3⋊C8)—22no—
55C4⋊(C3⋊C8)D414yescomplemented
56C4×C3⋊C8—22no—
57C22×D8—62no—
58C2×#127—31no—
59C2×#121—22no—
60C2×C4×D4C2×D312yes—
61#459—64no—
62#459—64no—
63C2×C4⋊C8—32no—
64C2×#102—62no—
65C2×#102—62no—
66C2×C4×C8—31no—
67#394—22no—
68#384C2×D311yescharacteristic
69#380—22no—
70D4⋊C8—64no—
71C24×C3—22no—
72C23×D3—41no—
73C2×C3×D4—22no—
74C2×C3×D4—22no—
75C2×C3×D4—22no—
76C2×C3×D4—22no—
77C2×C3×D4—22no—
78C2×C3×D4—48no—
79C2×C3×D4—82no—
80C2×C3×D4D814yescomplemented
81C22×C3×C4—22no—
82C22×C3×C4—22no—
83C22×C3×C4—42no—
84C22×C3×C4—42no—
85C22×C3×C4—42no—
86C22×C3×C4—42no—
87C22×C3×C4—42no—
88C22×C3×C4—42no—
89C22×C3×C4C2×D411yescharacteristic
90C22×C3×C4C2×D412yes—
91C2×D12—24no—
92C2×D12—22no—
93C2×D12—42no—
94C2×D12—42no—
95C2×D12—42no—
96C2×D12—42no—
97C2×D12—42no—
98C2×D12—42no—
99C2×D12—42no—
100C2×D12—41no—
101C3×C4⋊C4—22no—
102C3×C4⋊C4—22no—
103C3×C4⋊C4—42no—
104C3×C4⋊C4—42no—
105C3×C4⋊C4—42no—
106C3×C22⋊C4—48no—
107C3×C22⋊C4—42no—
108C3×C22⋊C4—42no—
109C3×C42C2×D411yescharacteristic
110C3×C42C2×D412yes—
111C3×C42#4111yescharacteristic
112#213—416no—
113C2×C3⋊C8—22no—
114C2×C3⋊C8—24no—
115C2×C3⋊C8—41no—
116C2×C3⋊C8—41no—
117C2×C3⋊C8—42no—
118C2×C3⋊C8—42no—
119C22×D4—22no—
120C22×D4—62no—
121C22×D4—61no—
122C22×D4#6712yes—
123C23×C4—22no—
124C2×D8—68no—
125C2×D8—124no—
126C2×D8—124no—
127C22×C8—62no—
128C22×C8—61no—
129#127—32no—
130#127—62no—
131#127—61no—
132#121—44no—
133#121—44no—
134C4×D4—22no—
135C4×D4—22no—
136C4×D4—22no—
137C4×D4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4#6712yes—
140C2×C4⋊C4#6712yes—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—22no—
143C2×C42C22×D311yescharacteristic
144C4⋊C8—34no—
145C4⋊C8—62no—
146#102—68no—
147#102—124no—
148C4×C8—62no—
149#95—22no—
150C23×C3—24no—
151C23×C3—42no—
152C23×C3—42no—
153C23×C3—42no—
154C23×C3—42no—
155C23×C3—42no—
156C23×C3—42no—
157C23×C3#12011yescharacteristic
158C22×D3—42no—
159C22×D3—82no—
160C22×D3—82no—
161C22×D3—81no—
162C22×D3—81no—
163C3×D4—24no—
164C3×D4—24no—
165C3×D4—42no—
166C3×D4—42no—
167C3×D4—88no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—42no—
171C2×C3×C4—42no—
172C2×C3×C4—44no—
173C2×C3×C4—42no—
174C2×C3×C4—42no—
175C2×C3×C4—82no—
176C2×C3×C4—82no—
177C2×C3×C4—82no—
178C2×C3×C4—82no—
179C2×C3×C4#12012yes—
180C2×C3×C4#12111yescharacteristic
181C2×C3×C4#12112yes—
182C2×C3×C4#12711yescharacteristic
183C2×C3×C4C2×D812yes—
184C2×C3×C4#13512yes—
185C2×C3×C4AGL(1,ℤ/8)12yes—
186C2×C3×C4AGL(1,ℤ/8)12yes—
187D12—44no—
188D12—44no—
189D12—44no—
190D12—82no—
191D12—82no—
192D12—82no—
193C3⋊C8—42no—
194C3⋊C8—44no—
195C24—22no—
196C24—121no—
197C2×D4—22no—
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—22no—
201C2×D4—22no—
202C2×D4—48no—
203C2×D4—64no—
204C2×D4—62no—
205C2×D4—82no—
206C2×D4—122no—
207C2×D4—122no—
208C2×D4—122no—
209C2×D4—122no—
210C2×D4—122no—
211C2×D4—121no—
212C2×D4—122no—
213C2×D4—122no—
214C2×D4#21314yescomplemented
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4C2×#6711yescharacteristic
224C22×C4C2×#6712yes—
225D8—1216no—
226C2×C8—62no—
227C2×C8—64no—
228C2×C8—121no—
229C2×C8—122no—
230C2×C8—121no—
231C2×C8—122no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C22⋊C4—48no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C42D3×D411yescharacteristic
241C42D3×D412yes—
242C42#23711yescharacteristic
243C22×C3—44no—
244C22×C3—44no—
245C22×C3—44no—
246C22×C3—82no—
247C22×C3—82no—
248C22×C3—82no—
249C22×C3—82no—
250C22×C3#39211yescharacteristic
251C22×C3#44712yes—
252C22×C3#44912yes—
253C22×C3#45311yescharacteristic
254C22×C3#45411yescharacteristic
255C2×D3—84no—
256C2×D3—82no—
257C2×D3—161no—
258C2×D3—161no—
259C2×D3—162no—
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—82no—
263C3×C4—82no—
264C3×C4#44712yes—
265C3×C4#44912yes—
266C3×C4#46612yes—
267C3×C4#46912yes—
268C23—24no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—122no—
276C23—242no—
277C23—242no—
278C23—241no—
279C23—241no—
280C23#95311yescharacteristic
281D4—24no—
282D4—24no—
283D4—42no—
284D4—42no—
285D4—88no—
286D4—124no—
287D4—124no—
288D4—124no—
289D4—242no—
290D4—242no—
291D4—242no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4C2×#21312yes—
304C2×C4#93212yes—
305C2×C4#93712yes—
306C2×C4#93811yescharacteristic
307C2×C4#93912yes—
308C2×C4#94011yescharacteristic
309C2×C4#94912yes—
310C2×C4#95012yes—
311C8—122no—
312C8—124no—
313C2×C3—84no—
314C2×C3#161911yescharacteristic
315C2×C3#162411yescharacteristic
316C2×C3#162512yes—
317C2×C3#199912yes—
318C2×C3#200611yescharacteristic
319D3—162no—
320C22—44no—
321C22—44no—
322C22—44no—
323C22—82no—
324C22—82no—
325C22—82no—
326C22—82no—
327C22—244no—
328C22—242no—
329C22—481no—
330C22—482no—
331C22—481no—
332C22#521712yes—
333C22#522311yescharacteristic
334C22#523812yes—
335C22#524111yescharacteristic
336C22#524511yescharacteristic
337C4—21no—
338C4—21no—
339C4—82no—
340C4—82no—
341C4#503612yes—
342C4#503712yes—
343C4#503812yes—
344C4#503912yes—
345C3#1022811yescharacteristic, complemented, Sylow
346C2—84no—
347C2—482no—
348C2#7168412yes—
349C2#7187911yescharacteristic
350C2#7190611yescharacteristic
351C2#7191012yes—
352C2#7192711yescharacteristic
subgroupG/H
centerC23#953
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2537C2
socleC23×C3#120
hypercenter#2537D3
hypocenterC3#10228
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
1. C3×#2537C2semidirect
2. C2×#5015C2semidirect
3. C2×D4⋊(C3⋊C8)C2semidirect
4. #69580C2semidirect
5. #67634C2semidirect
22. #2537D3semidirect
50. #874D4semidirect
55. C4⋊(C3⋊C8)D4semidirect
80. C2×C3×D4D8semidirect
214. C2×D4#213semidirect
345. C3#10228semidirect

Maximal subgroups

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

subgroupindex
C3×#25372
C2×#50152
C2×D4⋊(C3⋊C8)2
#695802
#676342
#102283