Finite Group Databasebeta

← #10695743

Subgroups — #10695743

Order 768.

1020 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 1020 conjugacy classes of subgroups — too many to draw. Only its 119 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
1C2×#5218C212yesmaximal
2#73361C211yesmaximal, characteristic, complemented
3C3×#1438C211yesmaximal, characteristic
4C2×#4452C212yesmaximal, complemented
5#67613C211yesmaximal, characteristic, complemented
6#9966—31nomaximal, Sylow
7C23×C3×D4C2211yescharacteristic
8C22×C3×C22⋊C4—22no—
9C22×#214—22no—
10C2×#924C2212yes—
11C2×C3×#95C2211yescharacteristic
12C2×C3×#95C2211yescharacteristic
13#5218—24no—
14#5218—24no—
15C2×#832—22no—
16#5212—22no—
17C2×#830C2212yes—
18#4661—22no—
19#4656—22no—
20#4452—24no—
21C2×#450—32no—
22#2659—31no—
23C2×#328—32no—
24#1438D311yescharacteristic
25#1280—31no—
26C25×C3—21no—
27C22×C3×D4—22no—
28C22×C3×D4—22no—
29C22×C3×D4—21no—
30C22×C3×D4—44no—
31C22×C3×D4D412yes—
32C23×C3×C4—21no—
33C23×C3×C4D412yes—
34C23×C3×C4C2311yescharacteristic
35C22×Dic6D412yes—
36C2×C3×C22⋊C4—44no—
37C2×C3×C22⋊C4—44no—
38C2×C3×C22⋊C4—44no—
39C2×#217—22no—
40C2×#214—28no—
41C2×#214—48no—
42C2×C4⋊Dic3—22no—
43C2×C4⋊Dic3—22no—
44C2×C4⋊Dic3D412yes—
45#924—22no—
46#924—22no—
47#924—22no—
48#924—22no—
49C2×Dic3⋊C4—22no—
50C22×C3⋊C8—22no—
51C3×#95—22no—
52C3×#95—22no—
53#832—28no—
54C3⋊#95—24no—
55#830—22no—
56#830D414yes—
57C23×D4C2×D311yescharacteristic
58C22×SD16—62no—
59C2×C22⋊Q8—32no—
60C22×C22⋊C4—22no—
61#450—64no—
62#450—64no—
63C2×#102—62no—
64C2×C22⋊C8—32no—
65#398—62no—
66#395—62no—
67#380—62no—
68C2×#95C2×D311yescharacteristic
69C2×#95C2×D311yescharacteristic
70#328—64no—
71C24×C3—24no—
72C24×C3—22no—
73C24×C3—42no—
74C24×C3—42no—
75C24×C3—42no—
76C24×C3—41no—
77C24×C3—41no—
78C24×C3—42no—
79C24×C3—42no—
80C24×C3C2×D411yescharacteristic
81C2×C3×D4—28no—
82C2×C3×D4—24no—
83C2×C3×D4—44no—
84C2×C3×D4—44no—
85C2×C3×D4—48no—
86C2×C3×D4—44no—
87C2×C3×D4—42no—
88C22×C3×C4—22no—
89C22×C3×C4—22no—
90C22×C3×C4—22no—
91C22×C3×C4—42no—
92C22×C3×C4—42no—
93C22×C3×C4—42no—
94C22×C3×C4—42no—
95C22×C3×C4—41no—
96C22×C3×C4—41no—
97C22×C3×C4—42no—
98C22×C3×C4SD1614yescomplemented
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4C2×D412yes—
101C22×C3×C4C2×D412yes—
102C22×C3×C4#4111yescharacteristic
103C22×Dic3—22no—
104C22×Dic3—22no—
105C2×Dic6—22no—
106C2×Dic6—22no—
107C2×Dic6—44no—
108C3×C22⋊C4—416no—
109#217—42no—
110#217—42no—
111#214—416no—
112C4⋊Dic3—22no—
113C4⋊Dic3—22no—
114C4⋊Dic3—82no—
115C4⋊Dic3—82no—
116Dic3⋊C4—42no—
117Dic3⋊C4—42no—
118C2×C3⋊C8—24no—
119C2×C3⋊C8—44no—
120C25—21no—
121C22×Q8—32no—
122C22×D4—21no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—44no—
126C22×D4D1212yes—
127C23×C4—21no—
128C23×C4#6712yes—
129C23×C4C22×D311yescharacteristic
130C2×SD16—68no—
131C2×SD16—128no—
132C22×C8—62no—
133C22⋊Q8—62no—
134C22⋊Q8—62no—
135C22⋊Q8—62no—
136C22⋊Q8—62no—
137C2×C4⋊C4—32no—
138C2×C4⋊C4—62no—
139C2×C4⋊C4—62no—
140C2×C4⋊C4—62no—
141C2×C22⋊C4—44no—
142C2×C22⋊C4—44no—
143C2×C22⋊C4—44no—
144C2×C22⋊C4—62no—
145#102—68no—
146C22⋊C8—34no—
147C22⋊C8—62no—
148#95—22no—
149#95—22no—
150#95—64no—
151C23×C3—21no—
152C23×C3—21no—
153C23×C3—22no—
154C23×C3—24no—
155C23×C3—44no—
156C23×C3—44no—
157C23×C3—42no—
158C23×C3—44no—
159C23×C3—44no—
160C23×C3—44no—
161C23×C3—44no—
162C23×C3—42no—
163C23×C3—44no—
164C23×C3—44no—
165C23×C3—82no—
166C23×C3—82no—
167C23×C3—82no—
168C23×C3—82no—
169C23×C3—82no—
170C23×C3—82no—
171C23×C3—81no—
172C23×C3—81no—
173C23×C3—81no—
174C23×C3—81no—
175C23×C3#12011yescharacteristic
176C23×C3#12111yescharacteristic
177C23×C3#12711yescharacteristic
178C23×C3C2×SD1612yes—
179C23×C3#13712yes—
180C3×D4—28no—
181C3×D4—44no—
182C3×D4—48no—
183C2×C3×C4—24no—
184C2×C3×C4—24no—
185C2×C3×C4—41no—
186C2×C3×C4—42no—
187C2×C3×C4—41no—
188C2×C3×C4—42no—
189C2×C3×C4—42no—
190C2×C3×C4—44no—
191C2×C3×C4—44no—
192C2×C3×C4—42no—
193C2×C3×C4—82no—
194C2×C3×C4—82no—
195C2×C3×C4—81no—
196C2×C3×C4—81no—
197C2×C3×C4#12012yes—
198C2×C3×C4#12112yes—
199C2×Dic3—42no—
200C2×Dic3—42no—
201C2×Dic3—42no—
202C2×Dic3—42no—
203C2×Dic3—44no—
204Dic6—44no—
205Dic6—82no—
206C3⋊C8—44no—
207C24—24no—
208C24—22no—
209C24—42no—
210C24—42no—
211C24—42no—
212C24—42no—
213C24—41no—
214C24—41no—
215C24—42no—
216C24C2×#6711yescharacteristic
217C2×Q8—62no—
218C2×Q8—62no—
219C2×Q8—124no—
220C2×D4—28no—
221C2×D4—24no—
222C2×D4—44no—
223C2×D4—44no—
224C2×D4—48no—
225C2×D4—44no—
226C2×D4—42no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—62no—
238C22×C4—62no—
239C22×C4#21414yescomplemented
240C22×C4C2×D1211yescharacteristic
241C22×C4#23511yescharacteristic
242C22×C4D3×D412yes—
243C22×C4D3×D412yes—
244SD16—1216no—
245C2×C8—64no—
246C2×C8—124no—
247C4⋊C4—62no—
248C4⋊C4—62no—
249C4⋊C4—122no—
250C4⋊C4—122no—
251C4⋊C4—242no—
252C4⋊C4—242no—
253C22⋊C4—416no—
254C22⋊C4—122no—
255C22⋊C4—122no—
256C22×C3—22no—
257C22×C3—22no—
258C22×C3—22no—
259C22×C3—22no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—48no—
265C22×C3—44no—
266C22×C3—84no—
267C22×C3—84no—
268C22×C3—84no—
269C22×C3—84no—
270C22×C3—84no—
271C22×C3—82no—
272C22×C3—82no—
273C22×C3#39211yescharacteristic
274C22×C3#44812yes—
275C22×C3#45311yescharacteristic
276C22×C3#45411yescharacteristic
277C22×C3B(4,𝔽2)12yes—
278C22×C3#46112yes—
279C22×C3#49211yescharacteristic
280C22×C3#49711yescharacteristic
281C3×C4—22no—
282C3×C4—22no—
283C3×C4—44no—
284C3×C4—82no—
285Dic3—82no—
286Dic3—82no—
287C23—22no—
288C23—21no—
289C23—21no—
290C23—24no—
291C23—44no—
292C23—44no—
293C23—42no—
294C23—44no—
295C23—44no—
296C23—44no—
297C23—44no—
298C23—42no—
299C23—44no—
300C23—44no—
301C23—82no—
302C23—82no—
303C23—82no—
304C23—82no—
305C23—82no—
306C23—82no—
307C23—81no—
308C23—81no—
309C23—81no—
310C23—81no—
311C23#93011yescharacteristic
312C23C2×#21412yes—
313C23#93711yescharacteristic
314C23#94011yescharacteristic
315C23#95112yes—
316Q8—124no—
317Q8—242no—
318D4—28no—
319D4—44no—
320D4—48no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—122no—
336C2×C4—122no—
337C2×C4—124no—
338C2×C4—122no—
339C2×C4—122no—
340C2×C4#88212yes—
341C2×C4#88412yes—
342C8—124no—
343C2×C3—21no—
344C2×C3—22no—
345C2×C3—21no—
346C2×C3—84no—
347C2×C3#159612yes—
348C2×C3#159912yes—
349C2×C3#200611yescharacteristic
350C2×C3#201311yescharacteristic
351C2×C3#202111yescharacteristic
352C22—21no—
353C22—21no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—21no—
359C22—21no—
360C22—48no—
361C22—44no—
362C22—84no—
363C22—84no—
364C22—84no—
365C22—82no—
366C22—82no—
367C22—84no—
368C22—84no—
369C22#496011yescharacteristic
370C22#501912yes—
371C22#503212yes—
372C22#504012yes—
373C22#506011yescharacteristic
374C22#506211yescharacteristic
375C22#507611yescharacteristic
376C22#507711yescharacteristic
377C4—22no—
378C4—22no—
379C4—44no—
380C4—82no—
381C4—242no—
382C4—242no—
383C3#996611yescharacteristic, complemented, Sylow
384C2—21no—
385C2—22no—
386C2—21no—
387C2—84no—
388C2#6950011yescharacteristic
389C2#6966111yescharacteristic
390C2#6966411yescharacteristic
391C2#6969812yes—
392C2#6970412yes—
subgroupG/H
centerC23#937
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1438C2
socleC23×C3#120
prime radicalC23×C3×D4C22
hypercenter#1438D3
hypocenterC3#9966
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1438

Complements

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

NH ≅ G/Ntype
2. #73361C2semidirect
4. C2×#4452C2semidirect
5. #67613C2semidirect
98. C22×C3×C4SD16semidirect
239. C22×C4#214semidirect
383. C3#9966semidirect

Maximal subgroups

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

subgroupindex
C2×#52182
#733612
C3×#14382
C2×#44522
#676132
#99663