Finite Group Databasebeta

← #10690915

Subgroups — #10690915

Order 768.

1010 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 1010 conjugacy classes of subgroups — too many to draw. Only its 143 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
1#84031C211yesmaximal, characteristic
2C2×C4×#217C211yesmaximal, characteristic
3C3×#1779C211yesmaximal, characteristic
4C2×#4467C212yesmaximal, complemented
5#67675C212yesmaximal, complemented
6#9146—31nomaximal, Sylow
7C24×C3×C4C2211yescharacteristic
8C22×#217—21no—
9C2×#924C412yescomplemented
10C2×#924C2211yescharacteristic
11C22×C4×Dic3—21no—
12C2×C3×C22⋊C8—22no—
13C2×C3×C22⋊C8C2212yes—
14#5247—21no—
15#5212C412yescomplemented
16#5212C2211yescharacteristic
17#5210—21no—
18#5210—22no—
19C4×#217—24no—
20C2×C3⋊#95C2212yes—
21#4467—24no—
22#2848—31no—
23C2×C4×C22⋊C4—31no—
24#1779D311yescharacteristic
25C2×#328—32no—
26#1294—32no—
27C25×C3D411yescharacteristic
28C23×C3×C4—22no—
29C23×C3×C4—22no—
30C23×C3×C4—44no—
31C23×C3×C4C2×C412yes—
32C23×C3×C4D411yescharacteristic
33C23×C3×C4D412yes—
34C23×C3×C4C2311yescharacteristic
35C23×Dic3—22no—
36C22×Dic6C2×C411yescharacteristic
37C22×C3×C8—22no—
38C2×#217—24no—
39C2×#217—28no—
40C2×#217—22no—
41C2×#217—22no—
42C2×#217—21no—
43C2×#217—21no—
44C2×C4⋊Dic3C2×C411yescharacteristic
45C2×C4⋊Dic3C2×C412yes—
46#924—22no—
47#924—22no—
48#924—44no—
49C2×Dic3⋊C4—22no—
50C2×Dic3⋊C4—21no—
51C2×C4×Dic3—24no—
52C2×C4×Dic3—44no—
53C3×C22⋊C8—22no—
54C3×C22⋊C8—22no—
55C3×C22⋊C8—44no—
56C3⋊#95—22no—
57C3⋊#95—22no—
58C3⋊#95—21no—
59C3⋊#95D414yes—
60C24×C4C2×D311yescharacteristic
61C2×C22⋊Q8—31no—
62C2×C22⋊Q8—32no—
63C22×C22⋊C4—61no—
64C22×C42—61no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8C2×D312yes—
67#385—61no—
68#385—62no—
69#380—32no—
70#380—31no—
71#379—61no—
72C4×C22⋊C4—64no—
73C2×#95—32no—
74#328—64no—
75C24×C3—24no—
76C24×C3—21no—
77C24×C3—21no—
78C24×C3—22no—
79C24×C3—22no—
80C24×C3—42no—
81C24×C3C22⋊C412yes—
82C24×C3C2×D411yescharacteristic
83C22×C3×C4—21no—
84C22×C3×C4—21no—
85C22×C3×C4—24no—
86C22×C3×C4—24no—
87C22×C3×C4—24no—
88C22×C3×C4—24no—
89C22×C3×C4—24no—
90C22×C3×C4—42no—
91C22×C3×C4—42no—
92C22×C3×C4—42no—
93C22×C3×C4—48no—
94C22×C3×C4—48no—
95C22×C3×C4C22⋊C412yes—
96C22×C3×C4C22×C411yescharacteristic
97C22×C3×C4C2×D412yes—
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4#4112yes—
100C22×Dic3—28no—
101C22×Dic3—24no—
102C22×Dic3—21no—
103C22×Dic3—22no—
104C22×Dic3—21no—
105C22×Dic3—44no—
106C22×Dic3—42no—
107C22×Dic3—42no—
108C2×Dic6—22no—
109C2×Dic6—42no—
110C2×C3×C8—44no—
111C2×C3×C8—42no—
112#217—216no—
113#217—44no—
114#217—44no—
115#217—41no—
116#217—41no—
117#217—42no—
118C4⋊Dic3—22no—
119C4⋊Dic3—42no—
120Dic3⋊C4—44no—
121Dic3⋊C4—42no—
122C4×Dic3—48no—
123C25D1211yescharacteristic
124C22×Q8—31no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—44no—
128C23×C4—62no—
129C23×C4C4×D312yes—
130C23×C4D1211yescharacteristic
131C23×C4#6712yes—
132C23×C4C22×D311yescharacteristic
133C22×C8—22no—
134C22⋊Q8—62no—
135C22⋊Q8—62no—
136C22⋊Q8—124no—
137C2×C4⋊C4—31no—
138C2×C4⋊C4—32no—
139C2×C4⋊C4—62no—
140C2×C4⋊C4—61no—
141C2×C22⋊C4—64no—
142C2×C22⋊C4—68no—
143C2×C22⋊C4—62no—
144C2×C22⋊C4—62no—
145C2×C22⋊C4—61no—
146C2×C22⋊C4—61no—
147C2×C42—64no—
148C2×C42—124no—
149C22⋊C8—22no—
150C22⋊C8—22no—
151C22⋊C8—44no—
152#95—34no—
153#95—62no—
154#95—62no—
155#95—61no—
156C23×C3—22no—
157C23×C3—22no—
158C23×C3—24no—
159C23×C3—24no—
160C23×C3—22no—
161C23×C3—22no—
162C23×C3—22no—
163C23×C3—22no—
164C23×C3—22no—
165C23×C3—22no—
166C23×C3—48no—
167C23×C3—48no—
168C23×C3—42no—
169C23×C3—42no—
170C23×C3—42no—
171C23×C3—42no—
172C23×C3#10414yes—
173C23×C3C2×C22⋊C411yescharacteristic
174C23×C3#12011yescharacteristic
175C23×C3#12311yescharacteristic
176C23×C3#13512yes—
177C23×C3#13712yes—
178C2×C3×C4—24no—
179C2×C3×C4—24no—
180C2×C3×C4—22no—
181C2×C3×C4—22no—
182C2×C3×C4—22no—
183C2×C3×C4—22no—
184C2×C3×C4—42no—
185C2×C3×C4—42no—
186C2×C3×C4—48no—
187C2×C3×C4—48no—
188C2×C3×C4#9914yes—
189C2×C3×C4C4×D412yes—
190C2×C3×C4#12112yes—
191C2×Dic3—28no—
192C2×Dic3—44no—
193C2×Dic3—48no—
194C2×Dic3—48no—
195C2×Dic3—44no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—42no—
199C2×Dic3—42no—
200C2×Dic3—42no—
201C2×Dic3—42no—
202C2×Dic3—42no—
203Dic6—82no—
204C3×C8—82no—
205C24—24no—
206C24—21no—
207C24—21no—
208C24—22no—
209C24—22no—
210C24—42no—
211C24(C2×D3)⋊C412yes—
212C24C2×D1211yescharacteristic
213C2×Q8—62no—
214C2×Q8—122no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—48no—
226C22×C4—48no—
227C22×C4—68no—
228C22×C4—64no—
229C22×C4—61no—
230C22×C4—62no—
231C22×C4—61no—
232C22×C4—124no—
233C22×C4—122no—
234C22×C4—122no—
235C22×C4(C2×D3)⋊C412yes—
236C22×C4C2×C4×D311yescharacteristic
237C22×C4D3×D412yes—
238C22×C4#23712yes—
239C22×C4C2×#6711yescharacteristic
240C2×C8—44no—
241C2×C8—42no—
242C4⋊C4—62no—
243C4⋊C4—122no—
244C4⋊C4—124no—
245C4⋊C4—122no—
246C22⋊C4—616no—
247C22⋊C4—124no—
248C22⋊C4—124no—
249C22⋊C4—121no—
250C22⋊C4—121no—
251C22⋊C4—122no—
252C42—128no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—21no—
257C22×C3—24no—
258C22×C3—24no—
259C22×C3—24no—
260C22×C3—24no—
261C22×C3—24no—
262C22×C3—48no—
263C22×C3—48no—
264C22×C3—42no—
265C22×C3—42no—
266C22×C3—42no—
267C22×C3—42no—
268C22×C3#38611yescharacteristic
269C22×C3C2×#9912yes—
270C22×C3#41611yescharacteristic
271C22×C3#41911yescharacteristic
272C22×C3C2×#10412yes—
273C22×C3#45212yes—
274C22×C3#48512yes—
275C3×C4—22no—
276C3×C4—22no—
277C3×C4—24no—
278Dic3—48no—
279Dic3—81no—
280Dic3—81no—
281Dic3—82no—
282C23—22no—
283C23—22no—
284C23—24no—
285C23—24no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—48no—
293C23—48no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23#82514yes—
299C23#88211yescharacteristic
300C23#88611yescharacteristic
301C23#90412yes—
302C23#90912yes—
303C23C2×(C2×D3)⋊C411yescharacteristic
304Q8—242no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—48no—
314C2×C4—48no—
315C2×C4—68no—
316C2×C4—124no—
317C2×C4—128no—
318C2×C4—128no—
319C2×C4—124no—
320C2×C4—122no—
321C2×C4—122no—
322C2×C4—122no—
323C2×C4—122no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4—122no—
327C2×C4#80614yes—
328C2×C4#67⋊C412yes—
329C2×C4#93912yes—
330C8—82no—
331C2×C3—22no—
332C2×C3—22no—
333C2×C3—44no—
334C2×C3#150112yes—
335C2×C3#150812yes—
336C2×C3#187311yescharacteristic
337C2×C3#189611yescharacteristic
338C2×C3#189811yescharacteristic
339C22—24no—
340C22—21no—
341C22—21no—
342C22—24no—
343C22—21no—
344C22—21no—
345C22—24no—
346C22—24no—
347C22—24no—
348C22—48no—
349C22—48no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—42no—
354C22#472312yes—
355C22#473312yes—
356C22C2×#80612yes—
357C22#495811yescharacteristic
358C22#511311yescharacteristic
359C22#512011yescharacteristic
360C22C2×#82512yes—
361C4—22no—
362C4—24no—
363C4—22no—
364C4—128no—
365C4—242no—
366C4—241no—
367C4—241no—
368C3#914611yescharacteristic, complemented, Sylow
369C2—22no—
370C2—22no—
371C2—44no—
372C2#6949611yescharacteristic
373C2#6993111yescharacteristic
374C2#6994212yes—
375C2#6996511yescharacteristic
376C2#6998012yes—
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1779C2
socleC23×C3C2×C22⋊C4
prime radicalC25×C3D4
cosocleC23×C3×C4C23
hypercenter#1779D3
hypocenterC3#9146
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
4. C2×#4467C2semidirect
5. #67675C2semidirect
9. C2×#924C4semidirect
15. #5212C4semidirect
368. C3#9146semidirect

Maximal subgroups

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

subgroupindex
#840312
C2×C4×#2172
C3×#17792
C2×#44672
#676752
#91463