Finite Group Databasebeta

← #10691228

Subgroups — #10691228

Order 768.

684 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 127 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#82117C211yesmaximal, characteristic
2#82094C211yesmaximal, characteristic
3C2×C3×#323C211yesmaximal, characteristic
4#69946C212yesmaximal, complemented
5#67676C212yesmaximal, complemented
6#9199—31nomaximal, Sylow
7C22×C3×C22⋊C4C2211yescharacteristic
8C22×#217—21no—
9C2×#924C2211yescharacteristic
10C2×C3×C22⋊C8C2212yes—
11#5247—21no—
12#5212—21no—
13#5212C2211yescharacteristic
14#5212C2212yes—
15#5210—21no—
16#4950—24no—
17#4949C412yescomplemented
18#4948—22no—
19#4947C412yescomplemented
20C3×#323—22no—
21#4467—24no—
22#2626—31no—
23#2315—31no—
24#1520—32no—
25C2×#323D311yescharacteristic
26#1289—32no—
27C25×C3D411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4C2311yescharacteristic
30C23×Dic3—22no—
31C22×Dic6C2×C411yescharacteristic, complemented
32C22×C3×C8—22no—
33C2×C3×C22⋊C4—22no—
34C2×C3×C22⋊C4—22no—
35C2×C3×C22⋊C4C2×C412yes—
36C2×C3×C22⋊C4D412yes—
37C2×#217—22no—
38C2×#217—21no—
39C2×#217—21no—
40C2×#217—22no—
41C2×#217—22no—
42C2×#217—22no—
43C2×#217—42no—
44C2×#217—42no—
45C2×C4⋊Dic3—22no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3—22no—
48C2×C4⋊Dic3C2×C411yescharacteristic, complemented
49#924—22no—
50#924—22no—
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—22no—
53C3×C22⋊C8—22no—
54C3×C22⋊C8—22no—
55C3⋊#95—22no—
56C3⋊#95—22no—
57C3⋊#95C2×C412yescomplemented
58C3⋊#95D414yes—
59C2×C22⋊Q8—31no—
60C22×C22⋊C4—61no—
61C22×C22⋊C4C2×D311yescharacteristic
62C2×C22⋊C8C2×D312yes—
63#399—32no—
64#396—62no—
65#393—32no—
66#385—64no—
67#385—61no—
68#380—31no—
69#380—32no—
70#380—61no—
71#379—61no—
72#328—64no—
73#323—22no—
74C24×C3—21no—
75C24×C3—21no—
76C24×C3—21no—
77C24×C3—21no—
78C24×C3—81no—
79C24×C3—81no—
80C24×C3C22⋊C412yes—
81C24×C3SD1612yes—
82C24×C3Q1612yes—
83C24×C3C2×D411yescharacteristic
84C22×C3×C4—21no—
85C22×C3×C4—21no—
86C22×C3×C4—81no—
87C22×C3×C4—81no—
88C22×C3×C4C22⋊C412yes—
89C22×C3×C4C22×C411yescharacteristic
90C22×C3×C4C2×D412yes—
91C22×C3×C4C2×D411yescharacteristic
92C22×C3×C4#4112yes—
93C22×Dic3—21no—
94C22×Dic3—22no—
95C22×Dic3—21no—
96C22×Dic3—24no—
97C22×Dic3—44no—
98C22×Dic3—82no—
99C22×Dic3—82no—
100C2×Dic6—22no—
101C2×Dic6—81no—
102C2×C3×C8—42no—
103C2×C3×C8—82no—
104C3×C22⋊C4—24no—
105C3×C22⋊C4—24no—
106#217—44no—
107#217—41no—
108#217—41no—
109#217—42no—
110#217—41no—
111#217—42no—
112#217—41no—
113#217—82no—
114#217—82no—
115C4⋊Dic3—22no—
116C4⋊Dic3—81no—
117C4⋊Dic3—82no—
118C4⋊Dic3—82no—
119Dic3⋊C4—42no—
120Dic3⋊C4—82no—
121C25D1211yescharacteristic
122C22×Q8—31no—
123C23×C4—62no—
124C23×C4D1211yescharacteristic
125C23×C4C22×D311yescharacteristic
126C22×C8—22no—
127C22⋊Q8—62no—
128C22⋊Q8—62no—
129C2×C4⋊C4—31no—
130C2×C4⋊C4—61no—
131C2×C4⋊C4—62no—
132C2×C4⋊C4—62no—
133C2×C4⋊C4—62no—
134C2×C4⋊C4—61no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—62no—
138C2×C22⋊C4—62no—
139C2×C22⋊C4—61no—
140C2×C22⋊C4—61no—
141C2×C22⋊C4—62no—
142C2×C22⋊C4—62no—
143C2×C22⋊C4—122no—
144C2×C22⋊C4—122no—
145C2×C22⋊C4C4×D312yes—
146C2×C22⋊C4#6712yes—
147C22⋊C8—22no—
148C22⋊C8—22no—
149#95—32no—
150#95—34no—
151#95—62no—
152#95—62no—
153C23×C3—22no—
154C23×C3—22no—
155C23×C3—22no—
156C23×C3—22no—
157C23×C3—22no—
158C23×C3—22no—
159C23×C3—41no—
160C23×C3—41no—
161C23×C3—42no—
162C23×C3—42no—
163C23×C3—42no—
164C23×C3—42no—
165C23×C3—81no—
166C23×C3—81no—
167C23×C3—82no—
168C23×C3—82no—
169C23×C3—82no—
170C23×C3—82no—
171C23×C3#10314yes—
172C23×C3C2×C22⋊C411yescharacteristic
173C23×C3#12011yescharacteristic
174C23×C3#12311yescharacteristic
175C23×C3C2×SD1611yescharacteristic
176C23×C3C2×Q1611yescharacteristic
177C23×C3AGL(1,ℤ/8)11yescharacteristic
178C23×C3#13711yescharacteristic
179C2×C3×C4—22no—
180C2×C3×C4—42no—
181C2×C3×C4—41no—
182C2×C3×C4—42no—
183C2×C3×C4—41no—
184C2×C3×C4—42no—
185C2×C3×C4—42no—
186C2×C3×C4—81no—
187C2×C3×C4C4×D412yes—
188C2×C3×C4#12112yes—
189C2×Dic3—42no—
190C2×Dic3—42no—
191C2×Dic3—42no—
192C2×Dic3—44no—
193C2×Dic3—44no—
194C2×Dic3—42no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—42no—
198C2×Dic3—44no—
199C2×Dic3—88no—
200Dic6—82no—
201C3×C8—82no—
202C24—21no—
203C24—21no—
204C24—21no—
205C24—21no—
206C24—81no—
207C24—81no—
208C24#20412yes—
209C24Dic1212yes—
210C24(C2×D3)⋊C412yes—
211C24C2×D1211yescharacteristic
212C2×Q8—62no—
213C2×Q8—241no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—64no—
217C22×C4—62no—
218C22×C4—61no—
219C22×C4—61no—
220C22×C4—81no—
221C22×C4—81no—
222C22×C4—124no—
223C22×C4—242no—
224C22×C4—242no—
225C22×C4(C2×D3)⋊C412yes—
226C22×C4C2×C4×D311yescharacteristic
227C22×C4D3×D412yes—
228C22×C4#23712yes—
229C22×C4C2×#6711yescharacteristic
230C2×C8—42no—
231C2×C8—82no—
232C4⋊C4—62no—
233C4⋊C4—122no—
234C4⋊C4—242no—
235C4⋊C4—242no—
236C4⋊C4—241no—
237C4⋊C4—242no—
238C22⋊C4—24no—
239C22⋊C4—24no—
240C22⋊C4—124no—
241C22⋊C4—121no—
242C22⋊C4—121no—
243C22⋊C4—122no—
244C22⋊C4—122no—
245C22⋊C4—121no—
246C22⋊C4—121no—
247C22⋊C4—242no—
248C22⋊C4—242no—
249C22×C3—21no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—24no—
256C22×C3—41no—
257C22×C3—41no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—44no—
261C22×C3—44no—
262C22×C3—44no—
263C22×C3—42no—
264C22×C3—42no—
265C22×C3—82no—
266C22×C3—82no—
267C22×C3—82no—
268C22×C3#38611yescharacteristic
269C22×C3C2×#10311yescharacteristic
270C22×C3#41711yescharacteristic
271C22×C3#45011yescharacteristic
272C22×C3#45111yescharacteristic
273C22×C3#48311yescharacteristic
274C22×C3#48411yescharacteristic
275C3×C4—22no—
276C3×C4—22no—
277C3×C4—42no—
278Dic3—82no—
279Dic3—84no—
280Dic3—81no—
281Dic3—81no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—41no—
289C23—41no—
290C23—42no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23—81no—
295C23—81no—
296C23—82no—
297C23—82no—
298C23—82no—
299C23—82no—
300C23#81614yes—
301C23#88211yescharacteristic
302C23#88611yescharacteristic
303C23C2×#20411yescharacteristic
304C23C2×Dic1211yescharacteristic
305C23#90811yescharacteristic
306C23#90911yescharacteristic
307C23C2×(C2×D3)⋊C411yescharacteristic
308Q8—242no—
309C2×C4—22no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—81no—
317C2×C4—122no—
318C2×C4—122no—
319C2×C4—122no—
320C2×C4—122no—
321C2×C4—124no—
322C2×C4—124no—
323C2×C4—122no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4—124no—
327C2×C4—248no—
328C2×C4#67⋊C412yes—
329C2×C4#93912yes—
330C8—82no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3—21no—
334C2×C3—21no—
335C2×C3—21no—
336C2×C3—21no—
337C2×C3—44no—
338C2×C3#187211yescharacteristic
339C2×C3#191111yescharacteristic
340C2×C3#191311yescharacteristic
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—24no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—44no—
353C22—44no—
354C22—44no—
355C22—42no—
356C22—42no—
357C22—82no—
358C22—82no—
359C22—82no—
360C22#472111yescharacteristic
361C22#472211yescharacteristic
362C22#473111yescharacteristic
363C22#473911yescharacteristic
364C22#495811yescharacteristic
365C22C2×#81611yescharacteristic
366C22#513311yescharacteristic
367C4—22no—
368C4—22no—
369C4—42no—
370C4—244no—
371C4—242no—
372C4—241no—
373C4—241no—
374C3#919911yescharacteristic, complemented, Sylow
375C2—21no—
376C2—21no—
377C2—21no—
378C2—21no—
379C2—21no—
380C2—21no—
381C2—44no—
382C2#6993011yescharacteristic
383C2#7002411yescharacteristic
384C2#7007111yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#323C2
socleC22×C3#386
prime radicalC25×C3D4
cosocleC23×C3×C4C23
hypercenterC2×#323D3
hypocenterC3#9199
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
4. #69946C2semidirect
5. #67676C2semidirect
17. #4949C4semidirect
19. #4947C4semidirect
31. C22×Dic6C2×C4semidirect
48. C2×C4⋊Dic3C2×C4semidirect
57. C3⋊#95C2×C4semidirect
374. C3#9199semidirect

Maximal subgroups

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

subgroupindex
#821172
#820942
C2×C3×#3232
#699462
#676762
#91993