Finite Group Databasebeta

← #10691231

Subgroups — #10691231

Order 768.

684 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 135 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×#2626C211yesmaximal, characteristic
2#82094C211yesmaximal, characteristic
3#71690C212yesmaximal, complemented
4C2×#4525C211yesmaximal, characteristic
5#67625C212yesmaximal, complemented
6#9199—31nomaximal, Sylow
7C2×C3×C22⋊Q8C2211yescharacteristic
8C22×C3×C22⋊C4C2211yescharacteristic
9C22×#217—21no—
10C3×#399C412yescomplemented
11C3×#396—22no—
12C3×#393C412yescomplemented
13C3×#385—21no—
14C3×#380C2211yescharacteristic
15C3×#379—21no—
16#5212—21no—
17#5212C2212yes—
18C2×#830C2212yes—
19#4950—24no—
20#4538—24no—
21#4525—22no—
22#2626D311yescharacteristic
23#2315—31no—
24#1520—32no—
25C2×#323—31no—
26#1289—32no—
27C25×C3D411yescharacteristic
28C22×C3×Q8C2×C411yescharacteristic, complemented
29C23×C3×C4D411yescharacteristic
30C23×C3×C4C2311yescharacteristic
31C23×Dic3—22no—
32C3×C22⋊Q8—22no—
33C3×C22⋊Q8—22no—
34C2×C3×C4⋊C4—21no—
35C2×C3×C4⋊C4—22no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4C2×C411yescharacteristic, complemented
38C2×C3×C22⋊C4—22no—
39C2×C3×C22⋊C4—22no—
40C2×C3×C22⋊C4—21no—
41C2×C3×C22⋊C4—21no—
42C2×C3×C22⋊C4—22no—
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4C2×C412yes—
45C2×C3×C22⋊C4D412yes—
46C2×#217—22no—
47C2×#217—22no—
48C2×#217—42no—
49C2×#217—42no—
50C2×C4⋊Dic3—22no—
51C2×C4⋊Dic3—22no—
52C22×C3⋊C8—22no—
53C3×#95—22no—
54C3×#95C2×C412yescomplemented
55C3⋊#95—22no—
56C3⋊#95D414yes—
57#830—22no—
58#830—22no—
59C2×C22⋊Q8C2×D311yescharacteristic
60C22×C22⋊C4—61no—
61C22×C22⋊C4C2×D311yescharacteristic
62C2×C22⋊C8—32no—
63#399Dic312yescomplemented
64#396—22no—
65#393Dic312yescomplemented
66#385—21no—
67#385—64no—
68#380—32no—
69#380—61no—
70#380C2×D311yescharacteristic
71#379—21no—
72#328—64no—
73#323—62no—
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
84C2×C3×Q8—22no—
85C2×C3×Q8—81no—
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4—22no—
90C22×C3×C4—21no—
91C22×C3×C4—81no—
92C22×C3×C4—81no—
93C22×C3×C4C22⋊C412yes—
94C22×C3×C4C22×C411yescharacteristic
95C22×C3×C4C2×D412yes—
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4#4112yes—
98C22×Dic3—24no—
99C22×Dic3—44no—
100C22×Dic3—82no—
101C22×Dic3—82no—
102C3×C4⋊C4—22no—
103C3×C4⋊C4—42no—
104C3×C4⋊C4—81no—
105C3×C4⋊C4—82no—
106C3×C22⋊C4—24no—
107C3×C22⋊C4—24no—
108C3×C22⋊C4—41no—
109C3×C22⋊C4—41no—
110C3×C22⋊C4—41no—
111C3×C22⋊C4—41no—
112C3×C22⋊C4—82no—
113C3×C22⋊C4—82no—
114#217—44no—
115#217—42no—
116#217—42no—
117C4⋊Dic3—82no—
118C4⋊Dic3—82no—
119C2×C3⋊C8—42no—
120C2×C3⋊C8—82no—
121C25#6711yescharacteristic
122C22×Q8C2×Dic311yescharacteristic, complemented
123C23×C4—62no—
124C23×C4#6711yescharacteristic
125C23×C4C22×D311yescharacteristic
126C22×C8—62no—
127C22⋊Q8—22no—
128C22⋊Q8—22no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—62no—
133C2×C4⋊C4—62no—
134C2×C4⋊C4C2×Dic311yescharacteristic, complemented
135C2×C22⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—21no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—62no—
142C2×C22⋊C4—62no—
143C2×C22⋊C4—122no—
144C2×C22⋊C4—122no—
145C2×C22⋊C4C2×Dic312yes—
146C2×C22⋊C4#6712yes—
147C22⋊C8—62no—
148C22⋊C8—62no—
149#95—22no—
150#95—34no—
151#95—62no—
152#95C2×Dic312yescomplemented
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
179C3×Q8—82no—
180C2×C3×C4—22no—
181C2×C3×C4—42no—
182C2×C3×C4—42no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—42no—
186C2×C3×C4—42no—
187C2×C3×C4—42no—
188C2×C3×C4—42no—
189C2×C3×C4—42no—
190C2×C3×C4—42no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—42no—
194C2×C3×C4—81no—
195C2×C3×C4C4×D412yes—
196C2×C3×C4#12112yes—
197C2×Dic3—44no—
198C2×Dic3—44no—
199C2×Dic3—44no—
200C2×Dic3—88no—
201C3⋊C8—82no—
202C24—21no—
203C24—21no—
204C24—21no—
205C24—21no—
206C24—81no—
207C24—81no—
208C24#21512yes—
209C24#21612yes—
210C24#21712yes—
211C24C2×#6711yescharacteristic
212C2×Q8—22no—
213C2×Q8—81no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—64no—
220C22×C4—81no—
221C22×C4—81no—
222C22×C4—124no—
223C22×C4—242no—
224C22×C4—242no—
225C22×C4#21712yes—
226C22×C4D3×D412yes—
227C22×C4#23712yes—
228C22×C4C22×Dic311yescharacteristic
229C22×C4C2×#6711yescharacteristic
230C2×C8—122no—
231C2×C8—242no—
232C4⋊C4—22no—
233C4⋊C4—42no—
234C4⋊C4—81no—
235C4⋊C4—82no—
236C4⋊C4—242no—
237C4⋊C4—242no—
238C22⋊C4—24no—
239C22⋊C4—24no—
240C22⋊C4—41no—
241C22⋊C4—41no—
242C22⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—82no—
245C22⋊C4—82no—
246C22⋊C4—124no—
247C22⋊C4—122no—
248C22⋊C4—122no—
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—
278C3×C4—82no—
279C3×C4—81no—
280C3×C4—81no—
281Dic3—84no—
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#83514yes—
301C23#93511yescharacteristic
302C23#93711yescharacteristic
303C23C2×#21511yescharacteristic
304C23C2×#21611yescharacteristic
305C23#94911yescharacteristic
306C23#95111yescharacteristic
307C23C2×#21711yescharacteristic
308Q8—82no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—81no—
324C2×C4—124no—
325C2×C4—124no—
326C2×C4—124no—
327C2×C4—248no—
328C2×C4D4×Dic312yes—
329C2×C4#93912yes—
330C8—242no—
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#504311yescharacteristic
361C22#504411yescharacteristic
362C22#504611yescharacteristic
363C22#505011yescharacteristic
364C22#522011yescharacteristic
365C22C2×#83511yescharacteristic
366C22#523211yescharacteristic
367C4—22no—
368C4—22no—
369C4—42no—
370C4—82no—
371C4—81no—
372C4—81no—
373C4—244no—
374C3#919911yescharacteristic, complemented, Sylow
375C2—21no—
376C2—21no—
377C2—21no—
378C2—21no—
379C2—21no—
380C2—21no—
381C2—44no—
382C2#7169211yescharacteristic
383C2#7184811yescharacteristic
384C2#7185311yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2626C2
socleC22×C3#386
prime radicalC25×C3D4
cosocleC23×C3×C4C23
hypercenter#2626D3
hypocenterC3#9199
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2626

Complements

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

NH ≅ G/Ntype
3. #71690C2semidirect
5. #67625C2semidirect
10. C3×#399C4semidirect
12. C3×#393C4semidirect
28. C22×C3×Q8C2×C4semidirect
37. C2×C3×C4⋊C4C2×C4semidirect
54. C3×#95C2×C4semidirect
63. #399Dic3semidirect
65. #393Dic3semidirect
122. C22×Q8C2×Dic3semidirect
134. C2×C4⋊C4C2×Dic3semidirect
152. #95C2×Dic3semidirect
374. C3#9199semidirect

Maximal subgroups

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

subgroupindex
C3×#26262
#820942
#716902
C2×#45252
#676252
#91993