Finite Group Databasebeta

← #10690176

Subgroups — #10690176

Order 768.

908 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 908 conjugacy classes of subgroups — too many to draw. Only its 147 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×#2315C211yesmaximal, characteristic
2C2×#4947C211yesmaximal, characteristic
3C2×#4525C211yesmaximal, characteristic
4#69525C212yesmaximal, complemented
5#67624C212yesmaximal, complemented
6#9013—31nomaximal, Sylow
7C23×Dic6C2211yescharacteristic
8C22×C3×C22⋊C4—21no—
9C22×C3×C22⋊C4C2211yescharacteristic
10C22×#217—21no—
11C3×#385—24no—
12C3×#380—21no—
13C3×#380C2212yes—
14C2×C3⋊#95—21no—
15C2×C3⋊#95C2211yescharacteristic
16C2×#830C2212yes—
17C2×#810—24no—
18#4947—22no—
19#4947C414yescomplemented
20#4525—22no—
21C2×#393—31no—
22#2315D311yescharacteristic
23#1787—32no—
24C2×#323—31no—
25#1294—32no—
26C25×C3D411yescharacteristic
27C23×C3×C4—22no—
28C23×C3×C4D411yescharacteristic
29C23×C3×C4C2311yescharacteristic
30C23×Dic3—21no—
31C23×Dic3—21no—
32C22×Dic6—22no—
33C22×Dic6—22no—
34C22×Dic6—21no—
35C22×Dic6—44no—
36C22×Dic6C2×C412yescomplemented
37C2×C3×C4⋊C4—22no—
38C2×C3×C4⋊C4D414yes—
39C2×C3×C22⋊C4—22no—
40C2×C3×C22⋊C4—22no—
41C2×C3×C22⋊C4—22no—
42C2×C3×C22⋊C4—22no—
43C2×C3×C22⋊C4—42no—
44C2×C3×C22⋊C4—42no—
45C2×C3×C22⋊C4C2×C412yes—
46C2×C3×C22⋊C4D412yes—
47C2×#217—24no—
48C2×#217—44no—
49C22×C3⋊C8—22no—
50C3×#95—22no—
51C3×#95—22no—
52C3⋊#95—24no—
53C3⋊#95—21no—
54C3⋊#95C2×C412yescomplemented
55#830—22no—
56#830—22no—
57#810—48no—
58C23×Q8—31no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—61no—
61C22×C22⋊C4C2×D311yescharacteristic
62C2×#103—64no—
63C2×C22⋊C8—32no—
64#393—34no—
65#393—62no—
66#385—24no—
67#380—21no—
68#380C2×D312yes—
69C2×#95—31no—
70C2×#95—61no—
71#323—62no—
72C24×C3—21no—
73C24×C3—21no—
74C24×C3—21no—
75C24×C3—21no—
76C24×C3—42no—
77C24×C3—42no—
78C24×C3C22⋊C412yes—
79C24×C3SD1612yes—
80C24×C3Q1612yes—
81C24×C3C2×D411yescharacteristic
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—24no—
85C22×C3×C4—42no—
86C22×C3×C4—42no—
87C22×C3×C4—44no—
88C22×C3×C4—82no—
89C22×C3×C4—82no—
90C22×C3×C4C22⋊C412yes—
91C22×C3×C4C22×C411yescharacteristic
92C22×C3×C4C2×D411yescharacteristic
93C22×C3×C4C2×D412yes—
94C22×C3×C4#4112yes—
95C22×Dic3—24no—
96C22×Dic3—22no—
97C22×Dic3—22no—
98C22×Dic3—42no—
99C22×Dic3—41no—
100C22×Dic3—41no—
101C22×Dic3—42no—
102C22×Dic3—42no—
103C22×Dic3—41no—
104C22×Dic3—41no—
105C2×Dic6—28no—
106C2×Dic6—24no—
107C2×Dic6—44no—
108C2×Dic6—48no—
109C2×Dic6—44no—
110C2×Dic6—44no—
111C2×Dic6—42no—
112C3×C4⋊C4—44no—
113C3×C4⋊C4—82no—
114C3×C22⋊C4—24no—
115C3×C22⋊C4—24no—
116C3×C22⋊C4—44no—
117C3×C22⋊C4—42no—
118C3×C22⋊C4—42no—
119#217—48no—
120C2×C3⋊C8—44no—
121C2×C3⋊C8—42no—
122C25#6711yescharacteristic
123C22×Q8—32no—
124C22×Q8—61no—
125C22×Q8—62no—
126C22×Q8—62no—
127C22×Q8—124no—
128C23×C4—22no—
129C23×C4—61no—
130C23×C4—61no—
131C23×C4#6711yescharacteristic
132C23×C4C22×D311yescharacteristic
133C22×C8—62no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4#6714yes—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—42no—
141C2×C22⋊C4—42no—
142C2×C22⋊C4—64no—
143C2×C22⋊C4—124no—
144C2×C22⋊C4C4×D312yes—
145C2×C22⋊C4D1212yes—
146#103—128no—
147C22⋊C8—62no—
148C22⋊C8—62no—
149#95—22no—
150#95—22no—
151#95—32no—
152#95—64no—
153#95—61no—
154C23×C3—22no—
155C23×C3—22no—
156C23×C3—22no—
157C23×C3—22no—
158C23×C3—22no—
159C23×C3—22no—
160C23×C3—22no—
161C23×C3—22no—
162C23×C3—42no—
163C23×C3—42no—
164C23×C3—44no—
165C23×C3—44no—
166C23×C3—42no—
167C23×C3—42no—
168C23×C3—82no—
169C23×C3—82no—
170C23×C3—81no—
171C23×C3—81no—
172C23×C3#10314yescomplemented
173C23×C3C2×C22⋊C411yescharacteristic
174C23×C3#12011yescharacteristic
175C23×C3#12311yescharacteristic
176C23×C3C2×SD1611yescharacteristic
177C23×C3C2×Q1611yescharacteristic
178C23×C3AGL(1,ℤ/8)11yescharacteristic
179C23×C3#13711yescharacteristic
180C2×C3×C4—24no—
181C2×C3×C4—22no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—42no—
185C2×C3×C4—42no—
186C2×C3×C4—42no—
187C2×C3×C4—44no—
188C2×C3×C4—44no—
189C2×C3×C4—44no—
190C2×C3×C4—81no—
191C2×C3×C4—88no—
192C2×C3×C4C4×D412yes—
193C2×C3×C4#12112yes—
194C2×Dic3—24no—
195C2×Dic3—42no—
196C2×Dic3—44no—
197C2×Dic3—42no—
198C2×Dic3—44no—
199C2×Dic3—42no—
200C2×Dic3—42no—
201C2×Dic3—42no—
202C2×Dic3—82no—
203C2×Dic3—82no—
204Dic6—28no—
205Dic6—48no—
206Dic6—44no—
207C3⋊C8—82no—
208C24—21no—
209C24—21no—
210C24—21no—
211C24—21no—
212C24—42no—
213C24—42no—
214C24(C2×D3)⋊C412yes—
215C24#21412yes—
216C24#21612yes—
217C24C2×#6711yescharacteristic
218C2×Q8—68no—
219C2×Q8—64no—
220C2×Q8—124no—
221C2×Q8—124no—
222C2×Q8—124no—
223C2×Q8—128no—
224C2×Q8—122no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—24no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—44no—
231C22×C4—64no—
232C22×C4—62no—
233C22×C4—62no—
234C22×C4—82no—
235C22×C4—82no—
236C22×C4—122no—
237C22×C4—121no—
238C22×C4—121no—
239C22×C4—122no—
240C22×C4—122no—
241C22×C4—121no—
242C22×C4—121no—
243C22×C4(C2×D3)⋊C412yes—
244C22×C4C2×C4×D311yescharacteristic
245C22×C4C2×D1211yescharacteristic
246C22×C4#23512yes—
247C22×C4C2×#6712yes—
248C2×C8—124no—
249C2×C8—122no—
250C4⋊C4—44no—
251C4⋊C4—82no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—44no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—128no—
258C22×C3—22no—
259C22×C3—22no—
260C22×C3—22no—
261C22×C3—22no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—24no—
267C22×C3—44no—
268C22×C3—44no—
269C22×C3—44no—
270C22×C3—42no—
271C22×C3—42no—
272C22×C3—82no—
273C22×C3—82no—
274C22×C3—82no—
275C22×C3#38611yescharacteristic
276C22×C3C2×#10311yescharacteristic
277C22×C3#41711yescharacteristic
278C22×C3#45011yescharacteristic
279C22×C3#45111yescharacteristic
280C22×C3#45512yes—
281C22×C3B(4,𝔽2)12yes—
282C22×C3#48311yescharacteristic
283C22×C3#48411yescharacteristic
284C3×C4—22no—
285C3×C4—22no—
286C3×C4—42no—
287C3×C4—84no—
288Dic3—44no—
289Dic3—82no—
290C23—22no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—42no—
299C23—42no—
300C23—44no—
301C23—44no—
302C23—42no—
303C23—42no—
304C23—82no—
305C23—82no—
306C23—81no—
307C23—81no—
308C23#81014yescomplemented
309C23C2×(C2×D3)⋊C411yescharacteristic
310C23#92911yescharacteristic
311C23#93211yescharacteristic
312C23C2×#21411yescharacteristic
313C23#94211yescharacteristic
314C23C2×#21611yescharacteristic
315C23#95311yescharacteristic
316Q8—68no—
317Q8—128no—
318Q8—124no—
319C2×C4—24no—
320C2×C4—22no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—64no—
330C2×C4—81no—
331C2×C4—88no—
332C2×C4—122no—
333C2×C4—124no—
334C2×C4—122no—
335C2×C4—124no—
336C2×C4—122no—
337C2×C4—122no—
338C2×C4—122no—
339C2×C4—242no—
340C2×C4—242no—
341C2×C4C4×#6712yes—
342C2×C4#93012yes—
343C8—242no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—44no—
349C2×C3#160112yes—
350C2×C3#160212yes—
351C2×C3#187211yescharacteristic
352C2×C3#188511yescharacteristic
353C2×C3#188911yescharacteristic
354C22—22no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—24no—
363C22—44no—
364C22—44no—
365C22—44no—
366C22—42no—
367C22—42no—
368C22—82no—
369C22—82no—
370C22—82no—
371C22#496611yescharacteristic
372C22C2×#81011yescharacteristic
373C22#497011yescharacteristic
374C22#497111yescharacteristic
375C22#521311yescharacteristic
376C22#521811yescharacteristic
377C22#522811yescharacteristic
378C22#524212yes—
379C22#524412yes—
380C4—22no—
381C4—22no—
382C4—42no—
383C4—84no—
384C4—124no—
385C4—242no—
386C3#901311yescharacteristic, complemented, Sylow
387C2—21no—
388C2—21no—
389C2—21no—
390C2—21no—
391C2—44no—
392C2#7155911yescharacteristic
393C2#7164911yescharacteristic
394C2#7165411yescharacteristic
395C2#7184612yes—
396C2#7189312yes—
subgroupG/H
centerC23#953
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2315C2
socleC23×C3#120
prime radicalC25×C3D4
cosocleC23×C3×C4C23
hypercenter#2315D3
hypocenterC3#9013
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2315

Complements

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

NH ≅ G/Ntype
4. #69525C2semidirect
5. #67624C2semidirect
19. #4947C4semidirect
36. C22×Dic6C2×C4semidirect
54. C3⋊#95C2×C4semidirect
172. C23×C3#103semidirect
308. C23#810semidirect
386. C3#9013semidirect

Maximal subgroups

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

subgroupindex
C3×#23152
C2×#49472
C2×#45252
#695252
#676242
#90133