Finite Group Databasebeta

← #10690212

Subgroups — #10690212

Order 768.

1224 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 1224 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
1C2×C3×#392C211yesmaximal, characteristic
2#82093C211yesmaximal, characteristic, complemented
3#72128C212yesmaximal, complemented
4C2×#4525C211yesmaximal, characteristic, complemented
5#67612C212yesmaximal, complemented
6#9020—31nomaximal, Sylow
7C23×C3×D4—21no—
8C23×C3×D4C2211yescharacteristic
9C22×C3×C22⋊C4—21no—
10C22×C3×C22⋊C4C2211yescharacteristic
11C3×#392—22no—
12C3×#392C414yescomplemented
13C2×C3×#95C2211yescharacteristic
14#5247—21no—
15C2×#832—24no—
16#5212C2212yes—
17C4×#217—21no—
18C2×#830C2212yes—
19#4951—24no—
20#4525—22no—
21C2×#392D311yescharacteristic
22#2312—31no—
23#1786—32no—
24C2×#323—31no—
25#1294—32no—
26C25×C3—21no—
27C25×C3D411yescharacteristic
28C22×C3×D4—24no—
29C22×C3×D4—22no—
30C22×C3×D4—21no—
31C22×C3×D4—22no—
32C22×C3×D4—44no—
33C22×C3×D4—44no—
34C22×C3×D4—44no—
35C22×C3×D4—44no—
36C22×C3×D4C2×C412yes—
37C23×C3×C4—21no—
38C23×C3×C4D411yescharacteristic
39C23×C3×C4C2311yescharacteristic
40C2×C3×C22⋊C4—22no—
41C2×C3×C22⋊C4—22no—
42C2×C3×C22⋊C4—24no—
43C2×C3×C22⋊C4—44no—
44C2×C3×C22⋊C4C2×C412yes—
45C2×C3×C22⋊C4D412yes—
46C2×#217—22no—
47C2×#217—22no—
48C2×#217—22no—
49C2×C4⋊Dic3D414yescomplemented
50C2×C4×Dic3—22no—
51C22×C3⋊C8—22no—
52C3×#95—21no—
53C3×#95C2×C412yes—
54#832—48no—
55C3⋊#95—22no—
56C3⋊#95—22no—
57#830—22no—
58#830—22no—
59C23×D4—21no—
60C23×D4C2×D311yescharacteristic
61C22×C22⋊C4—21no—
62C22×C22⋊C4C2×D311yescharacteristic
63C2×#102—64no—
64C2×C22⋊C8—32no—
65#392—22no—
66#392Dic314yescomplemented
67#388—64no—
68#380—32no—
69#379—61no—
70C4×C22⋊C4—61no—
71C2×#95C2×D311yescharacteristic
72#323—62no—
73C24×C3—24no—
74C24×C3—22no—
75C24×C3—21no—
76C24×C3—22no—
77C24×C3—21no—
78C24×C3—22no—
79C24×C3—42no—
80C24×C3—42no—
81C24×C3—42no—
82C24×C3—42no—
83C24×C3—42no—
84C24×C3—41no—
85C24×C3—41no—
86C24×C3—42no—
87C24×C3—42no—
88C24×C3C22⋊C412yes—
89C24×C3C2×D411yescharacteristic
90C2×C3×D4—28no—
91C2×C3×D4—24no—
92C2×C3×D4—44no—
93C2×C3×D4—48no—
94C2×C3×D4—44no—
95C2×C3×D4—48no—
96C2×C3×D4—44no—
97C2×C3×D4—48no—
98C2×C3×D4—48no—
99C2×C3×D4—432no—
100C2×C3×D4—42no—
101C22×C3×C4—24no—
102C22×C3×C4—22no—
103C22×C3×C4—21no—
104C22×C3×C4—21no—
105C22×C3×C4—42no—
106C22×C3×C4—42no—
107C22×C3×C4—42no—
108C22×C3×C4—42no—
109C22×C3×C4C22⋊C412yes—
110C22×C3×C4C22×C411yescharacteristic
111C22×C3×C4C2×D412yes—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4#4112yes—
114C22×Dic3—24no—
115C3×C22⋊C4—24no—
116C3×C22⋊C4—24no—
117C3×C22⋊C4—48no—
118#217—42no—
119#217—42no—
120#217—82no—
121#217—82no—
122C4⋊Dic3—44no—
123C4×Dic3—82no—
124C2×C3⋊C8—44no—
125C2×C3⋊C8—42no—
126C25—21no—
127C25#6711yescharacteristic
128C22×D4—24no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—21no—
132C22×D4—44no—
133C22×D4—44no—
134C22×D4—44no—
135C22×D4—44no—
136C22×D4C2×Dic312yes—
137C23×C4—21no—
138C23×C4#6711yescharacteristic
139C23×C4C22×D311yescharacteristic
140C22×C8—62no—
141C2×C4⋊C4—34no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—44no—
146C2×C22⋊C4—62no—
147C2×C22⋊C4—62no—
148C2×C22⋊C4—62no—
149C2×C22⋊C4C2×Dic312yes—
150C2×C22⋊C4#6712yes—
151C2×C42—62no—
152#102—128no—
153C22⋊C8—62no—
154C22⋊C8—62no—
155#95—21no—
156#95—62no—
157#95—62no—
158#95C2×Dic312yes—
159C23×C3—22no—
160C23×C3—22no—
161C23×C3—24no—
162C23×C3—22no—
163C23×C3—22no—
164C23×C3—22no—
165C23×C3—42no—
166C23×C3—42no—
167C23×C3—42no—
168C23×C3—48no—
169C23×C3—48no—
170C23×C3—42no—
171C23×C3—42no—
172C23×C3—44no—
173C23×C3—48no—
174C23×C3—42no—
175C23×C3—42no—
176C23×C3—42no—
177C23×C3—42no—
178C23×C3—42no—
179C23×C3—42no—
180C23×C3—44no—
181C23×C3—42no—
182C23×C3—81no—
183C23×C3—81no—
184C23×C3—84no—
185C23×C3—84no—
186C23×C3—84no—
187C23×C3—82no—
188C23×C3—82no—
189C23×C3—82no—
190C23×C3—82no—
191C23×C3C2×C22⋊C411yescharacteristic
192C23×C3#12011yescharacteristic
193C23×C3#12311yescharacteristic
194C23×C3#13512yes—
195C23×C3AGL(1,ℤ/8)12yes—
196C3×D4—28no—
197C3×D4—48no—
198C3×D4—44no—
199C3×D4—432no—
200C2×C3×C4—24no—
201C2×C3×C4—24no—
202C2×C3×C4—22no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—42no—
206C2×C3×C4—48no—
207C2×C3×C4—42no—
208C2×C3×C4—42no—
209C2×C3×C4—42no—
210C2×C3×C4—42no—
211C2×C3×C4—81no—
212C2×C3×C4C4×D412yes—
213C2×C3×C4#12112yes—
214C2×Dic3—44no—
215C2×Dic3—44no—
216C2×Dic3—44no—
217C3⋊C8—82no—
218C24—24no—
219C24—22no—
220C24—21no—
221C24—22no—
222C24—21no—
223C24—22no—
224C24—42no—
225C24—42no—
226C24—42no—
227C24—42no—
228C24—42no—
229C24—42no—
230C24—41no—
231C24—41no—
232C24—42no—
233C24#21712yes—
234C24C2×#6711yescharacteristic
235C2×D4—28no—
236C2×D4—24no—
237C2×D4—44no—
238C2×D4—48no—
239C2×D4—44no—
240C2×D4—48no—
241C2×D4—44no—
242C2×D4—48no—
243C2×D4—48no—
244C2×D4—432no—
245C2×D4—42no—
246C22×C4—24no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—64no—
255C22×C4#21712yes—
256C22×C4D3×D412yes—
257C22×C4#23712yes—
258C22×C4C22×Dic311yescharacteristic
259C22×C4C2×#6711yescharacteristic
260C2×C8—124no—
261C2×C8—122no—
262C4⋊C4—124no—
263C22⋊C4—24no—
264C22⋊C4—24no—
265C22⋊C4—48no—
266C22⋊C4—122no—
267C22⋊C4—122no—
268C22⋊C4—242no—
269C22⋊C4—242no—
270C42—242no—
271C22×C3—24no—
272C22×C3—24no—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—44no—
278C22×C3—44no—
279C22×C3—44no—
280C22×C3—42no—
281C22×C3—42no—
282C22×C3—44no—
283C22×C3—42no—
284C22×C3—82no—
285C22×C3—82no—
286C22×C3—82no—
287C22×C3—84no—
288C22×C3—88no—
289C22×C3—84no—
290C22×C3—88no—
291C22×C3—82no—
292C22×C3—82no—
293C22×C3#38611yescharacteristic
294C22×C3#41611yescharacteristic
295C22×C3#41811yescharacteristic
296C22×C3#44912yes—
297C22×C3#45312yes—
298C22×C3B(4,𝔽2)12yes—
299C22×C3#48212yes—
300C3×C4—22no—
301C3×C4—22no—
302C3×C4—44no—
303C3×C4—42no—
304Dic3—84no—
305C23—22no—
306C23—22no—
307C23—24no—
308C23—22no—
309C23—22no—
310C23—22no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—48no—
315C23—48no—
316C23—42no—
317C23—42no—
318C23—44no—
319C23—48no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—44no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—81no—
329C23—81no—
330C23—84no—
331C23—84no—
332C23—84no—
333C23—82no—
334C23—82no—
335C23—82no—
336C23—82no—
337C23#93212yes—
338C23#93511yescharacteristic
339C23#93711yescharacteristic
340C23#95012yes—
341C23C2×#21711yescharacteristic
342D4—28no—
343D4—48no—
344D4—44no—
345D4—432no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—48no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—81no—
358C2×C4—124no—
359C2×C4—124no—
360C2×C4—124no—
361C2×C4D4×Dic312yes—
362C2×C4#93912yes—
363C8—242no—
364C2×C3—21no—
365C2×C3—21no—
366C2×C3—22no—
367C2×C3—82no—
368C2×C3—84no—
369C2×C3#160014yes—
370C2×C3#187511yescharacteristic
371C2×C3#188711yescharacteristic
372C2×C3#188911yescharacteristic
373C22—24no—
374C22—24no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—44no—
380C22—44no—
381C22—44no—
382C22—42no—
383C22—44no—
384C22—42no—
385C22—42no—
386C22—82no—
387C22—82no—
388C22—82no—
389C22—84no—
390C22—88no—
391C22—84no—
392C22—88no—
393C22—82no—
394C22—82no—
395C22#503212yes—
396C22#503512yes—
397C22#503712yes—
398C22#507712yes—
399C22#521511yescharacteristic
400C22#522011yescharacteristic
401C22#523311yescharacteristic
402C4—22no—
403C4—22no—
404C4—44no—
405C4—42no—
406C4—244no—
407C3#902011yescharacteristic, complemented, Sylow
408C2—21no—
409C2—21no—
410C2—22no—
411C2—82no—
412C2—84no—
413C2#6969714yes—
414C2#7166911yescharacteristic
415C2#7167811yescharacteristic
416C2#7176311yescharacteristic
subgroupG/H
centerC23#937
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
Fitting, prime radicalC2×C3×#392C2
socleC23×C3#120
cosocleC23×C3×C4C23
hypercenterC2×#392D3
hypocenterC3#9020
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#392

Complements

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

NH ≅ G/Ntype
2. #82093C2semidirect
3. #72128C2semidirect
4. C2×#4525C2semidirect
5. #67612C2semidirect
12. C3×#392C4semidirect
49. C2×C4⋊Dic3D4semidirect
66. #392Dic3semidirect
407. C3#9020semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3922
#820932
#721282
C2×#45252
#676122
#90203