Finite Group Databasebeta

← #10690574

Subgroups — #10690574

Order 768.

628 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#387C211yesmaximal, characteristic
2C2×#4925C211yesmaximal, characteristic
3C2×#4528C212yesmaximal
4#71531C211yesmaximal, characteristic
5#67613C212yesmaximal
6#9082—31nomaximal, Sylow
7C22×C3×C4⋊C4C411yescharacteristic
8C22×C3×C4⋊C4C2211yescharacteristic
9C22×C3×C42C2211yescharacteristic
10C3×#387—21no—
11C3×#387—21no—
12C3×#387—21no—
13C3×#387—21no—
14C2×C3×#95C411yescharacteristic
15C2×C3×#95C411yescharacteristic
16C2×C3×#95C411yescharacteristic
17C2×C3×#95C2211yescharacteristic
18C2×#830C2212yes—
19C2×#830C2212yes—
20#4925—24no—
21C2×C4⋊(C3⋊C8)—21no—
22C2×C4×C3⋊C8—21no—
23#4528—24no—
24#4524—21no—
25#4524—21no—
26C2×#431—31no—
27C2×#387D311yescharacteristic
28#1817—31no—
29C2×#324—32no—
30#1280—32no—
31C23×C3×C4C2×C411yescharacteristic
32C23×C3×C4C2×C411yescharacteristic
33C23×C3×C4C2×C411yescharacteristic
34C23×C3×C4C2×C411yescharacteristic
35C23×C3×C4C2×C411yescharacteristic
36C23×C3×C4C2×C411yescharacteristic
37C23×C3×C4C2311yescharacteristic
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4—22no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4—22no—
44C2×C3×C4⋊C4—22no—
45C2×C3×C4⋊C4C814yescomplemented
46C2×C3×C42—22no—
47C2×C3×C42—22no—
48C2×C3×C42D411yescharacteristic
49C2×C3×C42D411yescharacteristic
50C2×C3×C42D411yescharacteristic
51C2×C3×C42D411yescharacteristic
52C22×C3⋊C8—22no—
53C22×C3⋊C8—22no—
54C3×#95—21no—
55C3×#95—21no—
56C3×#95—21no—
57C3×#95—21no—
58C3×#95—21no—
59C3×#95—21no—
60C3×#95—21no—
61C3×#95—21no—
62#830—22no—
63#830—22no—
64#830—22no—
65#830D414yes—
66C4⋊(C3⋊C8)—24no—
67C4×C3⋊C8—24no—
68C22×C4⋊C4Dic311yescharacteristic
69C22×C4⋊C4C2×D311yescharacteristic
70C22×C42C2×D311yescharacteristic
71#431—64no—
72C2×C4⋊C8—61no—
73C2×C22⋊C8—32no—
74C2×C22⋊C8—32no—
75C2×C4×C8—61no—
76#387—21no—
77#387—21no—
78#387—21no—
79#387—21no—
80C2×#95Dic311yescharacteristic
81C2×#95Dic311yescharacteristic
82C2×#95Dic311yescharacteristic
83C2×#95C2×D311yescharacteristic
84#336—61no—
85#336—61no—
86#324—64no—
87C24×C3C22×C411yescharacteristic
88C22×C3×C4—22no—
89C22×C3×C4—22no—
90C22×C3×C4—21no—
91C22×C3×C4—21no—
92C22×C3×C4—22no—
93C22×C3×C4—22no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—22no—
97C22×C3×C4—22no—
98C22×C3×C4—21no—
99C22×C3×C4—21no—
100C22×C3×C4—22no—
101C22×C3×C4—22no—
102C22×C3×C4—41no—
103C22×C3×C4—41no—
104C22×C3×C4—41no—
105C22×C3×C4—41no—
106C22×C3×C4—41no—
107C22×C3×C4—41no—
108C22×C3×C4—41no—
109C22×C3×C4—41no—
110C22×C3×C4—41no—
111C22×C3×C4C22⋊C411yescharacteristic
112C22×C3×C4C22⋊C411yescharacteristic
113C22×C3×C4C22⋊C411yescharacteristic
114C22×C3×C4C22⋊C411yescharacteristic
115C22×C3×C4C2×C812yes—
116C22×C3×C4C2×C812yes—
117C22×C3×C4C2×C812yes—
118C22×C3×C4M1612yes—
119C22×C3×C4C2×D412yes—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4#4112yes—
123C3×C4⋊C4—24no—
124C3×C4⋊C4—24no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—42no—
127C3×C4⋊C4—41no—
128C3×C42—21no—
129C3×C42—21no—
130C3×C42—22no—
131C3×C42—22no—
132C3×C42—21no—
133C3×C42—21no—
134C2×C3⋊C8—24no—
135C2×C3⋊C8—42no—
136C2×C3⋊C8—42no—
137C2×C3⋊C8—42no—
138C2×C3⋊C8—42no—
139C2×C3⋊C8—42no—
140C23×C4C2×Dic311yescharacteristic
141C23×C4C2×Dic311yescharacteristic
142C23×C4C2×Dic311yescharacteristic
143C23×C4C2×Dic311yescharacteristic
144C23×C4C2×Dic311yescharacteristic
145C23×C4C2×Dic311yescharacteristic
146C23×C4C22×D311yescharacteristic
147C22×C8—62no—
148C22×C8—62no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4C3⋊C814yescomplemented
157C2×C42—22no—
158C2×C42—22no—
159C2×C42#6711yescharacteristic
160C2×C42#6711yescharacteristic
161C2×C42#6711yescharacteristic
162C2×C42#6711yescharacteristic
163C4⋊C8—64no—
164C22⋊C8—34no—
165C22⋊C8—62no—
166C22⋊C8—62no—
167C22⋊C8—62no—
168C4×C8—64no—
169#95—21no—
170#95—21no—
171#95—21no—
172#95—21no—
173#95—21no—
174#95—21no—
175#95—21no—
176#95—21no—
177C23×C3—21no—
178C23×C3—21no—
179C23×C3—21no—
180C23×C3—21no—
181C23×C3C2×C22⋊C411yescharacteristic
182C23×C3C4×D412yes—
183C23×C3C22×C811yescharacteristic
184C23×C3C2×M1611yescharacteristic
185C23×C3#13112yes—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—24no—
197C2×C3×C4—24no—
198C2×C3×C4—21no—
199C2×C3×C4—22no—
200C2×C3×C4—22no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—24no—
204C2×C3×C4—42no—
205C2×C3×C4—42no—
206C2×C3×C4—44no—
207C2×C3×C4—42no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—82no—
221C2×C3×C4—82no—
222C2×C3×C4C22⋊C812yes—
223C2×C3×C4C22⋊C812yes—
224C2×C3×C4#10112yes—
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12112yes—
227C2×C3×C4#12311yescharacteristic
228C3⋊C8—44no—
229C3⋊C8—82no—
230C24C22×Dic311yescharacteristic
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4C2×C3⋊C812yes—
255C22×C4C2×C3⋊C812yes—
256C22×C4C2×C3⋊C812yes—
257C22×C4#20812yes—
258C22×C4#21711yescharacteristic
259C22×C4#21711yescharacteristic
260C22×C4#21711yescharacteristic
261C22×C4#21711yescharacteristic
262C22×C4D3×D412yes—
263C22×C4#23712yes—
264C22×C4C2×#6711yescharacteristic
265C22×C4C2×#6711yescharacteristic
266C2×C8—64no—
267C2×C8—122no—
268C2×C8—122no—
269C2×C8—122no—
270C2×C8—122no—
271C2×C8—122no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—42no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—22no—
282C42—22no—
283C22×C3—22no—
284C22×C3—22no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—22no—
290C22×C3—22no—
291C22×C3#38611yescharacteristic
292C22×C3C2×C22⋊C811yescharacteristic
293C22×C3#40811yescharacteristic
294C22×C3#41011yescharacteristic
295C22×C3#41011yescharacteristic
296C22×C3C2×#10111yescharacteristic
297C22×C3#41311yescharacteristic
298C22×C3C8×D412yes—
299C22×C3#43512yes—
300C3×C4—22no—
301C3×C4—22no—
302C3×C4—21no—
303C3×C4—22no—
304C3×C4—21no—
305C3×C4—22no—
306C3×C4—42no—
307C3×C4—41no—
308C3×C4—42no—
309C3×C4—81no—
310C3×C4—81no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23C22×C3⋊C811yescharacteristic
316C23C2×#20811yescharacteristic
317C23D4×Dic312yes—
318C23#94812yes—
319C23C2×#21711yescharacteristic
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—24no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4#83012yes—
357C2×C4#83012yes—
358C2×C4#83612yes—
359C2×C4#93511yescharacteristic
360C2×C4#93711yescharacteristic
361C2×C4#93912yes—
362C8—124no—
363C8—242no—
364C2×C3—22no—
365C2×C3—22no—
366C2×C3#145811yescharacteristic
367C2×C3#146111yescharacteristic
368C2×C3#147011yescharacteristic
369C2×C3#147311yescharacteristic
370C2×C3#186911yescharacteristic
371C2×C3#189911yescharacteristic
372C2×C3#190111yescharacteristic
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22D4×C3⋊C812yes—
382C22#501112yes—
383C22C2×#83011yescharacteristic
384C22#522011yescharacteristic
385C22C2×#83611yescharacteristic
386C22#523411yescharacteristic
387C22#523711yescharacteristic
388C22#524011yescharacteristic
389C22#524011yescharacteristic
390C4—22no—
391C4—21no—
392C4—22no—
393C4—21no—
394C4—22no—
395C4—22no—
396C4—41no—
397C4—42no—
398C4—42no—
399C4—81no—
400C4—81no—
401C3#908211yescharacteristic, complemented, Sylow
402C2—22no—
403C2—22no—
404C2#7157111yescharacteristic
405C2#7161011yescharacteristic
406C2#7162411yescharacteristic
407C2#7169811yescharacteristic
408C2#7170511yescharacteristic
409C2#7173711yescharacteristic
410C2#7174511yescharacteristic
subgroupG/H
centerC23C2×#217
commutatorC23×C3C22×C8
FrattiniC23×C4C22×D3
FittingC2×C3×#387C2
socleC23×C3C2×C22⋊C4
prime radicalC24×C3C22×C4
cosocleC23×C3×C4C23
hypercenterC2×#387D3
hypocenterC3#9082
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#387

Complements

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

NH ≅ G/Ntype
45. C2×C3×C4⋊C4C8semidirect
156. C2×C4⋊C4C3⋊C8semidirect
401. C3#9082semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3872
C2×#49252
C2×#45282
#715312
#676132
#90823