Finite Group Databasebeta

← #10691615

Subgroups — #10691615

Order 768.

552 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#483C211yesmaximal, characteristic
2#82159C211yesmaximal, characteristic
3#71701C211yesmaximal, characteristic
4#71690C211yesmaximal, characteristic
5#71553C211yesmaximal, characteristic
6#67677C211yesmaximal, characteristic
7#67676C211yesmaximal, characteristic
8#9256—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×C3×C4⋊C4C2211yescharacteristic
11C2×#926—21no—
12C3×#483—21no—
13C3×#483—21no—
14C3×#483—22no—
15C2×C3×#106C412yescomplemented
16C2×C3×#103C412yescomplemented
17C2×C3×C22⋊C8C2211yescharacteristic
18#5212C2211yescharacteristic
19#5212C2211yescharacteristic
20#5211—21no—
21C4×#217C2211yescharacteristic
22C2×#830C2211yescharacteristic
23#4984—21no—
24#4980—22no—
25Dic3×C4⋊C4—21no—
26#4538—22no—
27#4529—22no—
28#4526—21no—
29#4526—21no—
30C2×#483D311yescharacteristic
31#2344—31no—
32#1996—31no—
33#1823—31no—
34#1520—31no—
35#1289—31no—
36#1282—31no—
37C22×C3×Q8C2×C411yescharacteristic
38C23×C3×C4—21no—
39C23×C3×C4C2311yescharacteristic
40C22×C3×C8C2×C412yes—
41C3×C22⋊Q8—21no—
42C3×C22⋊Q8—21no—
43C3×C22⋊Q8—21no—
44C3×C22⋊Q8—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4C2×C411yescharacteristic
50C2×C3×C4⋊C4C2×C412yes—
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C4⋊C4D411yescharacteristic
53C2×C3×C4⋊C4D412yes—
54C2×C3×C22⋊C4—21no—
55C2×#217—21no—
56C2×#217—21no—
57C2×#217D411yescharacteristic
58C2×#217D411yescharacteristic
59#926—42no—
60#926—42no—
61C2×C4⋊Dic3—22no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3—21no—
64C2×C4×Dic3—22no—
65C2×C4×Dic3—21no—
66C22×C3⋊C8—21no—
67C3×#106—22no—
68C3×#106—22no—
69C3×#103—22no—
70C3×#103—22no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3⋊#95—22no—
74C3⋊#95—22no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95D412yes—
78#830—21no—
79#830—21no—
80C2×C22⋊Q8C2×D311yescharacteristic
81C2×#117—61no—
82C22×C4⋊C4C2×D311yescharacteristic
83#483—22no—
84#483—21no—
85#483—21no—
86C2×#106Dic312yescomplemented
87C2×#103Dic312yescomplemented
88C2×C22⋊C8—31no—
89C2×C22⋊C8C2×D311yescharacteristic
90#389—61no—
91#387—62no—
92#381—61no—
93#380—31no—
94#380—31no—
95C4×C4⋊C4—61no—
96C4×C22⋊C4—31no—
97#342—61no—
98#342—61no—
99#340—62no—
100#328—62no—
101C24×C3C2×D411yescharacteristic
102C2×C3×Q8—21no—
103C2×C3×Q8—21no—
104C2×C3×Q8—42no—
105C22×C3×C4—22no—
106C22×C3×C4—21no—
107C22×C3×C4—21no—
108C22×C3×C4—22no—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4—42no—
112C22×C3×C4—41no—
113C22×C3×C4—41no—
114C22×C3×C4—41no—
115C22×C3×C4—41no—
116C22×C3×C4C22×C411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C22×C3×C4#4111yescharacteristic
122C22×Dic3—21no—
123C22×Dic3—21no—
124C22×Dic3—22no—
125C22×Dic3—41no—
126C2×C3×C8—22no—
127C2×C3×C8—42no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4C22⋊C412yes—
137C3×C4⋊C4C22⋊C412yes—
138C3×C22⋊C4—41no—
139C3×C22⋊C4—41no—
140#217—21no—
141#217—21no—
142#217—21no—
143#217—41no—
144#217—41no—
145#217—41no—
146#217—41no—
147#217SD1612yes—
148C4⋊Dic3—44no—
149C4⋊Dic3—42no—
150C4⋊Dic3—42no—
151C4×Dic3—44no—
152C4×Dic3—42no—
153C2×C3⋊C8—42no—
154C2×C3⋊C8—41no—
155C22×Q8C2×Dic311yescharacteristic
156C23×C4—21no—
157C23×C4C22×D311yescharacteristic
158C22×C8—61no—
159C22×C8C2×Dic312yes—
160C22⋊Q8—21no—
161C22⋊Q8—21no—
162C22⋊Q8—21no—
163C22⋊Q8—21no—
164#117—122no—
165#117—122no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—62no—
171C2×C4⋊C4—61no—
172C2×C4⋊C4—61no—
173C2×C4⋊C4C2×Dic311yescharacteristic
174C2×C4⋊C4C2×Dic312yes—
175C2×C4⋊C4#6712yes—
176C2×C4⋊C4#6711yescharacteristic
177C2×C4⋊C4#6711yescharacteristic
178C2×C22⋊C4—21no—
179C2×C22⋊C4—31no—
180C2×C22⋊C4—31no—
181C2×C22⋊C4—61no—
182C2×C22⋊C4—61no—
183C2×C42—62no—
184C2×C42—61no—
185#106—22no—
186#106—22no—
187#103—22no—
188#103—22no—
189C22⋊C8—21no—
190C22⋊C8—21no—
191C22⋊C8—61no—
192C22⋊C8—61no—
193#95—32no—
194#95—62no—
195#95—62no—
196#95—61no—
197#95—61no—
198C23×C3—41no—
199C23×C3—41no—
200C23×C3C4×D411yescharacteristic
201C23×C3#12011yescharacteristic
202C23×C3#12111yescharacteristic
203C23×C3C2×SD1611yescharacteristic
204C23×C3#13511yescharacteristic
205C23×C3AGL(1,ℤ/8)11yescharacteristic
206C23×C3#13711yescharacteristic
207C3×Q8—42no—
208C3×Q8—81no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—42no—
224C2×C3×C4—42no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—81no—
228C2×C3×C4—81no—
229C2×C3×C4C2×C22⋊C411yescharacteristic
230C2×C3×C4C4×D411yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12311yescharacteristic
233C2×Dic3—22no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237C2×Dic3—42no—
238C2×Dic3—41no—
239C2×Dic3—42no—
240C2×Dic3—42no—
241C2×Dic3—41no—
242C2×Dic3—81no—
243C2×Dic3—81no—
244C2×Dic3—81no—
245C3×C8—42no—
246C3⋊C8—81no—
247C24D3×D411yescharacteristic
248C2×Q8—21no—
249C2×Q8—21no—
250C2×Q8—42no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—61no—
263C22×C4—62no—
264C22×C4—61no—
265C22×C4—121no—
266C22×C4D3×D411yescharacteristic
267C22×C4#23711yescharacteristic
268C22×C4#23711yescharacteristic
269C22×C4C22×Dic311yescharacteristic
270C22×C4C2×#6711yescharacteristic
271C22×C4C2×#6711yescharacteristic
272C2×C8—22no—
273C2×C8—42no—
274C2×C8—122no—
275C2×C8—121no—
276C4⋊C4—22no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—22no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—122no—
285C4⋊C4—122no—
286C4⋊C4—124no—
287C4⋊C4#21712yes—
288C4⋊C4#21712yes—
289C22⋊C4—32no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—61no—
293C22⋊C4—61no—
294C22⋊C4—61no—
295C22⋊C4—121no—
296C22⋊C4—121no—
297C22⋊C4—121no—
298C22⋊C4—121no—
299C42—122no—
300C42—124no—
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—41no—
308C22×C3—41no—
309C22×C3—41no—
310C22×C3—41no—
311C22×C3#38611yescharacteristic
312C22×C3C4×SD1611yescharacteristic
313C22×C3#44011yescharacteristic
314C22×C3#45011yescharacteristic
315C22×C3#45211yescharacteristic
316C22×C3#46111yescharacteristic
317C22×C3#46311yescharacteristic
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—42no—
322C3×C4—42no—
323C3×C4—81no—
324C3×C4—81no—
325C3×C4#41012yes—
326Dic3—42no—
327Dic3—82no—
328Dic3—81no—
329Dic3—161no—
330C23—41no—
331C23—41no—
332C23D3×SD1611yescharacteristic
333C23#91411yescharacteristic
334C23#91511yescharacteristic
335C23#91611yescharacteristic
336C23D4×Dic311yescharacteristic
337C23#93711yescharacteristic
338C23#93911yescharacteristic
339Q8—42no—
340Q8—81no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—62no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—121no—
363C2×C4—121no—
364C2×C4—121no—
365C2×C4—122no—
366C2×C4—122no—
367C2×C4—121no—
368C2×C4—122no—
369C2×C4—121no—
370C2×C4—241no—
371C2×C4—241no—
372C2×C4—241no—
373C2×C4D4×Dic311yescharacteristic
374C2×C4#93511yescharacteristic
375C2×C4#93911yescharacteristic
376C2×C4C2×#21711yescharacteristic
377C8—42no—
378C8—241no—
379C2×C3—21no—
380C2×C3—21no—
381C2×C3—21no—
382C2×C3—21no—
383C2×C3—21no—
384C2×C3—21no—
385C2×C3#187711yescharacteristic
386C2×C3#189911yescharacteristic
387C2×C3#190511yescharacteristic
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22Dic3×SD1611yescharacteristic
399C22#516211yescharacteristic
400C22#516411yescharacteristic
401C22#516511yescharacteristic
402C22#516911yescharacteristic
403C22#517311yescharacteristic
404C22#522011yescharacteristic
405C4—21no—
406C4—21no—
407C4—21no—
408C4—42no—
409C4—42no—
410C4—81no—
411C4—81no—
412C4—122no—
413C4—242no—
414C4—241no—
415C4—481no—
416C4#524012yes—
417C3#925611yescharacteristic, complemented, Sylow
418C2—21no—
419C2—21no—
420C2—21no—
421C2—21no—
422C2—21no—
423C2—21no—
424C2#7170511yescharacteristic
425C2#7204711yescharacteristic
426C2#7210111yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#483C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenterC2×#483D3
hypocenterC3#9256
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#483

Complements

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

NH ≅ G/Ntype
15. C2×C3×#106C4semidirect
16. C2×C3×#103C4semidirect
86. C2×#106Dic3semidirect
87. C2×#103Dic3semidirect
417. C3#9256semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4832
#821592
#717012
#716902
#715532
#676772
#676762
#92563