Finite Group Databasebeta

← #10901362

Subgroups — #10901362

Order 768.

1068 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5470C211yesmaximal, characteristic
2#82076C211yesmaximal, characteristic
3C2×#4851C212yesmaximal, complemented
4C3×#1797C211yesmaximal, characteristic
5#70485C212yesmaximal, complemented
6#69936C214yesmaximal
7#68633C212yesmaximal, complemented
8#68631C212yesmaximal, complemented
9#29688—31nomaximal, Sylow
10C22×C3×C42C2211yescharacteristic
11C2×#926—21no—
12C2×#924—21no—
13C2×#924—21no—
14C2×#924C2212yes—
15#5470—28no—
16C2×#870—21no—
17C2×#869C2211yescharacteristic
18C2×#869C2211yescharacteristic
19C2×#869C2211yescharacteristic
20C2×C4×Dic6—21no—
21C2×C4×Dic6C2212yes—
22C2×C3×C4⋊C8C2212yes—
23C2×C3×C22⋊C8C2212yes—
24#5212—21no—
25#5212C2212yes—
26C2×#817C2214yes—
27C2×#816C2214yes—
28C2×#816C2214yes—
29#4935C2211yescharacteristic
30#4929—21no—
31#4928—21no—
32#4928C2212yes—
33#4851—28no—
34C3×#336C2212yes—
35#4551C2214yes—
36C2×#533—31no—
37C2×#475—32no—
38#2857—31no—
39#2058—32no—
40#2043—32no—
41#2035—34no—
42#1803—32no—
43#1797D311yescharacteristic
44C23×C3×C4D411yescharacteristic
45C23×C3×C4D411yescharacteristic
46C23×C3×C4C2311yescharacteristic
47C22×Dic6—21no—
48C22×Dic6—21no—
49C22×Dic6C2312yes—
50C22×C3×C8C2314yes—
51C2×C3×C42—24no—
52C2×C3×C42D411yescharacteristic
53C2×C3×C42D411yescharacteristic
54C2×C3×C42C2311yescharacteristic
55C2×C3×C42C2311yescharacteristic
56C2×#217—22no—
57C2×#217—22no—
58#926—28no—
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3—21no—
64C2×C4⋊Dic3—21no—
65C2×C4⋊Dic3C2312yes—
66C2×C4⋊Dic3C2312yes—
67C2×C4⋊Dic3C2312yes—
68#924—28no—
69#924—24no—
70#924—24no—
71#924—44no—
72C2×Dic3⋊C4—22no—
73C2×Dic3⋊C4—22no—
74C2×Dic3⋊C4—22no—
75C2×C4×Dic3—22no—
76#870—28no—
77#869—24no—
78#869—24no—
79#869—24no—
80C4×Dic6—28no—
81C4×Dic6—24no—
82C4×Dic6D418yes—
83C3×C4⋊C8—24no—
84C3×C22⋊C8—42no—
85C3⋊#95—22no—
86C3⋊#95—22no—
87#817—28no—
88#816—28no—
89#816—44no—
90#533—68no—
91C2×C4⋊Q8—31no—
92C2×C4⋊Q8—31no—
93C2×C4⋊Q8—31no—
94C2×#125—61no—
95C2×C22⋊Q8—32no—
96C2×C22⋊Q8—61no—
97C2×C22⋊Q8—61no—
98C2×C4×Q8—32no—
99C2×C4×Q8—61no—
100C2×#117—61no—
101C22×C42C2×D311yescharacteristic
102#475—68no—
103C2×#106—34no—
104C2×C4⋊C8C2×D312yes—
105C2×#103—34no—
106C2×#103—34no—
107C2×C22⋊C8C2×D312yes—
108#391—61no—
109#389—32no—
110#389—61no—
111#384—31no—
112#380—32no—
113#380—61no—
114#340—34no—
115#336C2×D312yes—
116C24×C3C2×D411yescharacteristic
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—22no—
125C22×C3×C4—41no—
126C22×C3×C4SD1614yes—
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2411yescharacteristic
133C22×Dic3—22no—
134C22×Dic3—22no—
135C22×Dic3—22no—
136C22×Dic3—22no—
137C2×Dic6—22no—
138C2×Dic6—22no—
139C2×Dic6—22no—
140C2×Dic6—22no—
141C2×Dic6—24no—
142C2×Dic6—22no—
143C2×Dic6—42no—
144C2×Dic6—42no—
145C2×Dic6—42no—
146C2×Dic6—42no—
147C2×Dic6—42no—
148C2×Dic6C2×D414yes—
149C2×C3×C8—24no—
150C2×C3×C8—24no—
151C2×C3×C8—24no—
152C3×C42—22no—
153C3×C42—22no—
154C3×C42—22no—
155C3×C42C2×D414yes—
156#217—28no—
157#217—44no—
158C4⋊Dic3—24no—
159C4⋊Dic3—24no—
160C4⋊Dic3—24no—
161C4⋊Dic3—22no—
162C4⋊Dic3—24no—
163C4⋊Dic3—22no—
164C4⋊Dic3—22no—
165C4⋊Dic3—42no—
166C4⋊Dic3—42no—
167C4⋊Dic3—42no—
168C4⋊Dic3C2×D414yes—
169Dic3⋊C4—28no—
170Dic3⋊C4—28no—
171Dic3⋊C4—44no—
172C4×Dic3—28no—
173C22×Q8—32no—
174C22×Q8—61no—
175C22×Q8—61no—
176C23×C4D1211yescharacteristic
177C23×C4D1211yescharacteristic
178C23×C4C22×D311yescharacteristic
179C22×C8C22×D314yes—
180C4⋊Q8—64no—
181C4⋊Q8—64no—
182C4⋊Q8—64no—
183#125—68no—
184C22⋊Q8—64no—
185C22⋊Q8—64no—
186C22⋊Q8—68no—
187C22⋊Q8—124no—
188C4×Q8—38no—
189C4×Q8—68no—
190C4×Q8—64no—
191#117—68no—
192C2×C4⋊C4—32no—
193C2×C4⋊C4—32no—
194C2×C4⋊C4—32no—
195C2×C4⋊C4—61no—
196C2×C4⋊C4—61no—
197C2×C4⋊C4—62no—
198C2×C4⋊C4—62no—
199C2×C4⋊C4—61no—
200C2×C4⋊C4—61no—
201C2×C4⋊C4—62no—
202C2×C4⋊C4—61no—
203C2×C4⋊C4—61no—
204C2×C22⋊C4—62no—
205C2×C22⋊C4—62no—
206C2×C42—24no—
207C2×C42—62no—
208C2×C42D1211yescharacteristic
209C2×C42D1211yescharacteristic
210C2×C42C22×D311yescharacteristic
211C2×C42C22×D311yescharacteristic
212#106—68no—
213C4⋊C8—24no—
214#103—68no—
215#103—124no—
216C22⋊C8—42no—
217#95—62no—
218#95—62no—
219C23×C3—21no—
220C23×C3—21no—
221C23×C3—21no—
222C23×C3—21no—
223C23×C3—21no—
224C23×C3—21no—
225C23×C3C2×SD1612yes—
226C23×C3C22×D411yescharacteristic
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—28no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—44no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4#12012yes—
241C2×C3×C4#12012yes—
242C2×C3×C4C2×SD1612yes—
243C2×C3×C4C2×SD1612yes—
244C2×C3×C4#13712yes—
245C2×C3×C4#13712yes—
246C2×C3×C4C22×D412yes—
247C2×C3×C4H(5,𝔽2)11yescharacteristic
248C2×C3×C42−1+411yescharacteristic
249C2×C3×C42−1+412yes—
250C2×Dic3—24no—
251C2×Dic3—24no—
252C2×Dic3—42no—
253C2×Dic3—42no—
254C2×Dic3—42no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3—42no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262Dic6—24no—
263Dic6—24no—
264Dic6—48no—
265Dic6—42no—
266Dic6—42no—
267Dic6—84no—
268C3×C8—44no—
269C24C2×D1211yescharacteristic
270C2×Q8—34no—
271C2×Q8—64no—
272C2×Q8—62no—
273C2×Q8—62no—
274C2×Q8—62no—
275C2×Q8—62no—
276C2×Q8—62no—
277C2×Q8—122no—
278C2×Q8—122no—
279C2×Q8—122no—
280C2×Q8—122no—
281C2×Q8—122no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—41no—
291C22×C4—62no—
292C22×C4—62no—
293C22×C4—62no—
294C22×C4—62no—
295C22×C4#20414yes—
296C22×C4C2×D1211yescharacteristic
297C22×C4C2×D1211yescharacteristic
298C22×C4C2×D1211yescharacteristic
299C22×C4C2×D1211yescharacteristic
300C22×C4C2×D1211yescharacteristic
301C22×C4C23×D311yescharacteristic
302C2×C8—24no—
303C2×C8—24no—
304C2×C8—24no—
305C4⋊C4—34no—
306C4⋊C4—64no—
307C4⋊C4—64no—
308C4⋊C4—68no—
309C4⋊C4—64no—
310C4⋊C4—68no—
311C4⋊C4—62no—
312C4⋊C4—64no—
313C4⋊C4—62no—
314C4⋊C4—62no—
315C4⋊C4—122no—
316C4⋊C4—122no—
317C4⋊C4—124no—
318C4⋊C4—122no—
319C22⋊C4—68no—
320C22⋊C4—124no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—68no—
325C42D3×D414yes—
326C22×C3—22no—
327C22×C3—22no—
328C22×C3—22no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3—41no—
332C22×C3—41no—
333C22×C3C2×#12011yescharacteristic
334C22×C3#53611yescharacteristic
335C22×C3#53711yescharacteristic
336C22×C3C22×SD1611yescharacteristic
337C22×C3C2×#13711yescharacteristic
338C22×C3C2×#13711yescharacteristic
339C22×C3#57511yescharacteristic
340C3×C4—22no—
341C3×C4—22no—
342C3×C4—22no—
343C3×C4—41no—
344C3×C4#44814yes—
345C3×C4#53612yes—
346C3×C4#53712yes—
347Dic3—44no—
348Dic3—44no—
349Dic3—82no—
350Dic3—82no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23C2×#20412yes—
358C23C22×D1211yescharacteristic
359Q8—64no—
360Q8—64no—
361Q8—128no—
362Q8—122no—
363Q8—122no—
364Q8—244no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—28no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—64no—
379C2×C4—64no—
380C2×C4—122no—
381C2×C4—122no—
382C2×C4—122no—
383C2×C4—122no—
384C2×C4—122no—
385C2×C4—122no—
386C2×C4—122no—
387C2×C4—122no—
388C2×C4—122no—
389C2×C4—122no—
390C2×C4#88212yes—
391C2×C4#88212yes—
392C2×C4C2×#20412yes—
393C2×C4C2×#20412yes—
394C2×C4#90912yes—
395C2×C4#90912yes—
396C2×C4C2×D3×D412yes—
397C2×C4#100411yescharacteristic
398C2×C4#100711yescharacteristic
399C2×C4#101012yes—
400C8—44no—
401C2×C3—42no—
402C2×C3#261611yescharacteristic
403C2×C3C2×#44811yescharacteristic
404C2×C3#300711yescharacteristic
405C2×C3#322911yescharacteristic
406C2×C3#323711yescharacteristic
407C2×C3#324611yescharacteristic
408C2×C3#325511yescharacteristic
409C22—22no—
410C22—22no—
411C22—22no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22C2×#88211yescharacteristic
417C22#551011yescharacteristic
418C22#552611yescharacteristic
419C22C22×#20411yescharacteristic
420C22C2×#90911yescharacteristic
421C22C2×#90911yescharacteristic
422C22#574811yescharacteristic
423C4—22no—
424C4—22no—
425C4—22no—
426C4—41no—
427C4—124no—
428C4—124no—
429C4—242no—
430C4—242no—
431C4#473814yes—
432C4#560812yes—
433C4#562812yes—
434C3#2968811yescharacteristic, complemented, Sylow
435C2—42no—
436C2#7395011yescharacteristic
437C2C2×#473811yescharacteristic
438C2#7864011yescharacteristic
439C2#7902711yescharacteristic
440C2#7906911yescharacteristic
441C2#7927111yescharacteristic
442C2#7932111yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1797C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1797D3
hypocenterC3#29688
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
3. C2×#4851C2semidirect
5. #70485C2semidirect
7. #68633C2semidirect
8. #68631C2semidirect
434. C3#29688semidirect

Maximal subgroups

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

subgroupindex
C2×#54702
#820762
C2×#48512
C3×#17972
#704852
#699362
#686332
#686312
#296883