Finite Group Databasebeta

← #10901302

Subgroups — #10901302

Order 768.

1038 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3461C211yesmaximal, characteristic
2#82533C212yesmaximal, complemented
3#82475C211yesmaximal, characteristic, complemented
4#71905C212yesmaximal, complemented
5#71881C212yesmaximal
6#71868C214yesmaximal, complemented
7#71646C212yesmaximal, complemented
8#71637C211yesmaximal, characteristic, complemented
9#29681—31nomaximal, Sylow
10C3×#544C2211yescharacteristic
11C3×#542—22no—
12C3×#541—22no—
13C3×#539—22no—
14C3×#537C2211yescharacteristic
15C3×#536C2212yes—
16C3×#533—22no—
17C3×#529—22no—
18C3×#529—22no—
19C3×C4×#41C2212yes—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×#951C2214yes—
22C2×#944C2212yes—
23C2×#869C2211yescharacteristic
24C2×#837C2214yes—
25C2×#836C2212yes—
26#5031—22no—
27#5031—22no—
28#5019—22no—
29#5019—22no—
30#5004C2214yes—
31#4981—22no—
32#4973C2212yes—
33C4×C4⋊Dic3C2211yescharacteristic
34#4928—22no—
35#4924C2212yes—
36#4532C2214yes—
37#4530C2212yes—
38#3461D311yescharacteristic
39#3039—32no—
40#2546—31no—
41#2028—32no—
42#2007—32no—
43#2005—34no—
44#1806—32no—
45#1801—31no—
46C2×C3×#41C2312yes—
47C22×C3×Q8C2311yescharacteristic
48C22×Dic6C2312yes—
49C3×C4⋊Q8—22no—
50C3×C4⋊Q8—22no—
51C3×#126—24no—
52C3×#126—24no—
53C3×#125—22no—
54C3×#125—24no—
55C3×#125—22no—
56C3×#124—22no—
57C3×#124—22no—
58C3×#123—24no—
59C3×#123—24no—
60C3×C22⋊Q8—24no—
61C3×C22⋊Q8—22no—
62C3×C22⋊Q8—22no—
63C3×C22⋊Q8—22no—
64C3×C22⋊Q8—22no—
65C3×#121—22no—
66C3×#121—22no—
67C3×C4×Q8—22no—
68C3×C4×Q8—22no—
69C3×C4×Q8—22no—
70C3×C4×Q8D412yes—
71C3×C4×Q8D412yes—
72C3×C4×D4—22no—
73C3×C4×D4—22no—
74C3×C4×D4—22no—
75C3×C4×D4D412yes—
76C3×C4×D4D412yes—
77C3×#117—24no—
78C3×#117—22no—
79C3×#117C2312yes—
80C2×C3×C4⋊C4D412yes—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C42—22no—
83C2×C3×C42D412yes—
84C2×C3×C42C2311yescharacteristic
85#951—44no—
86#951—44no—
87#944—44no—
88C2×#216—24no—
89C2×#214—24no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3—21no—
93C2×Dic3⋊C4—24no—
94C2×C4×Dic3C2312yes—
95C2×#208C2314yes—
96#869—22no—
97#869—22no—
98#837—44no—
99#836—42no—
100C3⋊#95—24no—
101C3⋊#95—21no—
102#810—24no—
103#810—24no—
104C4⋊(C3⋊C8)—22no—
105C4⋊(C3⋊C8)—22no—
106C2×#137—34no—
107#544C2×D311yescharacteristic
108#542—22no—
109#541—22no—
110#539—22no—
111#537C2×D311yescharacteristic
112#536C2×D312yes—
113#533—22no—
114C2×C4⋊Q8—32no—
115C2×C4⋊Q8—31no—
116#529—22no—
117#529—22no—
118C4×#41C2×D312yes—
119C2×C4×Q8C2×D311yescharacteristic
120#462—62no—
121#462—62no—
122#461—62no—
123#461—62no—
124#423—32no—
125C2×#104—34no—
126#419—34no—
127C2×#101—32no—
128#398—62no—
129#391—32no—
130#389—62no—
131C4×C4⋊C4—31no—
132#343—32no—
133#339—34no—
134C3×#41—22no—
135C3×#41—22no—
136C3×#41—22no—
137C3×#41—22no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8—21no—
141C2×C3×Q8—22no—
142C2×C3×Q8—21no—
143C2×C3×Q8C2×D412yes—
144C2×C3×D4—22no—
145C2×C3×D4—22no—
146C2×C3×D4C2×D412yes—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2411yescharacteristic
155C22×Dic3—22no—
156C22×Dic3—22no—
157C2×Dic6—24no—
158C2×Dic6—24no—
159C2×Dic6—22no—
160C2×Dic6—22no—
161C3×C4⋊C4—24no—
162C3×C4⋊C4—24no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—22no—
165C3×C4⋊C4—24no—
166C3×C4⋊C4—24no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4C2×D412yes—
171C3×C4⋊C4C2×D412yes—
172C3×C22⋊C4—24no—
173C3×C22⋊C4—24no—
174C3×C22⋊C4—24no—
175C3×C22⋊C4—22no—
176C3×C22⋊C4—22no—
177C3×C22⋊C4—22no—
178C3×C42—22no—
179C3×C42—22no—
180C3×C42—22no—
181C3×C42—22no—
182C3×C42—22no—
183C3×C42C2×D412yes—
184C3×C42C2×D412yes—
185#216—48no—
186#214—48no—
187C4⋊Dic3—24no—
188C4⋊Dic3—41no—
189C4⋊Dic3—42no—
190C4⋊Dic3—41no—
191Dic3⋊C4—48no—
192C4×Dic3—24no—
193#208—24no—
194#208—24no—
195C2×C3⋊C8—24no—
196C2×#41C22×D312yes—
197C22×Q8—32no—
198C22×Q8C22×D311yescharacteristic
199#137—124no—
200#137—124no—
201C2×Q16—64no—
202C2×SD16—64no—
203C2×M16—34no—
204C4⋊Q8—22no—
205C4⋊Q8—22no—
206C4⋊Q8—62no—
207C4⋊Q8—62no—
208C4⋊Q8—124no—
209#126—24no—
210#126—24no—
211#125—22no—
212#125—24no—
213#125—22no—
214#124—22no—
215#124—22no—
216#123—24no—
217#123—24no—
218C22⋊Q8—24no—
219C22⋊Q8—22no—
220C22⋊Q8—22no—
221C22⋊Q8—22no—
222C22⋊Q8—22no—
223#121—22no—
224#121—22no—
225C4×Q8—22no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8#6712yes—
229C4×Q8#6712yes—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—22no—
233C4×D4#6712yes—
234C4×D4#6712yes—
235#117—24no—
236#117—22no—
237#117C22×D312yes—
238C2×C4⋊C4—61no—
239C2×C4⋊C4—61no—
240C2×C4⋊C4—64no—
241C2×C4⋊C4—61no—
242C2×C4⋊C4#6712yes—
243C2×C4⋊C4C22×D311yescharacteristic
244C2×C42—22no—
245C2×C42—32no—
246C2×C42#6712yes—
247C2×C42C22×D311yescharacteristic
248C4⋊C8—62no—
249C4⋊C8—62no—
250#104—124no—
251#103—64no—
252#103—64no—
253#101—122no—
254#95—61no—
255#95—64no—
256C23×C3—22no—
257C23×C3C22×D411yescharacteristic
258C3×Q8—22no—
259C3×Q8—22no—
260C3×Q8—42no—
261C3×Q8—42no—
262C3×Q8—82no—
263C3×D4—22no—
264C3×D4—22no—
265C3×D4—42no—
266C3×D4—42no—
267C3×D4—82no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—41no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—44no—
280C2×C3×C4—42no—
281C2×C3×C4—82no—
282C2×C3×C4#12011yescharacteristic
283C2×C3×C4#12012yes—
284C2×C3×C4#12011yescharacteristic
285C2×C3×C4C22×D412yes—
286C2×C3×C4H(5,𝔽2)11yescharacteristic
287C2×C3×C4H(5,𝔽2)12yes—
288C2×C3×C42−1+411yescharacteristic
289C2×Dic3—24no—
290C2×Dic3—42no—
291C2×Dic3—42no—
292C2×Dic3—42no—
293C2×Dic3—42no—
294C2×Dic3—42no—
295Dic6—44no—
296Dic6—44no—
297C3⋊C8—44no—
298#41—22no—
299#41—22no—
300#41—22no—
301#41—22no—
302C2×Q8—22no—
303C2×Q8—21no—
304C2×Q8—22no—
305C2×Q8—21no—
306C2×Q8—22no—
307C2×Q8—64no—
308C2×Q8—62no—
309C2×Q8—62no—
310C2×Q8—64no—
311C2×Q8C2×#6712yes—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4C2×#6712yes—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—62no—
319C22×C4—62no—
320C22×C4C2×#6712yes—
321C22×C4C2×#6711yescharacteristic
322C22×C4C2×#6712yes—
323C22×C4C2×#6711yescharacteristic
324C22×C4C23×D311yescharacteristic
325Q16—128no—
326SD16—128no—
327M16—64no—
328M16—64no—
329C2×C8—64no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—64no—
340C4⋊C4—128no—
341C4⋊C4—121no—
342C4⋊C4—122no—
343C4⋊C4—121no—
344C4⋊C4C2×#6712yes—
345C4⋊C4C2×#6712yes—
346C22⋊C4—24no—
347C22⋊C4—24no—
348C22⋊C4—24no—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C22⋊C4—22no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—64no—
358C42C2×#6712yes—
359C42C2×#6712yes—
360C22×C3—21no—
361C22×C3—21no—
362C22×C3—42no—
363C22×C3—42no—
364C22×C3—42no—
365C22×C3C2×#12011yescharacteristic
366C22×C3#53511yescharacteristic
367C22×C3#53711yescharacteristic
368C3×C4—21no—
369C3×C4—22no—
370C3×C4—42no—
371C3×C4—42no—
372C3×C4—42no—
373C3×C4—42no—
374C3×C4—42no—
375C3×C4#45612yes—
376C3×C4#53512yes—
377C3×C4#53712yes—
378Dic3—44no—
379Dic3—82no—
380C23—22no—
381C23C22×#6711yescharacteristic
382Q8—22no—
383Q8—22no—
384Q8—42no—
385Q8—42no—
386Q8—82no—
387Q8—124no—
388Q8—124no—
389D4—22no—
390D4—22no—
391D4—42no—
392D4—42no—
393D4—82no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—41no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—64no—
408C2×C4—82no—
409C2×C4—122no—
410C2×C4—122no—
411C2×C4—122no—
412C2×C4—122no—
413C2×C4—122no—
414C2×C4#95311yescharacteristic
415C2×C4#95312yes—
416C2×C4#95311yescharacteristic
417C2×C4#100911yescharacteristic
418C2×C4#100912yes—
419C2×C4#101011yescharacteristic
420C2×C4C22×#6712yes—
421C8—124no—
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—82no—
425C2×C3#261511yescharacteristic
426C2×C3C2×#45611yescharacteristic
427C2×C3#302211yescharacteristic
428C22—21no—
429C22—21no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22#583011yescharacteristic
434C22#583311yescharacteristic
435C22C2×#95311yescharacteristic
436C4—21no—
437C4—22no—
438C4—42no—
439C4—42no—
440C4—42no—
441C4—42no—
442C4—42no—
443C4—124no—
444C4—242no—
445C4#524612yes—
446C4#583012yes—
447C4#583312yes—
448C3#2968111yescharacteristic, complemented, Sylow
449C2—21no—
450C2—21no—
451C2—82no—
452C2#8409811yescharacteristic
453C2C2×#524611yescharacteristic
454C2#8412311yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3461C2
socleC22×C3C2×#120
prime radicalC3×#537C22
hypercenter#3461D3
hypocenterC3#29681
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3461

Complements

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

NH ≅ G/Ntype
2. #82533C2semidirect
3. #82475C2semidirect
4. #71905C2semidirect
6. #71868C2semidirect
7. #71646C2semidirect
8. #71637C2semidirect
448. C3#29681semidirect

Maximal subgroups

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

subgroupindex
C3×#34612
#825332
#824752
#719052
#718812
#718682
#716462
#716372
#296813