Finite Group Databasebeta

← #10944187

Subgroups — #10944187

Order 768.

1334 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×2−1+4C211yesmaximal, characteristic
2C2×#5246C212yesmaximal
3#82533C212yesmaximal, complemented
4#82475C211yesmaximal, characteristic, complemented
5#71666C214yesmaximal, complemented
6#71660C212yesmaximal, complemented
7#71646C212yesmaximal, complemented
8#71637C211yesmaximal, characteristic, complemented
9#32923—31nomaximal, Sylow
10C2×C3×2−1+4C2211yescharacteristic
11C3×#520—22no—
12C3×#520—22no—
13C3×#520C2212yes—
14C3×#519—22no—
15C3×#519C2211yescharacteristic
16C3×C4×#41—22no—
17C3×C4×#41—22no—
18C3×C4×#41C2212yes—
19C2×C3×C4×Q8—22no—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×#951C2214yes—
22C2×#944C2212yes—
23C2×#869C2211yescharacteristic
24#5246—28no—
25C2×#837C2214yes—
26C2×#836C2212yes—
27#5031—22no—
28#5031—22no—
29#5019—22no—
30#5019—22no—
31#5004C2214yes—
32#4981—22no—
33#4973C2212yes—
34C4×C4⋊Dic3C2211yescharacteristic
35#4928—22no—
36#4924C2212yes—
37#4532C2214yes—
38#4530C2212yes—
39C4×2−1+4D311yescharacteristic
40#3039—32no—
41C2×#456—32no—
42#2546—31no—
43#1886—32no—
44#1882—34no—
45#1806—32no—
46#1801—31no—
47C3×2−1+4—24no—
48C3×2−1+4—22no—
49C3×2−1+4D414yes—
50C2×C3×#41—22no—
51C2×C3×#41—22no—
52C2×C3×#41C2312yes—
53C22×C3×Q8—22no—
54C22×C3×Q8C2311yescharacteristic
55C22×Dic6C2312yes—
56C3×C4×Q8—22no—
57C3×C4×Q8—24no—
58C3×C4×Q8—22no—
59C3×C4×Q8—22no—
60C3×C4×Q8—22no—
61C3×C4×Q8—22no—
62C3×C4×Q8—22no—
63C3×C4×Q8—22no—
64C3×C4×Q8D412yes—
65C3×C4×Q8D412yes—
66C3×C4×D4—22no—
67C3×C4×D4—24no—
68C3×C4×D4—22no—
69C3×C4×D4—22no—
70C3×C4×D4—22no—
71C3×C4×D4—24no—
72C3×C4×D4—22no—
73C3×C4×D4D412yes—
74C3×C4×D4D412yes—
75C3×#117—24no—
76C3×#117—24no—
77C3×#117—22no—
78C3×#117—22no—
79C3×#117—22no—
80C3×#117C2312yes—
81C2×C3×C4⋊C4—24no—
82C2×C3×C4⋊C4—22no—
83C2×C3×C4⋊C4D412yes—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C42—24no—
86C2×C3×C42—22no—
87C2×C3×C42D412yes—
88C2×C3×C42C2311yescharacteristic
89#951—28no—
90#951—28no—
91#944—24no—
92#944—24no—
93C2×#216—24no—
94C2×#214—24no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3—21no—
98C2×Dic3⋊C4—24no—
99C2×C4×Dic3C2312yes—
100C2×#208C2314yes—
101#869—22no—
102#869—22no—
103#837—28no—
104#836—24no—
105C3⋊#95—24no—
106C3⋊#95—21no—
107#810—24no—
108#810—24no—
109C4⋊(C3⋊C8)—22no—
110C4⋊(C3⋊C8)—22no—
111C2×2−1+4C2×D311yescharacteristic
112C2×#137—34no—
113C2×C4⋊Q8—32no—
114C2×C4⋊Q8—31no—
115#520—22no—
116#520—22no—
117#520C2×D312yes—
118#519—22no—
119#519C2×D311yescharacteristic
120C4×#41—22no—
121C4×#41—22no—
122C4×#41C2×D312yes—
123C2×C4×Q8—22no—
124C2×C4×Q8C2×D311yescharacteristic
125#462—62no—
126#462—62no—
127#461—62no—
128#461—62no—
129#456—68no—
130#423—32no—
131C2×#104—34no—
132#419—34no—
133C2×#101—32no—
134#398—62no—
135#391—32no—
136#389—62no—
137C4×C4⋊C4—31no—
138#343—32no—
139#339—34no—
140C3×#41—28no—
141C3×#41—28no—
142C3×#41—28no—
143C3×#41—28no—
144C3×#41—22no—
145C3×#41—22no—
146C3×#41C2×D414yes—
147C3×#41#4114yes—
148C2×C3×Q8—22no—
149C2×C3×Q8—28no—
150C2×C3×Q8—28no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8C2×D412yes—
154C2×C3×Q8C2×D412yes—
155C2×C3×Q8#4112yes—
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C2×C3×D4C2×D412yes—
159C22×C3×C4—24no—
160C22×C3×C4—22no—
161C22×C3×C4—24no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2411yescharacteristic
169C22×Dic3—22no—
170C22×Dic3—22no—
171C2×Dic6—24no—
172C2×Dic6—22no—
173C2×Dic6—22no—
174C2×Dic6—24no—
175C3×C4⋊C4—24no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—24no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4C2×D412yes—
183C3×C4⋊C4C2×D412yes—
184C3×C22⋊C4—24no—
185C3×C22⋊C4—22no—
186C3×C22⋊C4—22no—
187C3×C22⋊C4—22no—
188C3×C42—24no—
189C3×C42—24no—
190C3×C42—22no—
191C3×C42—22no—
192C3×C42—22no—
193C3×C42—22no—
194C3×C42—22no—
195C3×C42C2×D412yes—
196C3×C42C2×D412yes—
197#216—48no—
198#214—48no—
199C4⋊Dic3—24no—
200C4⋊Dic3—41no—
201C4⋊Dic3—42no—
202C4⋊Dic3—41no—
203Dic3⋊C4—48no—
204C4×Dic3—22no—
205C4×Dic3—22no—
206#208—24no—
207#208—24no—
208C2×C3⋊C8—24no—
2092−1+4—24no—
2102−1+4—22no—
2112−1+4D1214yes—
212C2×#41—22no—
213C2×#41—22no—
214C2×#41C22×D312yes—
215C22×Q8—22no—
216C22×Q8—32no—
217C22×Q8C22×D311yescharacteristic
218#137—68no—
219#137—68no—
220C2×Q16—64no—
221C2×SD16—64no—
222C2×M16—34no—
223C4⋊Q8—64no—
224C4⋊Q8—64no—
225C4⋊Q8—62no—
226C4⋊Q8—62no—
227C4×Q8—22no—
228C4×Q8—24no—
229C4×Q8—22no—
230C4×Q8—22no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×Q8—22no—
235C4×Q8#6712yes—
236C4×Q8#6712yes—
237C4×D4—22no—
238C4×D4—24no—
239C4×D4—22no—
240C4×D4—22no—
241C4×D4—22no—
242C4×D4—24no—
243C4×D4—22no—
244C4×D4#6712yes—
245C4×D4#6712yes—
246#117—24no—
247#117—24no—
248#117—22no—
249#117—22no—
250#117—22no—
251#117C22×D312yes—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4—61no—
256C2×C4⋊C4—64no—
257C2×C4⋊C4—61no—
258C2×C4⋊C4#6712yes—
259C2×C4⋊C4C22×D311yescharacteristic
260C2×C42—24no—
261C2×C42—22no—
262C2×C42—32no—
263C2×C42#6712yes—
264C2×C42C22×D311yescharacteristic
265C4⋊C8—62no—
266C4⋊C8—62no—
267#104—68no—
268#103—64no—
269#103—64no—
270#101—64no—
271#95—61no—
272#95—64no—
273C23×C3—22no—
274C23×C3C22×D411yescharacteristic
275C3×Q8—28no—
276C3×Q8—24no—
277C3×Q8—22no—
278C3×Q8—24no—
279C3×Q8#12114yes—
280C3×D4—28no—
281C3×D4—24no—
282C3×D4—24no—
283C3×D4—22no—
284C3×D4#12114yes—
285C2×C3×C4—216no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—24no—
290C2×C3×C4—24no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—41no—
296C2×C3×C4—44no—
297C2×C3×C4—42no—
298C2×C3×C4—82no—
299C2×C3×C4—84no—
300C2×C3×C4#12011yescharacteristic
301C2×C3×C4#12012yes—
302C2×C3×C4#12011yescharacteristic
303C2×C3×C4#12114yes—
304C2×C3×C4C22×D411yescharacteristic
305C2×C3×C4C22×D412yes—
306C2×C3×C4C2×#4112yes—
307C2×C3×C4C2×#4111yescharacteristic
308C2×Dic3—24no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314Dic6—44no—
315Dic6—44no—
316C3⋊C8—44no—
317#41—28no—
318#41—28no—
319#41—28no—
320#41—28no—
321#41—22no—
322#41—22no—
323#41C2×D1214yes—
324#41#23514yes—
325C2×Q8—22no—
326C2×Q8—28no—
327C2×Q8—28no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—62no—
331C2×Q8—62no—
332C2×Q8—64no—
333C2×Q8—64no—
334C2×Q8C2×D1212yes—
335C2×Q8#23512yes—
336C2×Q8C2×#6712yes—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4C2×#6712yes—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—62no—
346C22×C4—62no—
347C22×C4C2×#6712yes—
348C22×C4C2×#6711yescharacteristic
349C22×C4C2×#6712yes—
350C22×C4C2×#6711yescharacteristic
351C22×C4C23×D311yescharacteristic
352Q16—128no—
353SD16—128no—
354M16—64no—
355M16—64no—
356C2×C8—64no—
357C4⋊C4—24no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—24no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—64no—
365C4⋊C4—128no—
366C4⋊C4—121no—
367C4⋊C4—122no—
368C4⋊C4—121no—
369C4⋊C4C2×#6712yes—
370C4⋊C4C2×#6712yes—
371C22⋊C4—24no—
372C22⋊C4—22no—
373C22⋊C4—22no—
374C22⋊C4—22no—
375C42—24no—
376C42—24no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—62no—
383C42—62no—
384C42C2×#6712yes—
385C42C2×#6712yes—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—24no—
389C22×C3—42no—
390C22×C3—42no—
391C22×C3C2×#12011yescharacteristic
392C22×C3C2×#12111yescharacteristic
393C22×C3#52511yescharacteristic
394C3×C4—24no—
395C3×C4—24no—
396C3×C4—21no—
397C3×C4—22no—
398C3×C4—42no—
399C3×C4—42no—
400C3×C4—42no—
401C3×C4#45612yescomplemented
402C3×C4C2×#12112yes—
403C3×C4#52512yes—
404Dic3—44no—
405Dic3—82no—
406C23—22no—
407C23C22×#6711yescharacteristic
408Q8—28no—
409Q8—24no—
410Q8—24no—
411Q8—22no—
412Q8—124no—
413Q8—124no—
414Q8#93014yes—
415D4—28no—
416D4—24no—
417D4—24no—
418D4—22no—
419D4#93014yes—
420C2×C4—216no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—41no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—64no—
434C2×C4—82no—
435C2×C4—84no—
436C2×C4—122no—
437C2×C4—122no—
438C2×C4—122no—
439C2×C4—122no—
440C2×C4—122no—
441C2×C4#93014yes—
442C2×C4#95311yescharacteristic
443C2×C4#95312yes—
444C2×C4#95311yescharacteristic
445C2×C4C22×D1211yescharacteristic
446C2×C4C2×#23512yes—
447C2×C4C2×#23511yescharacteristic
448C2×C4C22×#6712yes—
449C8—124no—
450C2×C3—21no—
451C2×C3—21no—
452C2×C3—44no—
453C2×C3#284711yescharacteristic
454C2×C3C2×#45611yescharacteristic
455C2×C3#302211yescharacteristic
456C22—21no—
457C22—21no—
458C22—24no—
459C22—42no—
460C22—42no—
461C22C2×#93011yescharacteristic
462C22#579111yescharacteristic
463C22C2×#95311yescharacteristic
464C4—24no—
465C4—24no—
466C4—21no—
467C4—22no—
468C4—42no—
469C4—42no—
470C4—42no—
471C4—124no—
472C4—242no—
473C4#524612yescomplemented
474C4C2×#93012yes—
475C4#579112yes—
476C3#3292311yescharacteristic, complemented, Sylow
477C2—21no—
478C2—21no—
479C2—44no—
480C2#8403211yescharacteristic
481C2C2×#524611yescharacteristic
482C2#8412311yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×2−1+4C2
socleC22×C3C2×#120
prime radicalC2×C3×2−1+4C22
hypercenterC4×2−1+4D3
hypocenterC3#32923
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×2−1+4

Complements

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

NH ≅ G/Ntype
3. #82533C2semidirect
4. #82475C2semidirect
5. #71666C2semidirect
6. #71660C2semidirect
7. #71646C2semidirect
8. #71637C2semidirect
401. C3×C4#456semidirect
473. C4#5246semidirect
476. C3#32923semidirect

Maximal subgroups

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

subgroupindex
C3×C4×2−1+42
C2×#52462
#825332
#824752
#716662
#716602
#716462
#716372
#329233