Finite Group Databasebeta

← #10831755

Subgroups — #10831755

Order 768.

1064 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
1C2×C3×#554C211yesmaximal, characteristic
2C2×#5094C211yesmaximal, characteristic, complemented
3#82480C211yesmaximal, characteristic, complemented
4C2×#5018C211yesmaximal, characteristic
5#71788C211yesmaximal, characteristic, complemented
6#71744C212yesmaximal, complemented
7#71739C212yesmaximal, complemented
8#71635C211yesmaximal, characteristic, complemented
9#71612C212yesmaximal
10#71607C212yesmaximal
11#71605C211yesmaximal, characteristic
12#24343—31nomaximal, Sylow
13C3×#554—24no—
14C3×#554—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#125C2212yes—
17C2×C3×C22⋊Q8—21no—
18C2×C3×C22⋊Q8—22no—
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×D4C2212yes—
21C2×C3×C4×D4C2211yescharacteristic
22C22×C3×C4⋊C4—21no—
23C22×#214C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25#5230—21no—
26C2×#832C2211yescharacteristic
27C2×#832C2212yes—
28#5094—24no—
29#5018—24no—
30#4985—21no—
31#4978—22no—
32C2×#810C2212yes—
33C2×#810C2211yescharacteristic
34C2×#808C2212yes—
35C2×#807C2211yescharacteristic
36#4936C2211yescharacteristic
37#4929C2212yes—
38#4928C2212yes—
39C2×C4⋊(C3⋊C8)C2211yescharacteristic
40C2×#803C2211yescharacteristic
41#4529C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#554D311yescharacteristic
47C2×#475—31no—
48C2×#440—31no—
49#2581—31no—
50#2059—32no—
51#2026—32no—
52#1993—31no—
53#1985—31no—
54#1877—32no—
55#1864—32no—
56#1804—31no—
57C22×C3×Q8—21no—
58C22×C3×Q8D411yescharacteristic
59C22×C3×D4C2311yescharacteristic
60C23×C3×C4—21no—
61C23×C3×C4—22no—
62C22×Dic6C2311yescharacteristic
63C3×C4⋊Q8—22no—
64C3×C4⋊Q8D414yes—
65C3×#125—24no—
66C3×#125—24no—
67C3×C22⋊Q8—28no—
68C3×C22⋊Q8—216no—
69C3×C4×Q8—22no—
70C3×C4×Q8—22no—
71C3×C4×D4—24no—
72C3×C4×D4—24no—
73C3×C4×D4—22no—
74C3×C4×D4D414yes—
75C2×C3×C4⋊C4—28no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4D412yes—
80C2×C3×C4⋊C4D411yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—21no—
85C2×C3×C42C2312yes—
86C2×C3×C42C2311yescharacteristic
87C2×#214—22no—
88C2×#214—22no—
89C2×#214Q814yescomplemented
90C2×C4⋊Dic3C2311yescharacteristic
91C2×C4⋊Dic3C2312yes—
92C2×Dic3⋊C4—22no—
93C2×Dic3⋊C4—21no—
94C2×C4×Dic3—21no—
95C22×C3⋊C8C2311yescharacteristic
96C22×C3⋊C8C2312yes—
97C22×C3⋊C8C2311yescharacteristic
98C4×Dic6—22no—
99C4×Dic6—22no—
100#832—22no—
101#832—24no—
102C3⋊#95—22no—
103C3⋊#95—22no—
104C3⋊#95—22no—
105C3⋊#95—21no—
106C3⋊#95—21no—
107#810—22no—
108#810—24no—
109#808—24no—
110#807—22no—
111C4⋊(C3⋊C8)—22no—
112#803—22no—
113C22×SD16—31no—
114#554—24no—
115#554—24no—
116C2×C4⋊Q8C2×D311yescharacteristic
117C2×#125C2×D312yes—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—22no—
120C2×C4×Q8—31no—
121C2×C4×Q8C2×D311yescharacteristic
122C2×C4×D4C2×D312yes—
123C2×C4×D4C2×D311yescharacteristic
124C22×C4⋊C4—21no—
125#475—64no—
126#440—64no—
127C2×#107—31no—
128C2×#106—32no—
129C2×C4⋊C8—31no—
130C2×#103—32no—
131C2×#103—31no—
132C2×#102—32no—
133C2×#102—31no—
134C2×#97—31no—
135#400—61no—
136#395—61no—
137#391—32no—
138#389—32no—
139#387—62no—
140#382—31no—
141#340—32no—
142#340—32no—
143#340—32no—
144#336—32no—
145#336—32no—
146C24×C3—21no—
147C2×C3×Q8—22no—
148C2×C3×Q8—24no—
149C2×C3×Q8—42no—
150C2×C3×Q8—82no—
151C2×C3×D4—21no—
152C2×C3×D4—21no—
153C2×C3×D4—21no—
154C2×C3×D4C2×D412yes—
155C2×C3×D4C2×Q812yes—
156C2×C3×D4#4112yes—
157C22×C3×C4—21no—
158C22×C3×C4—22no—
159C22×C3×C4—28no—
160C22×C3×C4—24no—
161C22×C3×C4—44no—
162C22×C3×C4—84no—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2411yescharacteristic
168C22×Dic3—21no—
169C22×Dic3—22no—
170C22×Dic3—21no—
171C2×Dic6—21no—
172C2×Dic6—21no—
173C2×Dic6—21no—
174C2×Dic6—21no—
175C2×Dic6—21no—
176C2×Dic6C2×Q812yes—
177C3×C4⋊C4—28no—
178C3×C4⋊C4—28no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—24no—
181C3×C4⋊C4—24no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4C2×D412yes—
185C3×C4⋊C4C2×D414yes—
186C3×C22⋊C4—28no—
187C3×C22⋊C4—24no—
188C3×C42—22no—
189C3×C42—22no—
190C3×C42C2×D412yes—
191C3×C42C2×D412yes—
192#214—28no—
193C4⋊Dic3—21no—
194C4⋊Dic3—21no—
195C4⋊Dic3—22no—
196C4⋊Dic3—22no—
197Dic3⋊C4—44no—
198Dic3⋊C4—42no—
199C4×Dic3—42no—
200C2×C3⋊C8—21no—
201C2×C3⋊C8—21no—
202C2×C3⋊C8—21no—
203C2×C3⋊C8—22no—
204C2×C3⋊C8—22no—
205C2×C3⋊C8—21no—
206C2×C3⋊C8—21no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8C2×Q812yes—
209C22×Q8—21no—
210C22×Q8—31no—
211C22×Q8#6711yescharacteristic
212C22×D4C22×D311yescharacteristic
213C23×C4—21no—
214C23×C4—22no—
215C2×SD16—34no—
216C2×SD16—62no—
217C2×SD16—62no—
218C22×C8—31no—
219C22×C8—31no—
220C22×C8—32no—
221C4⋊Q8—22no—
222C4⋊Q8#6714yes—
223#125—24no—
224#125—24no—
225C22⋊Q8—28no—
226C22⋊Q8—216no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×Q8—62no—
230C4×Q8—62no—
231C4×D4—24no—
232C4×D4—24no—
233C4×D4—22no—
234C4×D4#6714yes—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—31no—
240C2×C4⋊C4—32no—
241C2×C4⋊C4—61no—
242C2×C4⋊C4—62no—
243C2×C4⋊C4#6711yescharacteristic
244C2×C4⋊C4#6712yes—
245C2×C4⋊C4C22×D311yescharacteristic
246C2×C4⋊C4C22×D312yes—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—21no—
249C2×C42—61no—
250C2×C42C22×D312yes—
251C2×C42C22×D311yescharacteristic
252#107—62no—
253#106—64no—
254C4⋊C8—62no—
255#103—62no—
256#103—64no—
257#102—64no—
258#102—62no—
259#97—62no—
260#95—61no—
261#95—61no—
262#95—62no—
263#95—62no—
264#95—62no—
265C23×C3—22no—
266C23×C3—22no—
267C23×C3—22no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3C22×D411yescharacteristic
271C3×Q8—44no—
272C3×Q8—84no—
273C3×D4—22no—
274C3×D4—22no—
275C3×D4—22no—
276C3×D4C22⋊Q814yes—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—24no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—24no—
286C2×C3×C4—48no—
287C2×C3×C4—42no—
288C2×C3×C4—44no—
289C2×C3×C4—88no—
290C2×C3×C4#12012yes—
291C2×C3×C4#12012yes—
292C2×C3×C4C22×D411yescharacteristic
293C2×C3×C4C22×D411yescharacteristic
294C2×C3×C4C22×Q811yescharacteristic
295C2×C3×C4C2×#4111yescharacteristic
296C2×C3×C42−1+411yescharacteristic
297C2×C3×C42−1+411yescharacteristic
298C2×Dic3—22no—
299C2×Dic3—41no—
300C2×Dic3—41no—
301C2×Dic3—41no—
302C2×Dic3—42no—
303C2×Dic3—42no—
304C2×Dic3—42no—
305C2×Dic3—42no—
306Dic6—22no—
307Dic6—22no—
308Dic6—41no—
309Dic6—41no—
310C3⋊C8—22no—
311C3⋊C8—41no—
312C3⋊C8—42no—
313C24—21no—
314C2×Q8—22no—
315C2×Q8—24no—
316C2×Q8—32no—
317C2×Q8—42no—
318C2×Q8—61no—
319C2×Q8—61no—
320C2×Q8—61no—
321C2×Q8—61no—
322C2×Q8—61no—
323C2×Q8—82no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4D3×Q812yes—
328C2×D4#23912yes—
329C2×D4C2×#6712yes—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—28no—
333C22×C4—24no—
334C22×C4—44no—
335C22×C4—61no—
336C22×C4—62no—
337C22×C4—61no—
338C22×C4—84no—
339C22×C4C2×#6712yes—
340C22×C4C2×#6712yes—
341C22×C4C2×#6711yescharacteristic
342C22×C4C2×#6711yescharacteristic
343C22×C4C23×D311yescharacteristic
344SD16—68no—
345C2×C8—32no—
346C2×C8—61no—
347C2×C8—61no—
348C2×C8—62no—
349C2×C8—61no—
350C2×C8—61no—
351C2×C8—62no—
352C2×C8—61no—
353C2×C8—62no—
354C4⋊C4—28no—
355C4⋊C4—28no—
356C4⋊C4—22no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—21no—
360C4⋊C4—22no—
361C4⋊C4—62no—
362C4⋊C4—62no—
363C4⋊C4—61no—
364C4⋊C4—61no—
365C4⋊C4—124no—
366C4⋊C4—122no—
367C4⋊C4C2×#6714yes—
368C4⋊C4C2×#6712yes—
369C22⋊C4—28no—
370C22⋊C4—24no—
371C42—22no—
372C42—22no—
373C42—122no—
374C42C2×#6712yes—
375C42C2×#6712yes—
376C22×C3—24no—
377C22×C3—44no—
378C22×C3—44no—
379C22×C3—42no—
380C22×C3—42no—
381C22×C3C2×#12011yescharacteristic
382C22×C3#54811yescharacteristic
383C22×C3D4×Q811yescharacteristic
384C22×C3#57512yes—
385C22×C3#57712yes—
386C3×C4—22no—
387C3×C4—22no—
388C3×C4—42no—
389C3×C4—42no—
390C3×C4—42no—
391C3×C4—44no—
392C3×C4C2×C22⋊Q811yescharacteristic
393C3×C4#53611yescharacteristic
394C3×C4#54811yescharacteristic
395C3×C4D4×Q811yescharacteristic
396Dic3—42no—
397Dic3—82no—
398Dic3—81no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—42no—
403C23—42no—
404C23C22×#6711yescharacteristic
405Q8—44no—
406Q8—62no—
407Q8—62no—
408Q8—84no—
409Q8—121no—
410Q8—121no—
411D4—22no—
412D4—22no—
413D4—22no—
414D4#94614yes—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—48no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—62no—
428C2×C4—88no—
429C2×C4—121no—
430C2×C4—121no—
431C2×C4—122no—
432C2×C4—122no—
433C2×C4—122no—
434C2×C4—122no—
435C2×C4—121no—
436C2×C4#95312yes—
437C2×C4#95312yes—
438C2×C4C2×D3×Q811yescharacteristic
439C2×C4C2×#23911yescharacteristic
440C2×C4#100711yescharacteristic
441C2×C4#100711yescharacteristic
442C2×C4C22×#6711yescharacteristic
443C2×C4C22×#6711yescharacteristic
444C8—62no—
445C8—121no—
446C8—122no—
447C2×C3—44no—
448C2×C3#240911yescharacteristic
449C2×C3#300511yescharacteristic
450C2×C3#300911yescharacteristic
451C2×C3#337612yes—
452C2×C3#338812yes—
453C22—24no—
454C22—42no—
455C22—42no—
456C22—44no—
457C22—44no—
458C22Q8×#6711yescharacteristic
459C22#581711yescharacteristic
460C22#582212yes—
461C22#583612yes—
462C22C2×#95311yescharacteristic
463C4—22no—
464C4—22no—
465C4—42no—
466C4—42no—
467C4—42no—
468C4—44no—
469C4—122no—
470C4—241no—
471C4—242no—
472C4C2×#94611yescharacteristic
473C4Q8×#6711yescharacteristic
474C4#581611yescharacteristic
475C4#581711yescharacteristic
476C3#2434311yescharacteristic, complemented, Sylow
477C2—44no—
478C2#8363212yes—
479C2#8364512yes—
480C2#8406811yescharacteristic
481C2#8408111yescharacteristic
482C2#8410511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#554C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×#554D3
hypocenterC3#24343
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
2. C2×#5094C2semidirect
3. #82480C2semidirect
5. #71788C2semidirect
6. #71744C2semidirect
7. #71739C2semidirect
8. #71635C2semidirect
89. C2×#214Q8semidirect
476. C3#24343semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5542
C2×#50942
#824802
C2×#50182
#717882
#717442
#717392
#716352
#716122
#716072
#716052
#243433