Finite Group Databasebeta

← #10831777

Subgroups — #10831777

Order 768.

1020 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q82C211yesmaximal, characteristic
2C2×#5094C211yesmaximal, characteristic
3#82480C211yesmaximal, characteristic
4C2×#5030C211yesmaximal, characteristic
5#71794C212yesmaximal
6#71788C211yesmaximal, characteristic
7#71783C212yesmaximal
8#71635C211yesmaximal, characteristic
9#71632C212yesmaximal
10#71629C212yesmaximal
11#71627C211yesmaximal, characteristic
12#24345—31nomaximal, Sylow
13C3×Q82—24no—
14C3×Q82—24no—
15C2×C3×C4⋊Q8—22no—
16C2×C3×C4⋊Q8—21no—
17C2×C3×C4⋊Q8C2212yes—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×C4×Q8—21no—
20C2×C3×C4×Q8C2212yes—
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×Q8C2211yescharacteristic
23C22×#216C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25#5230—21no—
26C2×#835C2212yes—
27C2×#835C2211yescharacteristic
28#5094—24no—
29#5030—24no—
30#4985—21no—
31#4978—22no—
32C2×#810C2212yes—
33C2×#810C2211yescharacteristic
34C2×#808C2212yes—
35C2×#808C2211yescharacteristic
36#4936C2211yescharacteristic
37#4929C2212yes—
38#4928C2212yes—
39C2×C4⋊(C3⋊C8)C2211yescharacteristic
40C2×#803C2211yescharacteristic
41#4529C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×Q82D311yescharacteristic
47C2×#475—31no—
48C2×#441—31no—
49#2581—31no—
50#2058—32no—
51#2024—32no—
52#1993—31no—
53#1985—31no—
54#1878—32no—
55#1865—32no—
56#1805—31no—
57C22×C3×Q8—22no—
58C22×C3×Q8—21no—
59C22×C3×Q8D411yescharacteristic
60C22×C3×Q8C2311yescharacteristic
61C22×Dic6C2311yescharacteristic
62C3×C4⋊Q8—216no—
63C3×C4⋊Q8—28no—
64C3×C4⋊Q8—24no—
65C3×C4⋊Q8—24no—
66C3×C4⋊Q8—22no—
67C3×C4⋊Q8D414yes—
68C3×C4×Q8—28no—
69C3×C4×Q8—24no—
70C3×C4×Q8—24no—
71C3×C4×Q8—22no—
72C3×C4×Q8—22no—
73C3×C4×Q8—22no—
74C3×C4×Q8D414yes—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4D412yes—
80C2×C3×C4⋊C4D411yescharacteristic
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C42—22no—
84C2×C3×C42—21no—
85C2×C3×C42C2312yes—
86C2×C3×C42C2311yescharacteristic
87C2×#216—22no—
88C2×#216—22no—
89C2×#216Q814yescomplemented
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#835—22no—
101#835—24no—
102C3⋊#95—22no—
103C3⋊#95—22no—
104C3⋊#95—22no—
105C3⋊#95—21no—
106C3⋊#95—21no—
107#810—22no—
108#810—24no—
109#808—22no—
110#808—24no—
111C4⋊(C3⋊C8)—22no—
112#803—22no—
113C22×Q16—31no—
114Q82—24no—
115Q82—24no—
116C2×C4⋊Q8—22no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8C2×D312yes—
119C2×C4⋊Q8C2×D311yescharacteristic
120C2×C4×Q8—21no—
121C2×C4×Q8—31no—
122C2×C4×Q8C2×D312yes—
123C2×C4×Q8C2×D311yescharacteristic
124C2×C4×Q8C2×D311yescharacteristic
125#475—64no—
126#441—64no—
127C2×#106—32no—
128C2×#106—31no—
129C2×C4⋊C8—31no—
130C2×#103—32no—
131C2×#103—31no—
132C2×#103—32no—
133C2×#103—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—
146C2×C3×Q8—28no—
147C2×C3×Q8—22no—
148C2×C3×Q8—24no—
149C2×C3×Q8—24no—
150C2×C3×Q8—21no—
151C2×C3×Q8—21no—
152C2×C3×Q8—21no—
153C2×C3×Q8—82no—
154C2×C3×Q8—84no—
155C2×C3×Q8C2×D412yes—
156C2×C3×Q8C2×Q812yes—
157C2×C3×Q8#4112yes—
158C22×C3×C4—21no—
159C22×C3×C4—22no—
160C22×C3×C4—21no—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2411yescharacteristic
166C22×Dic3—21no—
167C22×Dic3—22no—
168C22×Dic3—21no—
169C2×Dic6—21no—
170C2×Dic6—21no—
171C2×Dic6—21no—
172C2×Dic6—21no—
173C2×Dic6—21no—
174C2×Dic6C2×Q812yes—
175C3×C4⋊C4—28no—
176C3×C4⋊C4—28no—
177C3×C4⋊C4—24no—
178C3×C4⋊C4—24no—
179C3×C4⋊C4—24no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4C2×D414yes—
184C3×C4⋊C4C2×D412yes—
185C3×C42—28no—
186C3×C42—24no—
187C3×C42—22no—
188C3×C42—22no—
189C3×C42C2×D412yes—
190C3×C42C2×D412yes—
191#216—28no—
192C4⋊Dic3—21no—
193C4⋊Dic3—21no—
194C4⋊Dic3—22no—
195C4⋊Dic3—22no—
196Dic3⋊C4—42no—
197Dic3⋊C4—44no—
198C4×Dic3—42no—
199C2×C3⋊C8—21no—
200C2×C3⋊C8—21no—
201C2×C3⋊C8—21no—
202C2×C3⋊C8—22no—
203C2×C3⋊C8—22no—
204C2×C3⋊C8—21no—
205C2×C3⋊C8—21no—
206C2×C3⋊C8—22no—
207C2×C3⋊C8C2×Q812yes—
208C22×Q8—22no—
209C22×Q8—21no—
210C22×Q8—31no—
211C22×Q8#6711yescharacteristic
212C22×Q8C22×D311yescharacteristic
213C2×Q16—34no—
214C2×Q16—62no—
215C2×Q16—62no—
216C22×C8—31no—
217C22×C8—31no—
218C22×C8—32no—
219C4⋊Q8—216no—
220C4⋊Q8—28no—
221C4⋊Q8—24no—
222C4⋊Q8—24no—
223C4⋊Q8—22no—
224C4⋊Q8#6714yes—
225C4×Q8—28no—
226C4×Q8—24no—
227C4×Q8—24no—
228C4×Q8—22no—
229C4×Q8—22no—
230C4×Q8—22no—
231C4×Q8—62no—
232C4×Q8—62no—
233C4×Q8#6714yes—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—31no—
239C2×C4⋊C4—32no—
240C2×C4⋊C4—61no—
241C2×C4⋊C4—62no—
242C2×C4⋊C4#6712yes—
243C2×C4⋊C4#6711yescharacteristic
244C2×C4⋊C4C22×D312yes—
245C2×C4⋊C4C22×D311yescharacteristic
246C2×C42—22no—
247C2×C42—21no—
248C2×C42—61no—
249C2×C42C22×D312yes—
250C2×C42C22×D311yescharacteristic
251#106—62no—
252#106—64no—
253C4⋊C8—62no—
254#103—64no—
255#103—62no—
256#103—62no—
257#103—64no—
258#97—62no—
259#95—61no—
260#95—61no—
261#95—62no—
262#95—62no—
263#95—62no—
264C23×C3C22×D411yescharacteristic
265C3×Q8—28no—
266C3×Q8—22no—
267C3×Q8—22no—
268C3×Q8—22no—
269C3×Q8—84no—
270C3×Q8—88no—
271C3×Q8C22⋊Q814yes—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—24no—
281C2×C3×C4—22no—
282C2×C3×C4—24no—
283C2×C3×C4—44no—
284C2×C3×C4—42no—
285C2×C3×C4#12012yes—
286C2×C3×C4#12012yes—
287C2×C3×C4#13714yes—
288C2×C3×C4C22×D411yescharacteristic
289C2×C3×C4C22×D411yescharacteristic
290C2×C3×C4C22×Q811yescharacteristic
291C2×C3×C4C2×#4111yescharacteristic
292C2×C3×C42−1+411yescharacteristic
293C2×C3×C42−1+411yescharacteristic
294C2×Dic3—22no—
295C2×Dic3—41no—
296C2×Dic3—41no—
297C2×Dic3—41no—
298C2×Dic3—42no—
299C2×Dic3—42no—
300C2×Dic3—42no—
301C2×Dic3—42no—
302Dic6—22no—
303Dic6—22no—
304Dic6—41no—
305Dic6—41no—
306C3⋊C8—22no—
307C3⋊C8—41no—
308C3⋊C8—42no—
309C2×Q8—28no—
310C2×Q8—24no—
311C2×Q8—22no—
312C2×Q8—24no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—32no—
317C2×Q8—61no—
318C2×Q8—61no—
319C2×Q8—61no—
320C2×Q8—61no—
321C2×Q8—61no—
322C2×Q8—82no—
323C2×Q8—84no—
324C2×Q8D3×Q812yes—
325C2×Q8#23912yes—
326C2×Q8C2×#6712yes—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—61no—
331C22×C4—62no—
332C22×C4—61no—
333C22×C4C2×#6712yes—
334C22×C4C2×#6712yes—
335C22×C4C2×#6711yescharacteristic
336C22×C4C2×#6711yescharacteristic
337C22×C4C23×D311yescharacteristic
338Q16—68no—
339C2×C8—32no—
340C2×C8—61no—
341C2×C8—61no—
342C2×C8—62no—
343C2×C8—61no—
344C2×C8—61no—
345C2×C8—62no—
346C2×C8—61no—
347C2×C8—62no—
348C4⋊C4—28no—
349C4⋊C4—28no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—21no—
356C4⋊C4—62no—
357C4⋊C4—62no—
358C4⋊C4—61no—
359C4⋊C4—61no—
360C4⋊C4—124no—
361C4⋊C4—122no—
362C4⋊C4C2×#6714yes—
363C4⋊C4C2×#6712yes—
364C42—28no—
365C42—24no—
366C42—22no—
367C42—22no—
368C42—122no—
369C42C2×#6712yes—
370C42C2×#6712yes—
371C22×C3C2×#12011yescharacteristic
372C22×C3#54811yescharacteristic
373C22×C3D4×Q811yescharacteristic
374C22×C3C2×#13712yes—
375C22×C3#57712yes—
376C3×C4—24no—
377C3×C4—22no—
378C3×C4—24no—
379C3×C4—22no—
380C3×C4—42no—
381C3×C4—44no—
382C3×C4—42no—
383C3×C4C2×C22⋊Q811yescharacteristic
384C3×C4#53611yescharacteristic
385C3×C4#54811yescharacteristic
386C3×C4D4×Q811yescharacteristic
387Dic3—42no—
388Dic3—82no—
389Dic3—81no—
390C23C22×#6711yescharacteristic
391Q8—28no—
392Q8—22no—
393Q8—22no—
394Q8—22no—
395Q8—62no—
396Q8—62no—
397Q8—84no—
398Q8—88no—
399Q8—121no—
400Q8—121no—
401Q8#94614yes—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—62no—
416C2×C4—121no—
417C2×C4—121no—
418C2×C4—122no—
419C2×C4—122no—
420C2×C4—122no—
421C2×C4—122no—
422C2×C4—121no—
423C2×C4#94214yes—
424C2×C4#95312yes—
425C2×C4#95312yes—
426C2×C4C2×D3×Q811yescharacteristic
427C2×C4C2×#23911yescharacteristic
428C2×C4#100711yescharacteristic
429C2×C4#100711yescharacteristic
430C2×C4C22×#6711yescharacteristic
431C2×C4C22×#6711yescharacteristic
432C8—62no—
433C8—121no—
434C8—122no—
435C2×C3#240911yescharacteristic
436C2×C3#300411yescharacteristic
437C2×C3#300911yescharacteristic
438C2×C3#336912yes—
439C2×C3#338712yes—
440C22C2×#94212yes—
441C22Q8×#6711yescharacteristic
442C22#581711yescharacteristic
443C22#583612yes—
444C22C2×#95311yescharacteristic
445C4—24no—
446C4—22no—
447C4—24no—
448C4—22no—
449C4—42no—
450C4—44no—
451C4—42no—
452C4—122no—
453C4—241no—
454C4—242no—
455C4C2×#94611yescharacteristic
456C4Q8×#6711yescharacteristic
457C4#581611yescharacteristic
458C4#581711yescharacteristic
459C3#2434511yescharacteristic, complemented, Sylow
460C2#8370412yes—
461C2#8370912yes—
462C2#8405911yescharacteristic
463C2#8406811yescharacteristic
464C2#8410511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×Q82C2
socle, prime radicalC23×C3C22×D4
hypercenterC2×Q82D3
hypocenterC3#24345
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×Q82

Complements

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

NH ≅ G/Ntype
89. C2×#216Q8semidirect
459. C3#24345semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q822
C2×#50942
#824802
C2×#50302
#717942
#717882
#717832
#716352
#716322
#716292
#716272
#243453