Finite Group Databasebeta

← #10831722

Subgroups — #10831722

Order 768.

1036 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#557C211yesmaximal, characteristic, complemented
2C2×#5084C211yesmaximal, characteristic
3#82501C211yesmaximal, characteristic
4C2×#5026C211yesmaximal, characteristic
5#71788C211yesmaximal, characteristic, complemented
6#71743C212yesmaximal
7#71739C212yesmaximal, complemented
8#71635C211yesmaximal, characteristic, complemented
9#71633C212yesmaximal
10#71629C212yesmaximal, complemented
11#71627C211yesmaximal, characteristic, complemented
12#24340—31nomaximal, Sylow
13C3×#557—24no—
14C3×#557—24no—
15C2×C3×C4⋊Q8—21no—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#125—22no—
18C2×C3×#125C2212yes—
19C2×C3×C4×Q8—21no—
20C2×C3×C4×Q8C2212yes—
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×Q8C2211yescharacteristic
23C22×#215C2211yescharacteristic
24C2×C4×D12C2211yescharacteristic
25#5231—21no—
26C2×#835C2211yescharacteristic
27C2×#835C2212yes—
28#5084—24no—
29#5026—24no—
30#4993—21no—
31#4989—22no—
32C2×#809C2211yescharacteristic
33C2×#809C2212yes—
34C2×#808C2212yes—
35C2×#807C2211yescharacteristic
36#4939C2212yes—
37#4936C2211yescharacteristic
38#4928C2212yes—
39C2×C4⋊(C3⋊C8)C2211yescharacteristic
40C2×#803C2211yescharacteristic
41#4529C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#557D311yescharacteristic, complemented
47C2×#476—31no—
48C2×#440—31no—
49#2580—31no—
50#2059—32no—
51#2025—32no—
52#1993—31no—
53#1985—31no—
54#1880—32no—
55#1865—32no—
56#1805—31no—
57C22×C3×Q8—21no—
58C22×C3×Q8D411yescharacteristic
59C22×C3×Q8C2311yescharacteristic
60C22×D12C2311yescharacteristic
61C22×C4×D3—21no—
62C3×C4⋊Q8—28no—
63C3×C4⋊Q8—22no—
64C3×C4⋊Q8D414yes—
65C3×#125—216no—
66C3×#125—24no—
67C3×#125—24no—
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⋊C4—22no—
80C2×C3×C4⋊C4D412yes—
81C2×C3×C4⋊C4D411yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C42—22no—
85C2×C3×C42—21no—
86C2×C3×C42C2312yes—
87C2×C3×C42C2311yescharacteristic
88C2×#215—22no—
89C2×#215—22no—
90C2×#215Q814yescomplemented
91C2×(C2×D3)⋊C4—22no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—22no—
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2312yes—
96C2×Dic3⋊C4—22no—
97C22×C3⋊C8C2311yescharacteristic
98C22×C3⋊C8C2312yes—
99C22×C3⋊C8C2311yescharacteristic
100C4×D12—22no—
101C4×D12—22no—
102#835—22no—
103#835—24no—
104C3⋊#95—22no—
105C3⋊#95—21no—
106C3⋊#95—21no—
107#809—22no—
108#809—24no—
109#808—24no—
110#807—22no—
111C4⋊(C3⋊C8)—22no—
112#803—22no—
113C22×SD16—31no—
114#557—24no—
115#557—24no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8C2×D311yescharacteristic
118C2×#125—22no—
119C2×#125C2×D312yes—
120C2×C4×Q8—21no—
121C2×C4×Q8C2×D312yes—
122C2×C4×Q8C2×D311yescharacteristic
123C2×C4×Q8C2×D311yescharacteristic
124C2×C4×D4—31no—
125#476—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#399—61no—
136#393—61no—
137#390—32no—
138#389—32no—
139#385—62no—
140#382—31no—
141#340—32no—
142#340—32no—
143#340—32no—
144#336—32no—
145#336—32no—
146C23×D3—21no—
147C2×C3×Q8—22no—
148C2×C3×Q8—24no—
149C2×C3×Q8—21no—
150C2×C3×Q8—21no—
151C2×C3×Q8—21no—
152C2×C3×Q8—42no—
153C2×C3×Q8—44no—
154C2×C3×Q8C2×D412yes—
155C2×C3×Q8C2×Q812yes—
156C2×C3×Q8#4112yes—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—22no—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2411yescharacteristic
165C22×Dic3—22no—
166C22×Dic3—21no—
167C2×D12—21no—
168C2×D12—21no—
169C2×D12—21no—
170C2×D12—21no—
171C2×D12—21no—
172C2×D12C2×Q812yes—
173C2×C4×D3—42no—
174C2×C4×D3—82no—
175C3×C4⋊C4—28no—
176C3×C4⋊C4—28no—
177C3×C4⋊C4—28no—
178C3×C4⋊C4—24no—
179C3×C4⋊C4—24no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—24no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4C2×D412yes—
185C3×C4⋊C4C2×D414yes—
186C3×C42—28no—
187C3×C42—24no—
188C3×C42—22no—
189C3×C42—22no—
190C3×C42C2×D412yes—
191C3×C42C2×D412yes—
192#215—28no—
193(C2×D3)⋊C4—44no—
194(C2×D3)⋊C4—42no—
195(C2×D3)⋊C4—82no—
196C4⋊Dic3—21no—
197C4⋊Dic3—21no—
198C4⋊Dic3—22no—
199C4⋊Dic3—22no—
200Dic3⋊C4—82no—
201C2×C3⋊C8—21no—
202C2×C3⋊C8—21no—
203C2×C3⋊C8—21no—
204C2×C3⋊C8—22no—
205C2×C3⋊C8—22no—
206C2×C3⋊C8—21no—
207C2×C3⋊C8—21no—
208C2×C3⋊C8—22no—
209C2×C3⋊C8C2×Q812yes—
210C22×Q8—21no—
211C22×Q8#6711yescharacteristic
212C22×Q8C22×D311yescharacteristic
213C22×D4—31no—
214C23×C4—61no—
215C2×SD16—34no—
216C2×SD16—62no—
217C2×SD16—62no—
218C22×C8—31no—
219C22×C8—31no—
220C22×C8—32no—
221C4⋊Q8—28no—
222C4⋊Q8—22no—
223C4⋊Q8#6714yes—
224#125—216no—
225#125—24no—
226#125—24no—
227C4×Q8—28no—
228C4×Q8—24no—
229C4×Q8—24no—
230C4×Q8—22no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×Q8#6714yes—
234C4×D4—62no—
235C4×D4—62no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—31no—
242C2×C4⋊C4—32no—
243C2×C4⋊C4—62no—
244C2×C4⋊C4#6711yescharacteristic
245C2×C4⋊C4#6712yes—
246C2×C4⋊C4C22×D311yescharacteristic
247C2×C4⋊C4C22×D312yes—
248C2×C22⋊C4—61no—
249C2×C22⋊C4—62no—
250C2×C22⋊C4—62no—
251C2×C42—22no—
252C2×C42—21no—
253C2×C42C22×D312yes—
254C2×C42C22×D311yescharacteristic
255#107—62no—
256#106—64no—
257C4⋊C8—62no—
258#103—64no—
259#103—62no—
260#102—62no—
261#102—64no—
262#97—62no—
263#95—61no—
264#95—61no—
265#95—62no—
266C23×C3C22×D411yescharacteristic
267C22×D3—22no—
268C22×D3—41no—
269C22×D3—41no—
270C22×D3—42no—
271C22×D3—42no—
272C3×Q8—22no—
273C3×Q8—22no—
274C3×Q8—22no—
275C3×Q8—44no—
276C3×Q8—48no—
277C3×Q8C22⋊Q814yes—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—24no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—44no—
292C2×C3×C4—42no—
293C2×C3×C4#12012yes—
294C2×C3×C4#12012yes—
295C2×C3×C4C22×D411yescharacteristic
296C2×C3×C4C22×D411yescharacteristic
297C2×C3×C4C22×Q811yescharacteristic
298C2×C3×C4C2×#4111yescharacteristic
299C2×C3×C42−1+411yescharacteristic
300C2×C3×C42−1+411yescharacteristic
301C2×Dic3—41no—
302C2×Dic3—42no—
303C2×Dic3—42no—
304C2×Dic3—42no—
305C2×Dic3—42no—
306D12—22no—
307D12—22no—
308D12—41no—
309D12—41no—
310C4×D3—84no—
311C3⋊C8—22no—
312C3⋊C8—41no—
313C3⋊C8—42no—
314C24—61no—
315C2×Q8—22no—
316C2×Q8—24no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—42no—
321C2×Q8—44no—
322C2×Q8D3×Q812yes—
323C2×Q8#23912yes—
324C2×Q8C2×#6712yes—
325C2×D4—32no—
326C2×D4—61no—
327C2×D4—61no—
328C2×D4—61no—
329C2×D4—61no—
330C2×D4—61no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—62no—
335C22×C4—61no—
336C22×C4—122no—
337C22×C4—242no—
338C22×C4C2×#6712yes—
339C22×C4C2×#6712yes—
340C22×C4C2×#6711yescharacteristic
341C22×C4C2×#6711yescharacteristic
342C22×C4C23×D311yescharacteristic
343SD16—68no—
344C2×C8—32no—
345C2×C8—61no—
346C2×C8—61no—
347C2×C8—62no—
348C2×C8—61no—
349C2×C8—61no—
350C2×C8—62no—
351C2×C8—61no—
352C2×C8—62no—
353C4⋊C4—28no—
354C4⋊C4—28no—
355C4⋊C4—28no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—22no—
359C4⋊C4—24no—
360C4⋊C4—21no—
361C4⋊C4—22no—
362C4⋊C4—62no—
363C4⋊C4—62no—
364C4⋊C4—61no—
365C4⋊C4—61no—
366C4⋊C4—242no—
367C4⋊C4C2×#6714yes—
368C4⋊C4C2×#6712yes—
369C22⋊C4—124no—
370C22⋊C4—122no—
371C22⋊C4—242no—
372C42—28no—
373C42—24no—
374C42—22no—
375C42—22no—
376C42C2×#6712yes—
377C42C2×#6712yes—
378C22×C3C2×#12011yescharacteristic
379C22×C3#54811yescharacteristic
380C22×C3D4×Q811yescharacteristic
381C22×C3#57512yes—
382C22×C3#57712yes—
383C2×D3—44no—
384C2×D3—42no—
385C2×D3—82no—
386C2×D3—81no—
387C2×D3—81no—
388C3×C4—24no—
389C3×C4—22no—
390C3×C4—22no—
391C3×C4—42no—
392C3×C4—44no—
393C3×C4—42no—
394C3×C4—42no—
395C3×C4C2×C22⋊Q811yescharacteristic
396C3×C4#53611yescharacteristic
397C3×C4#54811yescharacteristic
398C3×C4D4×Q811yescharacteristic
399Dic3—82no—
400Dic3—81no—
401C23—62no—
402C23—121no—
403C23—122no—
404C23—122no—
405C23—121no—
406C23C22×#6711yescharacteristic
407Q8—22no—
408Q8—22no—
409Q8—22no—
410Q8—44no—
411Q8—48no—
412Q8#94614yes—
413D4—62no—
414D4—62no—
415D4—121no—
416D4—121no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—122no—
433C2×C4—122no—
434C2×C4—122no—
435C2×C4—122no—
436C2×C4—121no—
437C2×C4—244no—
438C2×C4#95312yes—
439C2×C4#95312yes—
440C2×C4C2×D3×Q811yescharacteristic
441C2×C4C2×#23911yescharacteristic
442C2×C4#100711yescharacteristic
443C2×C4#100711yescharacteristic
444C2×C4C22×#6711yescharacteristic
445C2×C4C22×#6711yescharacteristic
446C8—62no—
447C8—121no—
448C8—122no—
449C2×C3#240911yescharacteristic
450C2×C3#300511yescharacteristic
451C2×C3#301011yescharacteristic
452C2×C3#337112yes—
453C2×C3#338812yes—
454D3—82no—
455C22—122no—
456C22—124no—
457C22—241no—
458C22—241no—
459C22—242no—
460C22Q8×#6711yescharacteristic
461C22#581711yescharacteristic
462C22#582212yes—
463C22#583712yes—
464C22C2×#95311yescharacteristic
465C4—24no—
466C4—22no—
467C4—22no—
468C4—42no—
469C4—44no—
470C4—42no—
471C4—42no—
472C4—241no—
473C4—242no—
474C4C2×#94611yescharacteristic
475C4Q8×#6711yescharacteristic
476C4#581611yescharacteristic
477C4#581711yescharacteristic
478C3#2434011yescharacteristic, complemented, Sylow
479C2—242no—
480C2#8369112yes—
481C2#8369512yes—
482C2#8406811yescharacteristic
483C2#8408111yescharacteristic
484C2#8411311yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#557C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenterC2×#557D3
hypocenterC3#24340
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
1. C2×C3×#557C2semidirect
5. #71788C2semidirect
7. #71739C2semidirect
8. #71635C2semidirect
10. #71629C2semidirect
11. #71627C2semidirect
46. C2×#557D3semidirect
90. C2×#215Q8semidirect
478. C3#24340semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5572
C2×#50842
#825012
C2×#50262
#717882
#717432
#717392
#716352
#716332
#716292
#716272
#243403