Finite Group Databasebeta

← #10772467

Subgroups — #10772467

Order 768.

960 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3511C211yesmaximal, characteristic
2#83270C211yesmaximal, characteristic, complemented
3C2×#5079C211yesmaximal, characteristic, complemented
4#82533C211yesmaximal, characteristic, complemented
5#71929C214yesmaximal, complemented
6#71926C214yesmaximal, complemented
7#71802C211yesmaximal, characteristic, complemented
8#71774C211yesmaximal, characteristic, complemented
9#71638C211yesmaximal, characteristic, complemented
10#18636—31nomaximal, Sylow
11C3×#556—21no—
12C3×#555C2212yes—
13C3×#554C2212yes—
14C3×#553—22no—
15C3×#548—22no—
16C3×#547—21no—
17C3×#542—22no—
18C3×#537—21no—
19C3×#533—21no—
20C3×#532—21no—
21C3×#532C2211yescharacteristic
22C2×C3×C4⋊Q8C2211yescharacteristic
23C3×#520—21no—
24C3×C4×#41C2211yescharacteristic
25C2×#951—21no—
26C2×#869C2211yescharacteristic
27C2×#835—22no—
28#5079—22no—
29#5079C2214yes—
30#5078—21no—
31#5070—21no—
32#5033—22no—
33#5031—21no—
34#5031C2212yes—
35#5019—21no—
36#5019C2212yes—
37#5007C2214yes—
38#5004—21no—
39#4925C2211yescharacteristic
40#4924C2211yescharacteristic
41C2×C4×C3⋊C8C2211yescharacteristic
42#4546C2214yes—
43#4535C2212yes—
44D4⋊(C3⋊C8)C2214yes—
45D4⋊(C3⋊C8)C2212yes—
46#3511D311yescharacteristic
47#3226—31no—
48C2×#487—31no—
49#3039—31no—
50#1646—34no—
51#1644—34no—
52#1548—31no—
53#1492—31no—
54#1476—31no—
55C2×C3×#41—22no—
56C2×C3×#41—21no—
57C2×C3×#41D411yescharacteristic
58C22×C3×Q8D412yes—
59C22×Dic6—21no—
60C3×C4⋊Q8—21no—
61C3×C4⋊Q8—21no—
62C3×C4⋊Q8D412yes—
63C3×C4⋊Q8D411yescharacteristic
64C3×C4⋊Q8C2312yes—
65C3×#127D411yescharacteristic
66C3×#126—24no—
67C3×#125—22no—
68C3×#124—22no—
69C3×#124—24no—
70C3×#124—21no—
71C3×#123—24no—
72C3×C22⋊Q8—22no—
73C3×C22⋊Q8—22no—
74C3×C22⋊Q8—22no—
75C3×C22⋊Q8—24no—
76C3×#121—22no—
77C3×#121—24no—
78C3×#121—22no—
79C3×#121—22no—
80C3×#121—21no—
81C3×#121D412yes—
82C3×C4×Q8—22no—
83C3×C4×Q8—22no—
84C3×C4×Q8D411yescharacteristic
85C3×C4×Q8C2311yescharacteristic
86C3×C4×D4—22no—
87C3×C4×D4—24no—
88C3×C4×D4—22no—
89C3×C4×D4—21no—
90C3×C4×D4—21no—
91C3×C4×D4—21no—
92C3×C4×D4D411yescharacteristic
93C3×C4×D4C2312yes—
94C3×C4×D4C2311yescharacteristic
95C3×#117—22no—
96C3×#117—22no—
97C3×#117—21no—
98C3×#117D411yescharacteristic
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—22no—
102C2×C3×C42—21no—
103C2×C3×C42C2311yescharacteristic
104#951—44no—
105C2×#216—22no—
106C2×#214—22no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3—21no—
109C2×#208—21no—
110C22×C3⋊C8—21no—
111#869—21no—
112#869—21no—
113#869—21no—
114#869C2312yes—
115#835—28no—
116#832—24no—
117#830—21no—
118#810—22no—
119#810—22no—
120#807—24no—
121C4⋊(C3⋊C8)—21no—
122C4⋊(C3⋊C8)C2312yes—
123C4⋊(C3⋊C8)C2312yes—
124C4×C3⋊C8—21no—
125C4×C3⋊C8—21no—
126C4×C3⋊C8C2312yes—
127C2×#137—61no—
128#556—21no—
129#555C2×D312yes—
130#554C2×D312yes—
131#553—22no—
132#548—22no—
133#547—21no—
134#542—22no—
135#537—21no—
136#533—21no—
137#532—21no—
138#532C2×D311yescharacteristic
139C2×C4⋊Q8—31no—
140C2×C4⋊Q8C2×D311yescharacteristic
141#520—21no—
142C4×#41C2×D311yescharacteristic
143#500—61no—
144#487—34no—
145#487—62no—
146#486—61no—
147#480—62no—
148#474—34no—
149#462—32no—
150#462—61no—
151#461—32no—
152#461—61no—
153#431—31no—
154#423—31no—
155#419—61no—
156C2×#103—62no—
157C2×C4×C8—31no—
158#333—34no—
159Q8⋊C8—32no—
160D4⋊C8—34no—
161D4⋊C8—32no—
162C3×#41—22no—
163C3×#41—22no—
164C3×#41—42no—
165C3×#41—42no—
166C3×#41—84no—
167C2×C3×Q8—21no—
168C2×C3×Q8—22no—
169C2×C3×Q8—44no—
170C2×C3×Q8C2×D411yescharacteristic
171C2×C3×Q8C2×D414yes—
172C2×C3×D4—21no—
173C2×C3×D4—22no—
174C2×C3×D4—24no—
175C2×C3×D4—21no—
176C2×C3×D4—21no—
177C2×C3×D4C2×D412yes—
178C2×C3×D4C2×D411yescharacteristic
179C22×C3×C4—24no—
180C22×C3×C4—22no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—22no—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×Dic3—41no—
189C2×Dic6—22no—
190C2×Dic6—42no—
191C2×Dic6—42no—
192C2×Dic6—41no—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—24no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4C2×D412yes—
203C3×C4⋊C4C2×D411yescharacteristic
204C3×C4⋊C4C2×D411yescharacteristic
205C3×C22⋊C4—24no—
206C3×C22⋊C4—24no—
207C3×C22⋊C4—24no—
208C3×C22⋊C4—21no—
209C3×C22⋊C4—21no—
210C3×C22⋊C4—22no—
211C3×C22⋊C4—21no—
212C3×C22⋊C4—22no—
213C3×C42—24no—
214C3×C42—21no—
215C3×C42—21no—
216C3×C42—21no—
217C3×C42C2×D411yescharacteristic
218C3×C42C2×D411yescharacteristic
219C3×C42C2×D411yescharacteristic
220C3×C42C2411yescharacteristic
221#216—44no—
222#214—44no—
223C4⋊Dic3—22no—
224C4⋊Dic3—22no—
225C4⋊Dic3—41no—
226C4⋊Dic3—41no—
227#208—42no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—42no—
232C2×#41—22no—
233C2×#41—21no—
234C2×#41#6711yescharacteristic
235C22×Q8—61no—
236C22×Q8#6712yes—
237#137—124no—
238C2×Q16—62no—
239C2×SD16—62no—
240C2×M16—61no—
241C22×C8—61no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—32no—
245C4⋊Q8—61no—
246C4⋊Q8—61no—
247C4⋊Q8—61no—
248C4⋊Q8#6712yes—
249C4⋊Q8#6711yescharacteristic
250C4⋊Q8C22×D312yes—
251#127#6711yescharacteristic
252#126—24no—
253#125—22no—
254#124—22no—
255#124—24no—
256#124—21no—
257#123—24no—
258C22⋊Q8—22no—
259C22⋊Q8—22no—
260C22⋊Q8—22no—
261C22⋊Q8—24no—
262#121—22no—
263#121—24no—
264#121—22no—
265#121—22no—
266#121—21no—
267#121#6712yes—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8#6711yescharacteristic
271C4×Q8C22×D311yescharacteristic
272C4×D4—22no—
273C4×D4—24no—
274C4×D4—22no—
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—21no—
278C4×D4#6711yescharacteristic
279C4×D4C22×D311yescharacteristic
280C4×D4C22×D312yes—
281#117—22no—
282#117—22no—
283#117—21no—
284#117#6711yescharacteristic
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C42—21no—
291C2×C42C22×D311yescharacteristic
292#107—64no—
293C4⋊C8—32no—
294C4⋊C8—32no—
295C4⋊C8—61no—
296#103—68no—
297#103—62no—
298#103—62no—
299#102—64no—
300C22⋊C8—61no—
301C4×C8—32no—
302C4×C8—61no—
303C4×C8—61no—
304C23×C3—21no—
305C23×C3—22no—
306C23×C3#12011yescharacteristic
307C3×Q8—21no—
308C3×Q8—21no—
309C3×Q8—44no—
310C3×Q8—42no—
311C3×Q8—44no—
312C3×D4—21no—
313C3×D4—21no—
314C3×D4—24no—
315C3×D4—42no—
316C3×D4—44no—
317C3×D4—42no—
318C3×D4—42no—
319C3×D4—84no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—22no—
325C2×C3×C4—21no—
326C2×C3×C4—24no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—82no—
333C2×C3×C4—84no—
334C2×C3×C4#12011yescharacteristic
335C2×C3×C4#12011yescharacteristic
336C2×C3×C4#12011yescharacteristic
337C2×C3×C4#13512yes—
338C2×C3×C4AGL(1,ℤ/8)12yes—
339C2×C3×C4C22×D411yescharacteristic
340C2×C3×C4C22×D411yescharacteristic
341C2×C3×C4C22×D411yescharacteristic
342C2×Dic3—42no—
343C2×Dic3—82no—
344Dic6—42no—
345Dic6—42no—
346Dic6—82no—
347C3⋊C8—42no—
348C3⋊C8—42no—
349C3⋊C8—42no—
350#41—22no—
351#41—22no—
352#41—42no—
353#41—42no—
354#41—84no—
355C2×Q8—21no—
356C2×Q8—22no—
357C2×Q8—44no—
358C2×Q8—62no—
359C2×Q8—121no—
360C2×Q8—122no—
361C2×Q8—122no—
362C2×Q8C2×#6711yescharacteristic
363C2×Q8C2×#6714yes—
364C2×D4—21no—
365C2×D4—22no—
366C2×D4—24no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4C2×#6712yes—
370C2×D4C2×#6711yescharacteristic
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—121no—
378C22×C4C2×#6711yescharacteristic
379C22×C4C2×#6711yescharacteristic
380C22×C4C2×#6711yescharacteristic
381Q16—124no—
382SD16—124no—
383M16—122no—
384C2×C8—62no—
385C2×C8—62no—
386C2×C8—62no—
387C2×C8—122no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—21no—
397C4⋊C4—62no—
398C4⋊C4—62no—
399C4⋊C4—121no—
400C4⋊C4—121no—
401C4⋊C4C2×#6711yescharacteristic
402C4⋊C4C2×#6711yescharacteristic
403C4⋊C4C2×#6712yes—
404C22⋊C4—24no—
405C22⋊C4—24no—
406C22⋊C4—24no—
407C22⋊C4—21no—
408C22⋊C4—21no—
409C22⋊C4—22no—
410C22⋊C4—21no—
411C22⋊C4—22no—
412C42—24no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42C2×#6711yescharacteristic
417C42C2×#6711yescharacteristic
418C42C2×#6711yescharacteristic
419C42C23×D311yescharacteristic
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—41no—
423C22×C3—42no—
424C22×C3—41no—
425C22×C3—42no—
426C22×C3—41no—
427C22×C3—42no—
428C22×C3#44912yes—
429C22×C3C2×#12011yescharacteristic
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—41no—
433C3×C4—42no—
434C3×C4—41no—
435C3×C4—41no—
436C3×C4—44no—
437C3×C4#44912yes—
438C3×C4#45512yes—
439C3×C4C2×#13511yescharacteristic
440C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
441C3×C4#57812yes—
442Dic3—82no—
443C23—21no—
444C23—22no—
445C23#95311yescharacteristic
446Q8—21no—
447Q8—21no—
448Q8—44no—
449Q8—42no—
450Q8—44no—
451Q8—122no—
452Q8—122no—
453Q8—242no—
454D4—21no—
455D4—21no—
456D4—24no—
457D4—42no—
458D4—44no—
459D4—42no—
460D4—42no—
461D4—84no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—24no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—82no—
475C2×C4—84no—
476C2×C4—122no—
477C2×C4—242no—
478C2×C4#94912yes—
479C2×C4#95012yes—
480C2×C4#95311yescharacteristic
481C2×C4#95311yescharacteristic
482C2×C4#95311yescharacteristic
483C2×C4C22×#6711yescharacteristic
484C2×C4C22×#6711yescharacteristic
485C2×C4C22×#6711yescharacteristic
486C8—122no—
487C8—122no—
488C8—122no—
489C2×C3—22no—
490C2×C3—81no—
491C2×C3—82no—
492C2×C3C2×#44911yescharacteristic
493C2×C3#301111yescharacteristic
494C2×C3C2×#45511yescharacteristic
495C22—21no—
496C22—21no—
497C22—41no—
498C22—42no—
499C22—41no—
500C22—42no—
501C22—41no—
502C22—42no—
503C22#523812yes—
504C22C2×#95311yescharacteristic
505C4—21no—
506C4—21no—
507C4—41no—
508C4—42no—
509C4—41no—
510C4—44no—
511C4—41no—
512C4—242no—
513C4#523812yes—
514C4#524212yes—
515C4C2×#94911yescharacteristic
516C4C2×#95011yescharacteristic
517C4#583812yes—
518C3#1863611yescharacteristic, complemented, Sylow
519C2—22no—
520C2—81no—
521C2—82no—
522C2C2×#523811yescharacteristic
523C2C2×#524211yescharacteristic
524C2#8411711yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
Fitting, prime radicalC3×#3511C2
socleC22×C3C2×#120
hypercenter#3511D3
hypocenterC3#18636
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3511

Complements

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

NH ≅ G/Ntype
2. #83270C2semidirect
3. C2×#5079C2semidirect
4. #82533C2semidirect
5. #71929C2semidirect
6. #71926C2semidirect
7. #71802C2semidirect
8. #71774C2semidirect
9. #71638C2semidirect
518. C3#18636semidirect

Maximal subgroups

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

subgroupindex
C3×#35112
#832702
C2×#50792
#825332
#719292
#719262
#718022
#717742
#716382
#186363