Finite Group Databasebeta

← #10695915

Subgroups — #10695915

Order 768.

686 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3036C211yesmaximal, characteristic, complemented
2#83298C211yesmaximal, characteristic, complemented
3#82416C211yesmaximal, characteristic, complemented
4#71814C211yesmaximal, characteristic, complemented
5#71597C211yesmaximal, characteristic, complemented
6#70496C211yesmaximal, characteristic, complemented
7#67713C211yesmaximal, characteristic, complemented
8#9996—31nomaximal, Sylow
9C2×C3×AGL(1,ℤ/8)—21no—
10C3×#532C2211yescharacteristic
11C2×C3×C4×D4C2211yescharacteristic
12C2×#938—21no—
13C2×C4×D12C2211yescharacteristic
14C3×#463—22no—
15C3×#461—21no—
16C3×#459—21no—
17C3×#423C2211yescharacteristic
18C3×#417—21no—
19#5213—22no—
20#5087—21no—
21#5072—21no—
22#5036—21no—
23#5033—21no—
24C4×#213—22no—
25#4987—21no—
26#4960—21no—
27#4953—21no—
28#4936C2211yescharacteristic
29#4925C2211yescharacteristic
30#4925C2211yescharacteristic
31#4537—22no—
32D4⋊(C3⋊C8)—22no—
33#4483—22no—
34#3100—31no—
35#3036D311yescharacteristic, complemented
36#2535—31no—
37#1517—31no—
38#1486—31no—
39#1478—31no—
40#1286—31no—
41C2×C3×#41—21no—
42C22×C3×D4D411yescharacteristic
43C23×C3×C4—21no—
44C22×#67—22no—
45C22×D12D411yescharacteristic
46C22×C4×D3—21no—
47C3×AGL(1,ℤ/8)—42no—
48C3×AGL(1,ℤ/8)—42no—
49C2×C3×SD16—22no—
50C2×C3×D8—22no—
51C2×C3×M16—21no—
52C3×C4⋊Q8D411yescharacteristic, complemented
53C3×#127D411yescharacteristic, complemented
54C3×#124—21no—
55C3×#121—21no—
56C3×#121—21no—
57C3×C4×D4—21no—
58C3×C4×D4—21no—
59C3×C4×D4—21no—
60C3×C4×D4—21no—
61C3×C4×D4—21no—
62C3×C4×D4—21no—
63C2×C3×C4⋊C4D411yescharacteristic
64C2×C3×C22⋊C4—21no—
65C2×C3×C42C2311yescharacteristic
66C2×#217—22no—
67#938—42no—
68#938—42no—
69C2×#213—22no—
70C2×(C2×D3)⋊C4—21no—
71C2×(C2×D3)⋊C4—22no—
72C2×C4⋊Dic3—21no—
73C2×C4⋊Dic3D411yescharacteristic
74C4×D12—21no—
75C4×D12—21no—
76C4×D12—21no—
77C4×D12—21no—
78C3×C4⋊C8—21no—
79C3×C4⋊C8D412yescomplemented
80C3×#103—22no—
81C3×#102—22no—
82#832—22no—
83C3⋊#95—22no—
84C3⋊#95—21no—
85C3⋊#95D412yes—
86#830—21no—
87#830—21no—
88#809—22no—
89#807—22no—
90C4⋊(C3⋊C8)—21no—
91C4⋊(C3⋊C8)—21no—
92C4×C3⋊C8—21no—
93C4×C3⋊C8D412yescomplemented
94C2×AGL(1,ℤ/8)—21no—
95#532C2×D311yescharacteristic
96C2×#121—61no—
97C2×C4×D4—31no—
98C2×C4×D4C2×D311yescharacteristic
99#480—61no—
100#474—61no—
101#463—22no—
102#461—21no—
103#459—21no—
104#459—61no—
105#447—61no—
106C4×D8—62no—
107#431—31no—
108#431—31no—
109#423C2×D311yescharacteristic
110#417—21no—
111#397—61no—
112#392—61no—
113#386—62no—
114#382—31no—
115#380—61no—
116#327—62no—
117D4⋊C8—62no—
118D4⋊C8—62no—
119C24×C3—21no—
120C23×D3—21no—
121C3×#41—41no—
122C3×#41—41no—
123C3×#41—41no—
124C3×#41—41no—
125C2×C3×Q8—21no—
126C2×C3×D4—21no—
127C2×C3×D4—21no—
128C2×C3×D4—21no—
129C2×C3×D4—21no—
130C2×C3×D4—21no—
131C2×C3×D4—21no—
132C2×C3×D4—21no—
133C2×C3×D4—21no—
134C2×C3×D4—41no—
135C2×C3×D4D812yescomplemented
136C22×C3×C4—21no—
137C22×C3×C4—41no—
138C22×C3×C4—41no—
139C22×C3×C4—41no—
140C22×C3×C4—82no—
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C2×#67—44no—
145C2×#67—44no—
146C2×#67—82no—
147C22×Dic3—22no—
148C22×Dic3—21no—
149C2×D12—21no—
150C2×D12—21no—
151C2×D12—42no—
152C2×C4×D3—41no—
153C2×C4×D3—41no—
154C2×C4×D3—41no—
155C2×C4×D3—41no—
156C2×C4×D3—41no—
157C2×C4×D3—41no—
158C3×SD16—44no—
159C3×D8—44no—
160C3×M16—41no—
161C3×M16—41no—
162C2×C3×C8—22no—
163C3×C4⋊C4—21no—
164C3×C4⋊C4—21no—
165C3×C4⋊C4—21no—
166C3×C4⋊C4—21no—
167C3×C22⋊C4—41no—
168C3×C22⋊C4—41no—
169C3×C22⋊C4—41no—
170C3×C22⋊C4—41no—
171C3×C42C2×D411yescharacteristic
172C3×C42C2×D411yescharacteristic
173C3×C42C2×D411yescharacteristic
174C3×C42#4111yescharacteristic
175#217—44no—
176#213—44no—
177(C2×D3)⋊C4—41no—
178(C2×D3)⋊C4—41no—
179(C2×D3)⋊C4—82no—
180C4⋊Dic3—21no—
181C4⋊Dic3—21no—
182C4⋊Dic3—41no—
183C4⋊Dic3—41no—
184C2×C3⋊C8—22no—
185C2×C3⋊C8—41no—
186C2×#41—21no—
187C22×D4—31no—
188C22×D4—62no—
189C22×D4#6711yescharacteristic
190C23×C4—21no—
191C23×C4—61no—
192AGL(1,ℤ/8)—42no—
193AGL(1,ℤ/8)—42no—
194C2×SD16—22no—
195C2×D8—22no—
196C2×D8—62no—
197C2×M16—21no—
198C4⋊Q8#6711yescharacteristic, complemented
199#127#6711yescharacteristic, complemented
200#124—21no—
201#121—21no—
202#121—21no—
203#121—122no—
204#121—122no—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4—61no—
212C4×D4—61no—
213C4×D4—61no—
214C4×D4—61no—
215C2×C4⋊C4—31no—
216C2×C4⋊C4—61no—
217C2×C4⋊C4#6711yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—62no—
220C2×C22⋊C4—61no—
221C2×C22⋊C4—62no—
222C2×C42C22×D311yescharacteristic
223#107—62no—
224C4⋊C8—21no—
225C4⋊C8—61no—
226C4⋊C8—61no—
227C4⋊C8#6712yescomplemented
228#103—22no—
229#102—22no—
230#102—62no—
231#102—62no—
232C22⋊C8—61no—
233C22⋊C8—61no—
234C4×C8—32no—
235C4×C8—61no—
236#95—32no—
237#95—62no—
238#95—61no—
239C23×C3—22no—
240C23×C3—42no—
241C23×C3—42no—
242C23×C3—41no—
243C23×C3—41no—
244C23×C3—41no—
245C23×C3#12011yescharacteristic
246C22×D3—22no—
247C22×D3—41no—
248C22×D3—41no—
249C22×D3—41no—
250C22×D3—41no—
251C22×D3—41no—
252C22×D3—41no—
253C3×Q8—41no—
254C3×Q8—41no—
255C3×D4—22no—
256C3×D4—22no—
257C3×D4—42no—
258C3×D4—41no—
259C3×D4—41no—
260C3×D4—41no—
261C3×D4—41no—
262C3×D4—81no—
263C3×D4—81no—
264C3×D4—81no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—42no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—82no—
271C2×C3×C4—82no—
272C2×C3×C4—81no—
273C2×C3×C4—81no—
274C2×C3×C4#12011yescharacteristic
275C2×C3×C4#12011yescharacteristic
276C2×C3×C4#12111yescharacteristic
277C2×C3×C4#12111yescharacteristic
278C2×C3×C4#12111yescharacteristic
279C2×C3×C4#12711yescharacteristic
280C2×C3×C4C2×D811yescharacteristic
281C2×C3×C4#13511yescharacteristic
282C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
283C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
284#67—88no—
285C2×Dic3—42no—
286C2×Dic3—42no—
287C2×Dic3—41no—
288C2×Dic3—42no—
289C2×Dic3—41no—
290C2×Dic3—41no—
291D12—22no—
292D12—22no—
293D12—41no—
294D12—41no—
295C4×D3—81no—
296C4×D3—81no—
297C4×D3—81no—
298C4×D3—81no—
299C3×C8—42no—
300C3⋊C8—42no—
301C3⋊C8—81no—
302C24—21no—
303C24—61no—
304#41—41no—
305#41—41no—
306#41—41no—
307#41—41no—
308C2×Q8—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—41no—
318C2×D4—61no—
319C2×D4—61no—
320C2×D4—122no—
321C2×D4—124no—
322C2×D4—124no—
323C2×D4—242no—
324C2×D4#21312yescomplemented
325C22×C4—21no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—62no—
330C22×C4—61no—
331C22×C4—82no—
332C22×C4—121no—
333C22×C4—121no—
334C22×C4—121no—
335C22×C4—121no—
336C22×C4—121no—
337C22×C4—121no—
338C22×C4D3×D411yescharacteristic
339C22×C4D3×D411yescharacteristic
340C22×C4C2×#6711yescharacteristic
341SD16—44no—
342D8—44no—
343D8—124no—
344M16—41no—
345M16—41no—
346C2×C8—22no—
347C2×C8—62no—
348C2×C8—121no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—61no—
354C4⋊C4—61no—
355C4⋊C4—121no—
356C4⋊C4—121no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—124no—
362C22⋊C4—121no—
363C22⋊C4—121no—
364C22⋊C4—242no—
365C42D3×D411yescharacteristic
366C42#23711yescharacteristic
367C42C2×#6711yescharacteristic
368C42C2×#6711yescharacteristic
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—44no—
372C22×C3—42no—
373C22×C3—82no—
374C22×C3—81no—
375C22×C3—81no—
376C22×C3—81no—
377C22×C3—81no—
378C22×C3—81no—
379C22×C3#39211yescharacteristic
380C22×C3#44711yescharacteristic
381C22×C3#44911yescharacteristic
382C2×D3—42no—
383C2×D3—42no—
384C2×D3—42no—
385C2×D3—81no—
386C2×D3—81no—
387C2×D3—81no—
388C2×D3—81no—
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—81no—
394C3×C4—81no—
395C3×C4#44711yescharacteristic
396C3×C4#44911yescharacteristic
397C3×C4#46611yescharacteristic
398C3×C4#46911yescharacteristic
399Dic3—82no—
400Dic3—81no—
401C23—22no—
402C23—42no—
403C23—42no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—62no—
408C23—121no—
409C23—121no—
410C23—121no—
411C23—121no—
412C23—121no—
413C23—121no—
414C23#93711yescharacteristic
415Q8—41no—
416Q8—41no—
417D4—22no—
418D4—22no—
419D4—42no—
420D4—41no—
421D4—41no—
422D4—41no—
423D4—41no—
424D4—62no—
425D4—62no—
426D4—81no—
427D4—81no—
428D4—81no—
429D4—121no—
430D4—121no—
431D4—248no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—122no—
442C2×C4—122no—
443C2×C4—121no—
444C2×C4—121no—
445C2×C4—122no—
446C2×C4—121no—
447C2×C4—241no—
448C2×C4—241no—
449C2×C4—241no—
450C2×C4—241no—
451C2×C4C2×#21311yescharacteristic
452C2×C4#93211yescharacteristic
453C2×C4#93711yescharacteristic
454C2×C4#93811yescharacteristic
455C2×C4#93911yescharacteristic
456C2×C4#93911yescharacteristic
457C2×C4#94011yescharacteristic
458C2×C4#94911yescharacteristic
459C2×C4#95011yescharacteristic
460C2×C4#95311yescharacteristic
461C8—42no—
462C8—122no—
463C8—241no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—82no—
467C2×C3—161no—
468C2×C3#199911yescharacteristic
469C2×C3#200811yescharacteristic
470C2×C3#201011yescharacteristic
471D3—82no—
472C22—21no—
473C22—21no—
474C22—44no—
475C22—42no—
476C22—82no—
477C22—81no—
478C22—81no—
479C22—81no—
480C22—81no—
481C22—81no—
482C22—122no—
483C22—122no—
484C22—122no—
485C22—241no—
486C22—241no—
487C22—241no—
488C22—241no—
489C22#503611yescharacteristic
490C22#503711yescharacteristic
491C22#522311yescharacteristic
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—81no—
497C4—81no—
498C4—242no—
499C4—241no—
500C4#503811yescharacteristic
501C4#503911yescharacteristic
502C4#521711yescharacteristic
503C4#523811yescharacteristic
504C3#999611yescharacteristic, complemented, Sylow
505C2—21no—
506C2—21no—
507C2—82no—
508C2—161no—
509C2—242no—
510C2#7168411yescharacteristic
511C2#7209911yescharacteristic
512C2#7211811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3036C2
socleC22×C3#392
hypercenter#3036D3
hypocenterC3#9996
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3036

Complements

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

NH ≅ G/Ntype
1. C3×#3036C2semidirect
2. #83298C2semidirect
3. #82416C2semidirect
4. #71814C2semidirect
5. #71597C2semidirect
6. #70496C2semidirect
7. #67713C2semidirect
35. #3036D3semidirect
52. C3×C4⋊Q8D4semidirect
53. C3×#127D4semidirect
79. C3×C4⋊C8D4semidirect
93. C4×C3⋊C8D4semidirect
135. C2×C3×D4D8semidirect
198. C4⋊Q8#67semidirect
199. #127#67semidirect
227. C4⋊C8#67semidirect
324. C2×D4#213semidirect
504. C3#9996semidirect

Maximal subgroups

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

subgroupindex
C3×#30362
#832982
#824162
#718142
#715972
#704962
#677132
#99963