Finite Group Databasebeta

← #10697497

Subgroups — #10697497

Order 768.

676 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82482C211yesmaximal, characteristic, complemented
2C2×#5027C211yesmaximal, characteristic
3C2×#4848C211yesmaximal, characteristic
4C2×C3×Q8⋊C8C211yesmaximal, characteristic, complemented
5#69772C211yesmaximal, characteristic
6C2×#4456C211yesmaximal, characteristic, complemented
7#67702C211yesmaximal, characteristic, complemented
8#10230—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C22×#215—21no—
11C22×#204—21no—
12C2×#874C2211yescharacteristic
13C2×C4×Dic6C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×C3×C4×C8C2211yescharacteristic
16#5229—21no—
17C2×#821—21no—
18C2×#821—22no—
19#5027—22no—
20#5027—22no—
21#4982—22no—
22#4972—21no—
23C2×#809—21no—
24#4935C2211yescharacteristic
25#4929—21no—
26#4928—21no—
27C2×C4⋊(C3⋊C8)C2211yescharacteristic
28#4848—22no—
29#4848—22no—
30C3×Q8⋊C8—22no—
31#4456—22no—
32C2×#460—31no—
33C2×#460—31no—
34#2582—31no—
35#1957—31no—
36C2×Q8⋊C8—31no—
37C2×Q8⋊C8D311yescharacteristic, complemented
38#1296—31no—
39C22×C3×Q8D411yescharacteristic
40C22×D12—21no—
41C22×D12—21no—
42C22×D12—21no—
43C22×Dic6D411yescharacteristic
44C22×C3×C8—21no—
45C22×C3×C8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C42—21no—
53C2×C3×C42C2311yescharacteristic
54C2×#215—24no—
55C2×#215—42no—
56C2×#215—42no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3D411yescharacteristic
60C2×C4⋊Dic3D412yes—
61C2×Dic3⋊C4—21no—
62C2×Dic3⋊C4—22no—
63C2×C4×Dic3—21no—
64C22×C3⋊C8—21no—
65C2×#204—24no—
66C2×#204—42no—
67C2×#204—42no—
68#874—21no—
69#874—21no—
70#874—21no—
71#874D412yes—
72C4×Dic6—21no—
73C4×Dic6—21no—
74C4×Dic6—21no—
75C4×Dic6—21no—
76C3×C4⋊C8—21no—
77C3×C4⋊C8D412yes—
78C3×C4×C8—21no—
79C3×C4×C8—21no—
80C3⋊#95—22no—
81C3⋊#95—22no—
82C3⋊#95—22no—
83#821—24no—
84#821—44no—
85#809—24no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)D412yes—
88C22×SD16—61no—
89C22×SD16—61no—
90C2×#127—31no—
91C2×C4×Q8—31no—
92C2×C4×Q8C2×D311yescharacteristic
93#460—62no—
94#460—62no—
95#460—62no—
96#460—62no—
97C2×C4⋊C8—31no—
98C2×C4⋊C8C2×D311yescharacteristic
99C2×#102—61no—
100C2×#102—61no—
101C2×#102—62no—
102C2×C4×C8C2×D311yescharacteristic
103#398—62no—
104#391—61no—
105#391—61no—
106#389—61no—
107#389—61no—
108#384—31no—
109Q8⋊C8—22no—
110Q8⋊C8—62no—
111C23×D3—41no—
112C2×C3×Q8—21no—
113C2×C3×Q8—21no—
114C2×C3×Q8—21no—
115C2×C3×Q8—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8SD1612yescomplemented
118C22×C3×C4—21no—
119C22×C3×C4—21no—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×Dic3—21no—
124C22×Dic3—22no—
125C22×Dic3—21no—
126C2×D12—22no—
127C2×D12—22no—
128C2×D12—22no—
129C2×D12—42no—
130C2×D12—42no—
131C2×D12—41no—
132C2×D12—41no—
133C2×D12—41no—
134C2×D12—41no—
135C2×D12—41no—
136C2×D12—41no—
137C2×D12—41no—
138C2×D12—41no—
139C2×D12—41no—
140C2×D12—41no—
141C2×D12—41no—
142C2×Dic6—21no—
143C2×Dic6—21no—
144C2×Dic6—21no—
145C2×Dic6—21no—
146C2×Dic6—21no—
147C2×Dic6SD1612yescomplemented
148C2×C3×C8—22no—
149C2×C3×C8—22no—
150C2×C3×C8—41no—
151C2×C3×C8—41no—
152C2×C3×C8—41no—
153C2×C3×C8—41no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—41no—
157C3×C4⋊C4—41no—
158C3×C42—41no—
159C3×C42—41no—
160C3×C42C2×D411yescharacteristic
161C3×C42C2×D411yescharacteristic
162C3×C42C2×D411yescharacteristic
163C3×C42#4111yescharacteristic
164#215—48no—
165C4⋊Dic3—21no—
166C4⋊Dic3—21no—
167C4⋊Dic3—41no—
168C4⋊Dic3—41no—
169C4⋊Dic3—41no—
170C4⋊Dic3—41no—
171C4⋊Dic3—42no—
172Dic3⋊C4—41no—
173Dic3⋊C4—41no—
174Dic3⋊C4—82no—
175C4×Dic3—41no—
176C4×Dic3—41no—
177C2×C3⋊C8—22no—
178C2×C3⋊C8—41no—
179C2×C3⋊C8—41no—
180#204—48no—
181C22×Q8—31no—
182C22×Q8D1211yescharacteristic
183C22×D4—61no—
184C22×D4—61no—
185C22×D4—61no—
186C2×SD16—64no—
187C2×SD16—64no—
188C2×SD16—122no—
189C2×SD16—122no—
190C2×SD16—122no—
191C2×SD16—122no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—61no—
195#127—32no—
196#127—61no—
197#127—61no—
198#127—61no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8—21no—
202C4×Q8—21no—
203C4×Q8—61no—
204C4×Q8—61no—
205C4×Q8—61no—
206C4×Q8—61no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—31no—
209C2×C4⋊C4—32no—
210C2×C4⋊C4—61no—
211C2×C4⋊C4—61no—
212C2×C4⋊C4—61no—
213C2×C4⋊C4—62no—
214C2×C4⋊C4D1211yescharacteristic
215C2×C42—21no—
216C2×C42—61no—
217C2×C42C22×D311yescharacteristic
218C4⋊C8—21no—
219C4⋊C8—32no—
220C4⋊C8—61no—
221C4⋊C8#6712yes—
222#102—64no—
223#102—64no—
224#102—124no—
225C4×C8—21no—
226C4×C8—21no—
227#95—62no—
228#95—62no—
229#95—62no—
230C23×C3#12011yescharacteristic
231C22×D3—42no—
232C22×D3—81no—
233C22×D3—81no—
234C22×D3—81no—
235C22×D3—81no—
236C22×D3—81no—
237C22×D3—81no—
238C3×Q8—22no—
239C3×Q8—22no—
240C3×Q8—41no—
241C3×Q8—41no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—22no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4#12011yescharacteristic
251C2×C3×C4#12011yescharacteristic
252C2×C3×C4#12111yescharacteristic
253C2×C3×C4#12111yescharacteristic
254C2×C3×C4#12111yescharacteristic
255C2×C3×C4#12711yescharacteristic
256C2×C3×C4C2×SD1611yescharacteristic
257C2×C3×C4C2×SD1611yescharacteristic
258C2×C3×C4#13511yescharacteristic
259C2×C3×C4#13511yescharacteristic
260C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
261C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
262C2×C3×C4#13711yescharacteristic
263C2×C3×C4#13711yescharacteristic
264C2×Dic3—22no—
265C2×Dic3—41no—
266C2×Dic3—41no—
267C2×Dic3—41no—
268C2×Dic3—42no—
269C2×Dic3—42no—
270C2×Dic3—41no—
271C2×Dic3—42no—
272C2×Dic3—41no—
273D12—44no—
274D12—42no—
275D12—42no—
276D12—42no—
277D12—42no—
278D12—82no—
279D12—81no—
280D12—81no—
281D12—81no—
282D12—81no—
283Dic6—22no—
284Dic6—22no—
285Dic6—41no—
286Dic6—41no—
287C3×C8—42no—
288C3×C8—42no—
289C3⋊C8—42no—
290C24—121no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—32no—
297C2×Q8—61no—
298C2×Q8—61no—
299C2×Q8—61no—
300C2×Q8—61no—
301C2×Q8—61no—
302C2×Q8#20412yescomplemented
303C2×D4—62no—
304C2×D4—62no—
305C2×D4—62no—
306C2×D4—122no—
307C2×D4—122no—
308C2×D4—121no—
309C2×D4—121no—
310C2×D4—121no—
311C2×D4—121no—
312C2×D4—121no—
313C2×D4—121no—
314C2×D4—121no—
315C2×D4—121no—
316C2×D4—121no—
317C2×D4—121no—
318C2×D4—121no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—61no—
322C22×C4—62no—
323C22×C4—61no—
324C22×C4C2×D1211yescharacteristic
325C22×C4D3×D411yescharacteristic
326C22×C4D3×D411yescharacteristic
327SD16—128no—
328SD16—128no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—62no—
336C2×C8—121no—
337C2×C8—121no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—61no—
343C4⋊C4—61no—
344C4⋊C4—122no—
345C4⋊C4—121no—
346C4⋊C4—121no—
347C4⋊C4—121no—
348C4⋊C4—121no—
349C4⋊C4—121no—
350C4⋊C4—121no—
351C4⋊C4—242no—
352C42—41no—
353C42—41no—
354C42—121no—
355C42—121no—
356C42#23511yescharacteristic
357C42D3×D411yescharacteristic
358C42D3×D411yescharacteristic
359C42C2×#6711yescharacteristic
360C22×C3#39211yescharacteristic
361C22×C3#44811yescharacteristic
362C22×C3#44811yescharacteristic
363C22×C3#44911yescharacteristic
364C22×C3#44911yescharacteristic
365C22×C3#45311yescharacteristic
366C22×C3#45411yescharacteristic
367C2×D3—82no—
368C2×D3—82no—
369C2×D3—82no—
370C2×D3—161no—
371C2×D3—161no—
372C2×D3—161no—
373C2×D3—161no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—42no—
377C3×C4—81no—
378C3×C4#45011yescharacteristic
379C3×C4#45011yescharacteristic
380C3×C4#45211yescharacteristic
381C3×C4#45211yescharacteristic
382C3×C4#46511yescharacteristic
383C3×C4#46511yescharacteristic
384C3×C4#46811yescharacteristic
385C3×C4#46811yescharacteristic
386Dic3—42no—
387Dic3—82no—
388Dic3—81no—
389C23—122no—
390C23—241no—
391C23—241no—
392C23—241no—
393C23—241no—
394C23—241no—
395C23—241no—
396C23#88211yescharacteristic
397Q8—22no—
398Q8—22no—
399Q8—41no—
400Q8—41no—
401Q8—62no—
402Q8—62no—
403Q8—121no—
404Q8—121no—
405D4—124no—
406D4—122no—
407D4—122no—
408D4—122no—
409D4—122no—
410D4—242no—
411D4—241no—
412D4—241no—
413D4—241no—
414D4—241no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—62no—
424C2×C4—121no—
425C2×C4—121no—
426C2×C4—121no—
427C2×C4—122no—
428C2×C4—122no—
429C2×C4—122no—
430C2×C4—121no—
431C2×C4—121no—
432C2×C4#88211yescharacteristic
433C2×C4#88411yescharacteristic
434C2×C4#88411yescharacteristic
435C2×C4C2×#20411yescharacteristic
436C2×C4#90411yescharacteristic
437C2×C4#90811yescharacteristic
438C2×C4#90911yescharacteristic
439C2×C4D3×SD1611yescharacteristic
440C2×C4#91411yescharacteristic
441C2×C4#91811yescharacteristic
442C2×C4#91911yescharacteristic
443C2×C4#93011yescharacteristic
444C2×C4#93711yescharacteristic
445C2×C4#94011yescharacteristic
446C8—42no—
447C8—42no—
448C8—122no—
449C2×C3#162011yescharacteristic
450C2×C3#162311yescharacteristic
451C2×C3#162611yescharacteristic
452C2×C3#162611yescharacteristic
453C2×C3#200011yescharacteristic
454C2×C3#200011yescharacteristic
455C2×C3#200611yescharacteristic
456D3—162no—
457C22—242no—
458C22—242no—
459C22—242no—
460C22—481no—
461C22—481no—
462C22—481no—
463C22—481no—
464C22#473411yescharacteristic
465C22#473811yescharacteristic
466C22#480611yescharacteristic
467C22#481011yescharacteristic
468C22#482211yescharacteristic
469C22#482711yescharacteristic
470C22#496011yescharacteristic
471C4—21no—
472C4—21no—
473C4—42no—
474C4—81no—
475C4—122no—
476C4—242no—
477C4—241no—
478C4#473111yescharacteristic
479C4#473311yescharacteristic
480C4#480711yescharacteristic
481C4#481211yescharacteristic
482C4#511411yescharacteristic
483C4#513411yescharacteristic
484C4#516911yescharacteristic
485C4#518711yescharacteristic
486C3#1023011yescharacteristic, complemented, Sylow
487C2—482no—
488C2#6825211yescharacteristic
489C2#6825811yescharacteristic
490C2#6826411yescharacteristic
491C2#6826511yescharacteristic
492C2#6996211yescharacteristic
493C2#7027211yescharacteristic
494C2#7039211yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×Q8⋊C8C2
socleC23×C3#120
prime radicalC2×#874C22
hypercenterC2×Q8⋊C8D3
hypocenterC3#10230
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
1. #82482C2semidirect
4. C2×C3×Q8⋊C8C2semidirect
6. C2×#4456C2semidirect
7. #67702C2semidirect
37. C2×Q8⋊C8D3semidirect
117. C2×C3×Q8SD16semidirect
147. C2×Dic6SD16semidirect
302. C2×Q8#204semidirect
486. C3#10230semidirect

Maximal subgroups

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

subgroupindex
#824822
C2×#50272
C2×#48482
C2×C3×Q8⋊C82
#697722
C2×#44562
#677022
#102303