Finite Group Databasebeta

← #10698701

Subgroups — #10698701

Order 768.

630 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82417C211yesmaximal, characteristic
2C3×#1802C211yesmaximal, characteristic, complemented
3C2×D4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
4C2×#4483C211yesmaximal, characteristic, complemented
5#69584C211yesmaximal, characteristic, complemented
6#69576C211yesmaximal, characteristic, complemented
7#67702C211yesmaximal, characteristic, complemented
8#10432—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×#938—21no—
11C2×#930—21no—
12C2×C4×D12C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×C3×#102—21no—
15C3×#384C2211yescharacteristic
16#5212—21no—
17#4987—21no—
18C2×#809—21no—
19C2×#808—22no—
20#4961—22no—
21#4935C2211yescharacteristic
22C2×C4⋊(C3⋊C8)C2211yescharacteristic
23C2×C4×C3⋊C8C2211yescharacteristic
24C3×#340—21no—
25D4⋊(C3⋊C8)—22no—
26#4529—21no—
27#4483—22no—
28#2537—31no—
29#1844—31no—
30#1802—31no—
31#1802D311yescharacteristic, complemented
32C2×D4⋊C8—31no—
33C2×D4⋊C8—31no—
34#1296—31no—
35C22×C3×D4D411yescharacteristic
36C23×C3×C4—21no—
37C22×#67—22no—
38C22×D12D411yescharacteristic
39C22×C4×D3—21no—
40C22×C3×C8—21no—
41C3×C4×D4—21no—
42C3×C4×D4—21no—
43C3×C4×D4—21no—
44C3×C4×D4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4D411yescharacteristic
49C2×C3×C22⋊C4—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#217—22no—
52#938—42no—
53#938—42no—
54#930—42no—
55#930—42no—
56C2×(C2×D3)⋊C4—22no—
57C2×(C2×D3)⋊C4—21no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3D411yescharacteristic
61C2×C4⋊Dic3D412yes—
62C22×C3⋊C8—21no—
63C22×C3⋊C8—21no—
64C4×D12—21no—
65C4×D12—21no—
66C4×D12—21no—
67C4×D12—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×#102—42no—
71C3⋊#95—22no—
72#809—42no—
73#808—44no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4×C3⋊C8—21no—
77C4×C3⋊C8D412yescomplemented
78C2×#121—61no—
79C2×#121—61no—
80C2×C4×D4—31no—
81C2×C4×D4C2×D311yescharacteristic
82C2×#106—62no—
83C2×C4⋊C8—31no—
84C2×C4⋊C8C2×D311yescharacteristic
85C2×#102—21no—
86C2×#102—61no—
87C2×C4×C8—31no—
88#394—62no—
89#384—31no—
90#384C2×D311yescharacteristic
91#380—61no—
92#380—61no—
93#340—21no—
94#340—61no—
95D4⋊C8—62no—
96D4⋊C8—62no—
97C24×C3—21no—
98C23×D3—21no—
99C2×C3×D4—21no—
100C2×C3×D4—21no—
101C2×C3×D4—21no—
102C2×C3×D4—21no—
103C2×C3×D4—21no—
104C2×C3×D4SD1612yes—
105C22×C3×C4—21no—
106C22×C3×C4—41no—
107C22×C3×C4—41no—
108C22×C3×C4—41no—
109C22×C3×C4—41no—
110C22×C3×C4—41no—
111C22×C3×C4—41no—
112C22×C3×C4—41no—
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C2×#67—44no—
117C2×#67—44no—
118C2×#67—82no—
119C22×Dic3—21no—
120C22×Dic3—22no—
121C2×D12—21no—
122C2×D12—21no—
123C2×D12—21no—
124C2×D12—21no—
125C2×D12—21no—
126C2×D12SD1612yes—
127C2×C4×D3—41no—
128C2×C4×D3—41no—
129C2×C4×D3—41no—
130C2×C4×D3—41no—
131C2×C4×D3—41no—
132C2×C4×D3—41no—
133C2×C3×C8—41no—
134C2×C3×C8—41no—
135C2×C3×C8—41no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—42no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C42C2×D411yescharacteristic
146C3×C42#4111yescharacteristic
147C3×C42#4111yescharacteristic
148C3×C42#4111yescharacteristic
149#217—44no—
150(C2×D3)⋊C4—44no—
151(C2×D3)⋊C4—41no—
152(C2×D3)⋊C4—41no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155C4⋊Dic3—41no—
156C4⋊Dic3—41no—
157C4⋊Dic3—41no—
158C4⋊Dic3—41no—
159C4⋊Dic3—42no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—41no—
164C2×C3⋊C8—41no—
165C2×C3⋊C8—41no—
166C22×D4—31no—
167C22×D4—62no—
168C22×D4#6711yescharacteristic
169C23×C4—21no—
170C23×C4—61no—
171C22×C8—21no—
172C22×C8—61no—
173C22×C8—61no—
174#121—122no—
175#121—122no—
176#121—122no—
177#121—122no—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—61no—
183C4×D4—61no—
184C4×D4—61no—
185C4×D4—61no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—31no—
190C2×C4⋊C4—32no—
191C2×C4⋊C4—61no—
192C2×C4⋊C4—61no—
193C2×C4⋊C4#6711yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—62no—
196C2×C22⋊C4—62no—
197C2×C22⋊C4—61no—
198C2×C42C22×D311yescharacteristic
199#106—124no—
200C4⋊C8—21no—
201C4⋊C8—21no—
202C4⋊C8—61no—
203C4⋊C8—61no—
204#102—42no—
205#102—122no—
206C4×C8—32no—
207C4×C8—61no—
208#95—62no—
209C23×C3—22no—
210C23×C3—41no—
211C23×C3—41no—
212C23×C3—41no—
213C23×C3—41no—
214C23×C3—41no—
215C23×C3—41no—
216C23×C3#12011yescharacteristic
217C22×D3—22no—
218C22×D3—41no—
219C22×D3—41no—
220C22×D3—41no—
221C22×D3—41no—
222C22×D3—41no—
223C22×D3—41no—
224C3×D4—22no—
225C3×D4—22no—
226C3×D4—41no—
227C3×D4—41no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—81no—
234C2×C3×C4—81no—
235C2×C3×C4—81no—
236C2×C3×C4—81no—
237C2×C3×C4—81no—
238C2×C3×C4—81no—
239C2×C3×C4—81no—
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#12311yescharacteristic
244C2×C3×C4#12311yescharacteristic
245C2×C3×C4#12411yescharacteristic
246C2×C3×C4C2×SD1611yescharacteristic
247C2×C3×C4C2×SD1611yescharacteristic
248C2×C3×C4#13511yescharacteristic
249C2×C3×C4#13511yescharacteristic
250C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
251C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
252C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
253C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
254#67—88no—
255C2×Dic3—41no—
256C2×Dic3—41no—
257C2×Dic3—42no—
258C2×Dic3—42no—
259C2×Dic3—41no—
260C2×Dic3—42no—
261D12—22no—
262D12—22no—
263D12—41no—
264D12—41no—
265C4×D3—81no—
266C4×D3—81no—
267C4×D3—81no—
268C4×D3—81no—
269C3×C8—81no—
270C3⋊C8—24no—
271C3⋊C8—81no—
272C24—21no—
273C24—61no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—32no—
280C2×D4—61no—
281C2×D4—61no—
282C2×D4—61no—
283C2×D4—61no—
284C2×D4—61no—
285C2×D4—124no—
286C2×D4—124no—
287C2×D4—242no—
288C2×D4#21512yes—
289C22×C4—21no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—61no—
298C22×C4—62no—
299C22×C4—121no—
300C22×C4—121no—
301C22×C4—121no—
302C22×C4—121no—
303C22×C4—121no—
304C22×C4—121no—
305C22×C4D3×D411yescharacteristic
306C22×C4D3×D411yescharacteristic
307C22×C4C2×#6711yescharacteristic
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—62no—
312C2×C8—62no—
313C2×C8—62no—
314C2×C8—121no—
315C2×C8—121no—
316C2×C8—121no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—42no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—61no—
325C4⋊C4—61no—
326C4⋊C4—122no—
327C4⋊C4—121no—
328C4⋊C4—121no—
329C4⋊C4—121no—
330C4⋊C4—121no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—124no—
334C22⋊C4—124no—
335C22⋊C4—121no—
336C22⋊C4—121no—
337C42#23511yescharacteristic
338C42#23511yescharacteristic
339C42D3×D411yescharacteristic
340C42#23711yescharacteristic
341C22×C3—42no—
342C22×C3—42no—
343C22×C3—42no—
344C22×C3—81no—
345C22×C3—81no—
346C22×C3—81no—
347C22×C3—81no—
348C22×C3#39411yescharacteristic
349C22×C3#44911yescharacteristic
350C22×C3#44911yescharacteristic
351C22×C3#45011yescharacteristic
352C22×C3#45011yescharacteristic
353C22×C3#45311yescharacteristic
354C22×C3#45611yescharacteristic
355C2×D3—42no—
356C2×D3—42no—
357C2×D3—42no—
358C2×D3—81no—
359C2×D3—81no—
360C2×D3—81no—
361C2×D3—81no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—81no—
365C3×C4—161no—
366C3×C4#46511yescharacteristic
367C3×C4#46511yescharacteristic
368C3×C4#46911yescharacteristic
369C3×C4#46911yescharacteristic
370C3×C4#48111yescharacteristic
371C3×C4#48111yescharacteristic
372C3×C4#48211yescharacteristic
373C3×C4#48211yescharacteristic
374Dic3—82no—
375Dic3—81no—
376C23—22no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—62no—
384C23—121no—
385C23—121no—
386C23—121no—
387C23—121no—
388C23—121no—
389C23—121no—
390C23#93711yescharacteristic
391D4—22no—
392D4—22no—
393D4—41no—
394D4—41no—
395D4—62no—
396D4—62no—
397D4—121no—
398D4—121no—
399D4—248no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—122no—
413C2×C4—122no—
414C2×C4—121no—
415C2×C4—121no—
416C2×C4—121no—
417C2×C4—122no—
418C2×C4—241no—
419C2×C4—241no—
420C2×C4—241no—
421C2×C4—241no—
422C2×C4#88311yescharacteristic
423C2×C4#88411yescharacteristic
424C2×C4#88411yescharacteristic
425C2×C4#88511yescharacteristic
426C2×C4#91111yescharacteristic
427C2×C4#91211yescharacteristic
428C2×C4D3×SD1611yescharacteristic
429C2×C4#91411yescharacteristic
430C2×C4#92911yescharacteristic
431C2×C4#93211yescharacteristic
432C2×C4#93911yescharacteristic
433C2×C4C2×#21511yescharacteristic
434C2×C4#94911yescharacteristic
435C2×C4#95011yescharacteristic
436C8—64no—
437C8—81no—
438C8—241no—
439C2×C3—82no—
440C2×C3#162411yescharacteristic
441C2×C3#162911yescharacteristic
442C2×C3#165411yescharacteristic
443C2×C3#165411yescharacteristic
444C2×C3#203411yescharacteristic
445C2×C3#203411yescharacteristic
446C2×C3#203911yescharacteristic
447D3—82no—
448C22—42no—
449C22—42no—
450C22—42no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22—81no—
455C22—122no—
456C22—122no—
457C22—122no—
458C22—241no—
459C22—241no—
460C22—241no—
461C22—241no—
462C22#495911yescharacteristic
463C22#503711yescharacteristic
464C22#504611yescharacteristic
465C22#515811yescharacteristic
466C22#516911yescharacteristic
467C22#519811yescharacteristic
468C22#520511yescharacteristic
469C4—21no—
470C4—21no—
471C4—81no—
472C4—161no—
473C4—241no—
474C4—242no—
475C4#477711yescharacteristic
476C4#477811yescharacteristic
477C4#478211yescharacteristic
478C4#478411yescharacteristic
479C4#496811yescharacteristic
480C4#500311yescharacteristic
481C4#503911yescharacteristic
482C4#504811yescharacteristic
483C3#1043211yescharacteristic, complemented, Sylow
484C2—82no—
485C2—242no—
486C2#6952411yescharacteristic
487C2#7020711yescharacteristic
488C2#7031111yescharacteristic
489C2#7101711yescharacteristic
490C2#7105111yescharacteristic
491C2#7111111yescharacteristic
492C2#7114311yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1802C2
socleC23×C3#120
prime radical#82417C2
hypercenter#1802D3
hypocenterC3#10432
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. C3×#1802C2semidirect
3. C2×D4⋊(C3⋊C8)C2semidirect
4. C2×#4483C2semidirect
5. #69584C2semidirect
6. #69576C2semidirect
7. #67702C2semidirect
31. #1802D3semidirect
77. C4×C3⋊C8D4semidirect
483. C3#10432semidirect

Maximal subgroups

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

subgroupindex
#824172
C3×#18022
C2×D4⋊(C3⋊C8)2
C2×#44832
#695842
#695762
#677022
#104323