Finite Group Databasebeta

← #10698696

Subgroups — #10698696

Order 768.

614 conjugacy classes of subgroups.

The subgroup lattice has 492 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#69574C211yesmaximal, characteristic, complemented
7#67702C211yesmaximal, characteristic, complemented
8#10431—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—
19#4961—22no—
20#4935C2211yescharacteristic
21C2×C4⋊(C3⋊C8)C2211yescharacteristic
22C2×C4×C3⋊C8C2211yescharacteristic
23C3×#340—21no—
24D4⋊(C3⋊C8)—22no—
25#4529—22no—
26#4529—21no—
27#4483—22no—
28#2537—31no—
29#1834—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—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75C4×C3⋊C8—21no—
76C4×C3⋊C8D412yescomplemented
77C2×#121—61no—
78C2×#121—61no—
79C2×C4×D4—31no—
80C2×C4×D4C2×D311yescharacteristic
81C2×C4⋊C8—31no—
82C2×C4⋊C8C2×D311yescharacteristic
83C2×#102—21no—
84C2×#102—61no—
85C2×C4×C8—31no—
86#394—62no—
87#384—31no—
88#384C2×D311yescharacteristic
89#380—61no—
90#380—61no—
91#340—21no—
92#340—61no—
93#340—62no—
94D4⋊C8—62no—
95D4⋊C8—62no—
96C24×C3—21no—
97C23×D3—21no—
98C2×C3×D4—21no—
99C2×C3×D4—21no—
100C2×C3×D4—21no—
101C2×C3×D4—21no—
102C2×C3×D4—21no—
103C2×C3×D4SD1612yes—
104C22×C3×C4—21no—
105C22×C3×C4—41no—
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×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C2×#67—44no—
116C2×#67—44no—
117C2×#67—82no—
118C22×Dic3—21no—
119C22×Dic3—22no—
120C2×D12—21no—
121C2×D12—21no—
122C2×D12—21no—
123C2×D12—21no—
124C2×D12—21no—
125C2×D12D812yes—
126C2×C4×D3—41no—
127C2×C4×D3—41no—
128C2×C4×D3—41no—
129C2×C4×D3—41no—
130C2×C4×D3—41no—
131C2×C4×D3—41no—
132C2×C3×C8—41no—
133C2×C3×C8—41no—
134C2×C3×C8—41no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—42no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C42C2×D411yescharacteristic
145C3×C42#4111yescharacteristic
146C3×C42#4111yescharacteristic
147C3×C42#4111yescharacteristic
148#217—44no—
149(C2×D3)⋊C4—44no—
150(C2×D3)⋊C4—41no—
151(C2×D3)⋊C4—41no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154C4⋊Dic3—41no—
155C4⋊Dic3—41no—
156C4⋊Dic3—41no—
157C4⋊Dic3—41no—
158C4⋊Dic3—42no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—41no—
161C2×C3⋊C8—41no—
162C2×C3⋊C8—41no—
163C2×C3⋊C8—41no—
164C2×C3⋊C8—41no—
165C22×D4—31no—
166C22×D4—62no—
167C22×D4#6711yescharacteristic
168C23×C4—21no—
169C23×C4—61no—
170C22×C8—21no—
171C22×C8—61no—
172C22×C8—61no—
173#121—122no—
174#121—122no—
175#121—122no—
176#121—122no—
177C4×D4—21no—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—61no—
182C4×D4—61no—
183C4×D4—61no—
184C4×D4—61no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—31no—
189C2×C4⋊C4—32no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4—61no—
192C2×C4⋊C4#6711yescharacteristic
193C2×C22⋊C4—21no—
194C2×C22⋊C4—62no—
195C2×C22⋊C4—62no—
196C2×C22⋊C4—61no—
197C2×C42C22×D311yescharacteristic
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8—61no—
201C4⋊C8—61no—
202#102—42no—
203#102—122no—
204C4×C8—32no—
205C4×C8—61no—
206#95—62no—
207C23×C3—22no—
208C23×C3—41no—
209C23×C3—41no—
210C23×C3—41no—
211C23×C3—41no—
212C23×C3—41no—
213C23×C3—41no—
214C23×C3#12011yescharacteristic
215C22×D3—22no—
216C22×D3—41no—
217C22×D3—41no—
218C22×D3—41no—
219C22×D3—41no—
220C22×D3—41no—
221C22×D3—41no—
222C3×D4—22no—
223C3×D4—22no—
224C3×D4—41no—
225C3×D4—41no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—81no—
232C2×C3×C4—81no—
233C2×C3×C4—81no—
234C2×C3×C4—81no—
235C2×C3×C4—81no—
236C2×C3×C4—81no—
237C2×C3×C4—81no—
238C2×C3×C4#12111yescharacteristic
239C2×C3×C4#12111yescharacteristic
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12311yescharacteristic
242C2×C3×C4#12311yescharacteristic
243C2×C3×C4#12411yescharacteristic
244C2×C3×C4C2×D811yescharacteristic
245C2×C3×C4C2×SD1611yescharacteristic
246C2×C3×C4#13511yescharacteristic
247C2×C3×C4#13511yescharacteristic
248C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
249C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
250C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
251C2×C3×C4#13711yescharacteristic
252#67—88no—
253C2×Dic3—41no—
254C2×Dic3—42no—
255C2×Dic3—41no—
256C2×Dic3—42no—
257C2×Dic3—41no—
258C2×Dic3—42no—
259D12—22no—
260D12—22no—
261D12—41no—
262D12—41no—
263C4×D3—81no—
264C4×D3—81no—
265C4×D3—81no—
266C4×D3—81no—
267C3×C8—81no—
268C3⋊C8—42no—
269C3⋊C8—81no—
270C24—21no—
271C24—61no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—32no—
278C2×D4—61no—
279C2×D4—61no—
280C2×D4—61no—
281C2×D4—61no—
282C2×D4—61no—
283C2×D4—124no—
284C2×D4—124no—
285C2×D4—242no—
286C2×D4#21512yes—
287C22×C4—21no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—61no—
296C22×C4—62no—
297C22×C4—121no—
298C22×C4—121no—
299C22×C4—121no—
300C22×C4—121no—
301C22×C4—121no—
302C22×C4—121no—
303C22×C4D3×D411yescharacteristic
304C22×C4D3×D411yescharacteristic
305C22×C4C2×#6711yescharacteristic
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—62no—
310C2×C8—121no—
311C2×C8—121no—
312C2×C8—121no—
313C2×C8—121no—
314C2×C8—121no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—42no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—61no—
323C4⋊C4—61no—
324C4⋊C4—122no—
325C4⋊C4—121no—
326C4⋊C4—121no—
327C4⋊C4—121no—
328C4⋊C4—121no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—124no—
332C22⋊C4—124no—
333C22⋊C4—121no—
334C22⋊C4—121no—
335C42#23511yescharacteristic
336C42#23511yescharacteristic
337C42D3×D411yescharacteristic
338C42#23711yescharacteristic
339C22×C3—42no—
340C22×C3—42no—
341C22×C3—42no—
342C22×C3—81no—
343C22×C3—81no—
344C22×C3—81no—
345C22×C3—81no—
346C22×C3#39411yescharacteristic
347C22×C3#44711yescharacteristic
348C22×C3#44911yescharacteristic
349C22×C3#45011yescharacteristic
350C22×C3#45211yescharacteristic
351C22×C3#45411yescharacteristic
352C22×C3#45511yescharacteristic
353C2×D3—42no—
354C2×D3—42no—
355C2×D3—42no—
356C2×D3—81no—
357C2×D3—81no—
358C2×D3—81no—
359C2×D3—81no—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—81no—
363C3×C4—161no—
364C3×C4#46511yescharacteristic
365C3×C4#46611yescharacteristic
366C3×C4#46811yescharacteristic
367C3×C4#46911yescharacteristic
368C3×C4#48011yescharacteristic
369C3×C4#48111yescharacteristic
370C3×C4#48211yescharacteristic
371C3×C4#48211yescharacteristic
372Dic3—82no—
373Dic3—81no—
374C23—22no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—62no—
382C23—121no—
383C23—121no—
384C23—121no—
385C23—121no—
386C23—121no—
387C23—121no—
388C23#93711yescharacteristic
389D4—22no—
390D4—22no—
391D4—41no—
392D4—41no—
393D4—62no—
394D4—62no—
395D4—121no—
396D4—121no—
397D4—248no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—122no—
411C2×C4—122no—
412C2×C4—121no—
413C2×C4—121no—
414C2×C4—121no—
415C2×C4—122no—
416C2×C4—241no—
417C2×C4—241no—
418C2×C4—241no—
419C2×C4—241no—
420C2×C4#88311yescharacteristic
421C2×C4#88411yescharacteristic
422C2×C4#88411yescharacteristic
423C2×C4#88511yescharacteristic
424C2×C4D3×D811yescharacteristic
425C2×C4#91111yescharacteristic
426C2×C4#91511yescharacteristic
427C2×C4#91611yescharacteristic
428C2×C4#92911yescharacteristic
429C2×C4#93211yescharacteristic
430C2×C4#93911yescharacteristic
431C2×C4C2×#21511yescharacteristic
432C2×C4#94911yescharacteristic
433C2×C4#95011yescharacteristic
434C8—81no—
435C8—122no—
436C8—241no—
437C2×C3—82no—
438C2×C3#162511yescharacteristic
439C2×C3#163511yescharacteristic
440C2×C3#163911yescharacteristic
441C2×C3#164611yescharacteristic
442C2×C3#203311yescharacteristic
443C2×C3#203411yescharacteristic
444C2×C3#204011yescharacteristic
445D3—82no—
446C22—42no—
447C22—42no—
448C22—42no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22—81no—
453C22—122no—
454C22—122no—
455C22—122no—
456C22—241no—
457C22—241no—
458C22—241no—
459C22—241no—
460C22#495911yescharacteristic
461C22#503711yescharacteristic
462C22#504611yescharacteristic
463C22#515611yescharacteristic
464C22#517311yescharacteristic
465C22#520111yescharacteristic
466C22#520211yescharacteristic
467C4—21no—
468C4—21no—
469C4—81no—
470C4—161no—
471C4—241no—
472C4—242no—
473C4#477411yescharacteristic
474C4#477511yescharacteristic
475C4#477911yescharacteristic
476C4#478011yescharacteristic
477C4#496811yescharacteristic
478C4#500311yescharacteristic
479C4#503911yescharacteristic
480C4#504811yescharacteristic
481C3#1043111yescharacteristic, complemented, Sylow
482C2—82no—
483C2—242no—
484C2#6952411yescharacteristic
485C2#7020611yescharacteristic
486C2#7031211yescharacteristic
487C2#7101211yescharacteristic
488C2#7104711yescharacteristic
489C2#7110811yescharacteristic
490C2#7114611yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1802C2
socleC23×C3#120
prime radical#82417C2
hypercenter#1802D3
hypocenterC3#10431
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. #69574C2semidirect
7. #67702C2semidirect
31. #1802D3semidirect
76. C4×C3⋊C8D4semidirect
481. C3#10431semidirect

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
#695742
#677022
#104313