Finite Group Databasebeta

← #10698695

Subgroups — #10698695

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
2C2×C3×D4⋊C8C211yesmaximal, characteristic, complemented
3#71604C211yesmaximal, characteristic, complemented
4#70479C211yesmaximal, characteristic, complemented
5#69762C211yesmaximal, characteristic, complemented
6C2×#4459C211yesmaximal, 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×C4×C8C2211yescharacteristic
15C2×#832—21no—
16#5212—21no—
17C2×#821—21no—
18#4987—21no—
19#4961—22no—
20#4935C2211yescharacteristic
21#4935C2211yescharacteristic
22C2×C4⋊(C3⋊C8)C2211yescharacteristic
23C3×D4⋊C8—22no—
24#4551—21no—
25#4551—22no—
26#4529—21no—
27#4459—22no—
28#2537—31no—
29#1834—31no—
30#1802—31no—
31#1802—31no—
32C2×D4⋊C8—31no—
33C2×D4⋊C8D311yescharacteristic, complemented
34#1296—31no—
35C22×C3×D4D411yescharacteristic
36C23×C3×C4—21no—
37C22×#67—22no—
38C22×D12D411yescharacteristic
39C22×C4×D3—21no—
40C22×C3×C8—21no—
41C22×C3×C8—21no—
42C3×C4×D4—21no—
43C3×C4×D4—21no—
44C3×C4×D4—21no—
45C3×C4×D4—21no—
46C2×C3×C4⋊C4D411yescharacteristic
47C2×C3×C22⋊C4—21no—
48C2×C3×C42C2311yescharacteristic
49C2×#217—22no—
50#938—42no—
51#938—42no—
52#930—42no—
53#930—42no—
54C2×(C2×D3)⋊C4—22no—
55C2×(C2×D3)⋊C4—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3D411yescharacteristic
62C2×C4⋊Dic3D412yes—
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×C4×C8—21no—
71C3×C4×C8D412yescomplemented
72#832—42no—
73C3⋊#95—22no—
74#821—42no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77C2×#121—61no—
78C2×#121—61no—
79C2×C4×D4—31no—
80C2×C4×D4C2×D311yescharacteristic
81C2×C4⋊C8—31no—
82C2×C4⋊C8C2×D311yescharacteristic
83C2×#102—61no—
84C2×#102—61no—
85C2×C4×C8C2×D311yescharacteristic
86#394—62no—
87#384—31no—
88#384—31no—
89#380—61no—
90#380—61no—
91#340—61no—
92#340—62no—
93#340—61no—
94D4⋊C8—22no—
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×D4D812yes—
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×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C2×#67—44no—
115C2×#67—44no—
116C2×#67—82no—
117C22×Dic3—21no—
118C22×Dic3—22no—
119C22×Dic3—41no—
120C2×D12—21no—
121C2×D12—21no—
122C2×D12—21no—
123C2×D12—21no—
124C2×D12—21no—
125C2×D12SD1612yes—
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—22no—
133C2×C3×C8—41no—
134C2×C3×C8—41no—
135C2×C3×C8—41no—
136C2×C3×C8—41no—
137C2×C3×C8—41no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C22⋊C4—41no—
141C3×C22⋊C4—41no—
142C3×C42C2×D411yescharacteristic
143C3×C42#4111yescharacteristic
144C3×C42#4111yescharacteristic
145C3×C42#4111yescharacteristic
146#217—44no—
147(C2×D3)⋊C4—44no—
148(C2×D3)⋊C4—41no—
149(C2×D3)⋊C4—41no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—41no—
153C4⋊Dic3—41no—
154C4⋊Dic3—41no—
155C4⋊Dic3—41no—
156C4⋊Dic3—42no—
157C4⋊Dic3—42no—
158C4⋊Dic3—41no—
159C4⋊Dic3—41no—
160C4⋊Dic3—41no—
161C4⋊Dic3—41no—
162C2×C3⋊C8—41no—
163C2×C3⋊C8—41no—
164C2×C3⋊C8—41no—
165C22×D4—31no—
166C22×D4—62no—
167C22×D4D1211yescharacteristic
168C23×C4—21no—
169C23×C4—61no—
170C22×C8—21no—
171C22×C8—21no—
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—31no—
186C2×C4⋊C4—32no—
187C2×C4⋊C4—61no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4—61no—
192C2×C4⋊C4D1211yescharacteristic
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—122no—
203#102—122no—
204C4×C8—21no—
205C4×C8#6712yescomplemented
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#12111yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12311yescharacteristic
239C2×C3×C4#12311yescharacteristic
240C2×C3×C4#12411yescharacteristic
241C2×C3×C4C2×D811yescharacteristic
242C2×C3×C4C2×SD1611yescharacteristic
243C2×C3×C4#13511yescharacteristic
244C2×C3×C4#13511yescharacteristic
245C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
246C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
247C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
248C2×C3×C4#13711yescharacteristic
249#67—88no—
250C2×Dic3—41no—
251C2×Dic3—42no—
252C2×Dic3—41no—
253C2×Dic3—42no—
254C2×Dic3—41no—
255C2×Dic3—42no—
256C2×Dic3—81no—
257C2×Dic3—81no—
258C2×Dic3—81no—
259D12—22no—
260D12—22no—
261D12—41no—
262D12—41no—
263C4×D3—81no—
264C4×D3—81no—
265C4×D3—81no—
266C4×D3—81no—
267C3×C8—42no—
268C3×C8—81no—
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×D4D2412yes—
287C22×C4—21no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—61no—
295C22×C4—62no—
296C22×C4—121no—
297C22×C4—121no—
298C22×C4—121no—
299C22×C4—121no—
300C22×C4—121no—
301C22×C4—121no—
302C22×C4—121no—
303C22×C4C2×D1211yescharacteristic
304C22×C4D3×D411yescharacteristic
305C22×C4D3×D411yescharacteristic
306C2×C8—22no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—121no—
313C2×C8—121no—
314C2×C8—121no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—61no—
318C4⋊C4—61no—
319C4⋊C4—122no—
320C4⋊C4—122no—
321C4⋊C4—121no—
322C4⋊C4—121no—
323C4⋊C4—121no—
324C4⋊C4—121no—
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#23711yescharacteristic
337C42#23711yescharacteristic
338C42C2×#6711yescharacteristic
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#46511yescharacteristic
364C3×C4#46611yescharacteristic
365C3×C4#46811yescharacteristic
366C3×C4#46911yescharacteristic
367C3×C4#48011yescharacteristic
368C3×C4#48111yescharacteristic
369C3×C4#48211yescharacteristic
370C3×C4#48211yescharacteristic
371Dic3—82no—
372Dic3—81no—
373Dic3—161no—
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#88211yescharacteristic
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—121no—
408C2×C4—121no—
409C2×C4—122no—
410C2×C4—121no—
411C2×C4—122no—
412C2×C4—122no—
413C2×C4—241no—
414C2×C4—241no—
415C2×C4—241no—
416C2×C4—241no—
417C2×C4—241no—
418C2×C4—241no—
419C2×C4—241no—
420C2×C4#88311yescharacteristic
421C2×C4#88511yescharacteristic
422C2×C4#88611yescharacteristic
423C2×C4C2×D2411yescharacteristic
424C2×C4#90411yescharacteristic
425C2×C4#90811yescharacteristic
426C2×C4#90911yescharacteristic
427C2×C4#91111yescharacteristic
428C2×C4#91111yescharacteristic
429C2×C4D3×SD1611yescharacteristic
430C2×C4#91611yescharacteristic
431C2×C4#93011yescharacteristic
432C2×C4#93811yescharacteristic
433C2×C4#93911yescharacteristic
434C8—42no—
435C8—81no—
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#473211yescharacteristic
461C22#473311yescharacteristic
462C22#477611yescharacteristic
463C22#478111yescharacteristic
464C22#482511yescharacteristic
465C22#482711yescharacteristic
466C22#496111yescharacteristic
467C4—21no—
468C4—21no—
469C4—81no—
470C4—241no—
471C4—242no—
472C4—481no—
473C4#473611yescharacteristic
474C4#473711yescharacteristic
475C4#477711yescharacteristic
476C4#477911yescharacteristic
477C4#511511yescharacteristic
478C4#513411yescharacteristic
479C4#515911yescharacteristic
480C4#517111yescharacteristic
481C3#1043111yescharacteristic, complemented, Sylow
482C2—82no—
483C2—242no—
484C2#6818111yescharacteristic
485C2#6818311yescharacteristic
486C2#6819611yescharacteristic
487C2#6820011yescharacteristic
488C2#6993411yescharacteristic
489C2#7017611yescharacteristic
490C2#7036211yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×D4⋊C8C2
socleC23×C3#120
prime radical#82417C2
hypercenterC2×D4⋊C8D3
hypocenterC3#10431
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×D4⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×C3×D4⋊C8C2semidirect
3. #71604C2semidirect
4. #70479C2semidirect
5. #69762C2semidirect
6. C2×#4459C2semidirect
7. #67702C2semidirect
33. C2×D4⋊C8D3semidirect
71. C3×C4×C8D4semidirect
205. C4×C8#67semidirect
481. C3#10431semidirect

Maximal subgroups

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

subgroupindex
#824172
C2×C3×D4⋊C82
#716042
#704792
#697622
C2×#44592
#677022
#104313