Finite Group Databasebeta

← #10698693

Subgroups — #10698693

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
1C3×#2537C211yesmaximal, characteristic
2#71604C211yesmaximal, characteristic, complemented
3#71604C211yesmaximal, characteristic, complemented
4C2×D4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
5C2×D4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6#69574C211yesmaximal, characteristic, complemented
7#67634C211yesmaximal, characteristic, complemented
8#10431—31nomaximal, Sylow
9C2×C3×#121—21no—
10C2×C3×#121—21no—
11C2×C3×C4×D4C2211yescharacteristic
12C2×C3×C4×D4C2211yescharacteristic
13C3×#394—22no—
14C3×#384C2211yescharacteristic
15C3×#380—21no—
16C3×#380—21no—
17C2×#832—21no—
18C2×#832—21no—
19#4935C2211yescharacteristic
20C2×C4⋊(C3⋊C8)C2211yescharacteristic
21C2×C4⋊(C3⋊C8)C2211yescharacteristic
22C2×C4×C3⋊C8C2211yescharacteristic
23D4⋊(C3⋊C8)—22no—
24D4⋊(C3⋊C8)—22no—
25#4529—21no—
26#4529—21no—
27#4529—22no—
28#2537D311yescharacteristic
29#1834—31no—
30#1802—31no—
31#1802—31no—
32C2×D4⋊C8—31no—
33C2×D4⋊C8—31no—
34#1296—31no—
35C22×C3×D4—22no—
36C22×C3×D4D411yescharacteristic
37C22×C3×D4D411yescharacteristic
38C23×C3×C4—21no—
39C23×C3×C4—21no—
40C3×#121—42no—
41C3×#121—42no—
42C3×#121—42no—
43C3×#121—42no—
44C3×C4×D4—21no—
45C3×C4×D4—21no—
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C4⋊C4D412yes—
57C2×C3×C22⋊C4—22no—
58C2×C3×C22⋊C4—22no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4—21no—
61C2×C3×C42C2311yescharacteristic
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3—21no—
64C2×C4⋊Dic3—21no—
65C22×C3⋊C8—21no—
66C22×C3⋊C8—21no—
67C22×C3⋊C8—21no—
68C3×#95—22no—
69#832—42no—
70#832—42no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75C4×C3⋊C8—21no—
76C4×C3⋊C8D412yescomplemented
77C2×#121—21no—
78C2×#121—21no—
79C2×C4×D4C2×D311yescharacteristic
80C2×C4×D4C2×D311yescharacteristic
81C2×C4⋊C8—31no—
82C2×C4⋊C8—31no—
83C2×#102—61no—
84C2×#102—61no—
85C2×C4×C8—31no—
86#394—22no—
87#384—31no—
88#384C2×D311yescharacteristic
89#380—21no—
90#380—21no—
91#340—61no—
92#340—61no—
93#340—62no—
94D4⋊C8—62no—
95D4⋊C8—62no—
96C24×C3—21no—
97C24×C3—21no—
98C2×C3×D4—21no—
99C2×C3×D4—21no—
100C2×C3×D4—21no—
101C2×C3×D4—21no—
102C2×C3×D4—21no—
103C2×C3×D4—21no—
104C2×C3×D4—21no—
105C2×C3×D4—21no—
106C2×C3×D4—21no—
107C2×C3×D4—21no—
108C2×C3×D4—44no—
109C2×C3×D4—44no—
110C2×C3×D4—82no—
111C2×C3×D4D812yes—
112C2×C3×D4SD1612yes—
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4—22no—
116C22×C3×C4—41no—
117C22×C3×C4—41no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4—41no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4—41no—
126C22×C3×C4—41no—
127C22×C3×C4—41no—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×Dic3—41no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—42no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C22⋊C4—44no—
142C3×C22⋊C4—44no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C22⋊C4—41no—
146C3×C22⋊C4—41no—
147C3×C42C2×D411yescharacteristic
148C3×C42#4111yescharacteristic
149C3×C42#4111yescharacteristic
150C3×C42#4111yescharacteristic
151C4⋊Dic3—42no—
152C4⋊Dic3—41no—
153C4⋊Dic3—41no—
154C4⋊Dic3—41no—
155C4⋊Dic3—41no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159C2×C3⋊C8—41no—
160C2×C3⋊C8—41no—
161C2×C3⋊C8—41no—
162C2×C3⋊C8—41no—
163C2×C3⋊C8—41no—
164C2×C3⋊C8—41no—
165C22×D4—22no—
166C22×D4#6711yescharacteristic
167C22×D4#6711yescharacteristic
168C23×C4—21no—
169C23×C4—21no—
170C22×C8—61no—
171C22×C8—61no—
172C22×C8—61no—
173#121—42no—
174#121—42no—
175#121—42no—
176#121—42no—
177C4×D4—21no—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—61no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4#6711yescharacteristic
191C2×C4⋊C4#6711yescharacteristic
192C2×C4⋊C4#6712yes—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C42C22×D311yescharacteristic
198C4⋊C8—61no—
199C4⋊C8—61no—
200C4⋊C8—61no—
201C4⋊C8—61no—
202#102—122no—
203#102—122no—
204C4×C8—32no—
205C4×C8—61no—
206#95—22no—
207C23×C3—22no—
208C23×C3—22no—
209C23×C3—41no—
210C23×C3—41no—
211C23×C3—41no—
212C23×C3—41no—
213C23×C3—41no—
214C23×C3—41no—
215C23×C3—41no—
216C23×C3—41no—
217C23×C3—41no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3#12011yescharacteristic
222C3×D4—22no—
223C3×D4—22no—
224C3×D4—22no—
225C3×D4—22no—
226C3×D4—41no—
227C3×D4—41no—
228C3×D4—41no—
229C3×D4—41no—
230C3×D4—88no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—42no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—42no—
240C2×C3×C4—41no—
241C2×C3×C4—42no—
242C2×C3×C4—81no—
243C2×C3×C4—81no—
244C2×C3×C4—81no—
245C2×C3×C4—81no—
246C2×C3×C4—81no—
247C2×C3×C4—81no—
248C2×C3×C4—81no—
249C2×C3×C4—81no—
250C2×C3×C4#12111yescharacteristic
251C2×C3×C4#12111yescharacteristic
252C2×C3×C4#12111yescharacteristic
253C2×C3×C4#12311yescharacteristic
254C2×C3×C4#12311yescharacteristic
255C2×C3×C4#12411yescharacteristic
256C2×C3×C4C2×D811yescharacteristic
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×C4AGL(1,ℤ/8)11yescharacteristic
263C2×C3×C4#13711yescharacteristic
264C2×Dic3—81no—
265C2×Dic3—81no—
266C2×Dic3—81no—
267C3⋊C8—42no—
268C3⋊C8—81no—
269C3⋊C8—81no—
270C24—21no—
271C24—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—44no—
283C2×D4—44no—
284C2×D4—82no—
285C2×D4#21312yes—
286C2×D4#21412yes—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—121no—
303C22×C4C2×#6711yescharacteristic
304C22×C4C2×#6711yescharacteristic
305C22×C4C2×#6711yescharacteristic
306C2×C8—62no—
307C2×C8—121no—
308C2×C8—121no—
309C2×C8—121no—
310C2×C8—121no—
311C2×C8—121no—
312C2×C8—121no—
313C2×C8—121no—
314C2×C8—121no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—42no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—122no—
325C4⋊C4—121no—
326C4⋊C4—121no—
327C4⋊C4—121no—
328C4⋊C4—121no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C42D3×D411yescharacteristic
336C42#23711yescharacteristic
337C42#23711yescharacteristic
338C42#23711yescharacteristic
339C22×C3—42no—
340C22×C3—42no—
341C22×C3—42no—
342C22×C3—42no—
343C22×C3—42no—
344C22×C3—42no—
345C22×C3—81no—
346C22×C3—81no—
347C22×C3—81no—
348C22×C3—81no—
349C22×C3—81no—
350C22×C3—81no—
351C22×C3—81no—
352C22×C3—81no—
353C22×C3#39411yescharacteristic
354C22×C3#44711yescharacteristic
355C22×C3#44911yescharacteristic
356C22×C3#45011yescharacteristic
357C22×C3#45211yescharacteristic
358C22×C3#45411yescharacteristic
359C22×C3#45511yescharacteristic
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—82no—
363C3×C4—81no—
364C3×C4—81no—
365C3×C4#46511yescharacteristic
366C3×C4#46611yescharacteristic
367C3×C4#46811yescharacteristic
368C3×C4#46911yescharacteristic
369C3×C4#48011yescharacteristic
370C3×C4#48111yescharacteristic
371C3×C4#48211yescharacteristic
372C3×C4#48211yescharacteristic
373Dic3—161no—
374C23—22no—
375C23—22no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23#95311yescharacteristic
389D4—22no—
390D4—22no—
391D4—22no—
392D4—22no—
393D4—41no—
394D4—41no—
395D4—41no—
396D4—41no—
397D4—88no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—241no—
418C2×C4—241no—
419C2×C4—241no—
420C2×C4C2×#21311yescharacteristic
421C2×C4#93211yescharacteristic
422C2×C4#93211yescharacteristic
423C2×C4C2×#21411yescharacteristic
424C2×C4#93511yescharacteristic
425C2×C4#93511yescharacteristic
426C2×C4#93611yescharacteristic
427C2×C4#93811yescharacteristic
428C2×C4#93911yescharacteristic
429C2×C4#93911yescharacteristic
430C2×C4#94911yescharacteristic
431C2×C4#95011yescharacteristic
432C2×C4#95011yescharacteristic
433C2×C4#95111yescharacteristic
434C8—122no—
435C8—241no—
436C8—241no—
437C2×C3—82no—
438C2×C3—82no—
439C2×C3#162511yescharacteristic
440C2×C3#163511yescharacteristic
441C2×C3#163911yescharacteristic
442C2×C3#164611yescharacteristic
443C2×C3#203311yescharacteristic
444C2×C3#203411yescharacteristic
445C2×C3#204011yescharacteristic
446C22—42no—
447C22—42no—
448C22—42no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—81no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22#521711yescharacteristic
461C22#521811yescharacteristic
462C22#522211yescharacteristic
463C22#523811yescharacteristic
464C22#523911yescharacteristic
465C22#524211yescharacteristic
466C22#524511yescharacteristic
467C4—21no—
468C4—21no—
469C4—81no—
470C4—82no—
471C4—81no—
472C4—481no—
473C4#503311yescharacteristic
474C4#503411yescharacteristic
475C4#503511yescharacteristic
476C4#503511yescharacteristic
477C4#503811yescharacteristic
478C4#503911yescharacteristic
479C4#504111yescharacteristic
480C4#504211yescharacteristic
481C3#1043111yescharacteristic, complemented, Sylow
482C2—82no—
483C2—82no—
484C2#7167911yescharacteristic
485C2#7168011yescharacteristic
486C2#7187711yescharacteristic
487C2#7190811yescharacteristic
488C2#7191011yescharacteristic
489C2#7191111yescharacteristic
490C2#7192911yescharacteristic
subgroupG/H
centerC23#953
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#2537C2
socleC23×C3#120
hypercenter#2537D3
hypocenterC3#10431
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2537

Complements

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

NH ≅ G/Ntype
2. #71604C2semidirect
3. #71604C2semidirect
4. C2×D4⋊(C3⋊C8)C2semidirect
5. C2×D4⋊(C3⋊C8)C2semidirect
6. #69574C2semidirect
7. #67634C2semidirect
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
C3×#25372
#716042
C2×D4⋊(C3⋊C8)2
#695742
#676342
#104313