Finite Group Databasebeta

← #10823203

Subgroups — #10823203

Order 768.

1458 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#909C211yesmaximal, characteristic
2C2×#5118C211yesmaximal, characteristic, complemented
3#82228C211yesmaximal, characteristic
4C2×#5004C211yesmaximal, characteristic, complemented
5C2×#4973C211yesmaximal, characteristic, complemented
6C2×#4966C211yesmaximal, characteristic
7C3×#1736C211yesmaximal, characteristic, complemented
8#68864C214yesmaximal, complemented
9#68863C214yesmaximal, complemented
10#23547—31nomaximal, Sylow
11C22×#235C2211yescharacteristic
12C23×Dic6C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C4×Dic3C2211yescharacteristic
17C22×#208C2211yescharacteristic
18C2×#909—24no—
19C2×#909—24no—
20C2×#909—24no—
21C22×Dic12C2212yes—
22C22×#204C2212yes—
23C2×C3⋊#95—22no—
24#5118—24no—
25C2×Dic3⋊C8C2214yes—
26#5004—24no—
27#4988C2212yes—
28#4973—21no—
29#4973—21no—
30#4973C2212yes—
31#4973C2214yes—
32C2×#810C2212yes—
33C2×#810C2214yes—
34#4966—24no—
35C2×#809C2212yes—
36#4958—22no—
37C4×(C2×D3)⋊C4—21no—
38C4×(C2×D3)⋊C4—21no—
39C3×#340C2214yes—
40C22×#137—31no—
41C2×#423—31no—
42C2×#417—31no—
43C2×#419—31no—
44#2297—31no—
45C2×#391—31no—
46#2056—34no—
47#2055—34no—
48#1736D311yescharacteristic, complemented
49C23×C3×C4—21no—
50C23×C3×C4C2311yescharacteristic
51C22×#67—21no—
52C23×Dic3—21no—
53C23×Dic3D412yes—
54C2×#235—22no—
55C2×#235—22no—
56C2×#235—22no—
57C2×#235—22no—
58C2×#235—22no—
59C2×#235D414yes—
60C22×D12C2311yescharacteristic
61C22×C4×D3—21no—
62C22×Dic6—22no—
63C22×Dic6—22no—
64C22×Dic6—22no—
65C22×Dic6—22no—
66C22×Dic6—21no—
67C22×Dic6—22no—
68C22×Dic6D414yes—
69C22×Dic6C2312yes—
70C22×Dic6C2311yescharacteristic
71C2×C3×M16—22no—
72C2×C3×M16—22no—
73C2×C3×M16D414yes—
74C22×C3×C8C2312yes—
75C3×#117—22no—
76C3×#117—22no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C22⋊C4—21no—
83C2×C3×C42—21no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—22no—
86C2×C4×Dic3—22no—
87C2×C4×Dic3—22no—
88C2×C4×Dic3D411yescharacteristic
89C2×C4×Dic3D411yescharacteristic
90C2×C4×Dic3C2312yes—
91C2×#208—22no—
92C2×#208—22no—
93C2×#208—22no—
94C22×C3⋊C8C2312yes—
95#909—232no—
96C2×Dic12—24no—
97C2×Dic12—24no—
98C2×Dic12—24no—
99C2×#204—24no—
100C2×#204—24no—
101C2×#204—24no—
102C3⋊#95—22no—
103C3⋊#95—22no—
104C3⋊#95—24no—
105C3⋊#95—24no—
106Dic3⋊C8—28no—
107#810—24no—
108#810—28no—
109#809—24no—
110C22×#41—31no—
111C23×Q8—31no—
112C2×#137—64no—
113C2×#137—64no—
114C2×#137—64no—
115C22×Q16—32no—
116C22×SD16—32no—
117C22×M16—31no—
118C22×M16C2×D311yescharacteristic
119C2×#117C2×D311yescharacteristic
120C22×C4⋊C4C2×D311yescharacteristic
121C22×C42—31no—
122#423—64no—
123C2×C4⋊C8—34no—
124#419—64no—
125#417—64no—
126C2×#103—34no—
127C2×#103—32no—
128C2×#102—32no—
129#391—34no—
130#391—32no—
131#391—61no—
132#391—61no—
133#390—32no—
134#386—62no—
135C4×C22⋊C4—61no—
136C4×C22⋊C4—61no—
137C2×#95—62no—
138#340C2×D314yes—
139C24×C3C2×D411yescharacteristic
140C23×D3—21no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—44no—
144C22×C3×C4—42no—
145C22×C3×C4C2×D412yes—
146C22×C3×C4C2×D412yes—
147C22×C3×C4C2×D412yes—
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4#4112yes—
150C22×C3×C4#4112yes—
151C22×C3×C4#4112yes—
152C22×C3×C4C2411yescharacteristic
153C2×#67—24no—
154C2×#67—42no—
155C2×#67—42no—
156C22×Dic3—24no—
157C22×Dic3—24no—
158C22×Dic3—22no—
159C22×Dic3—21no—
160C22×Dic3—24no—
161C22×Dic3—42no—
162C22×Dic3—42no—
163C22×Dic3—42no—
164C22×Dic3C2×D412yes—
165C22×Dic3C2×D412yes—
166#235—28no—
167#235—28no—
168#235—44no—
169#235—44no—
170C2×D12—21no—
171C2×D12—21no—
172C2×D12—21no—
173C2×D12—21no—
174C2×D12—21no—
175C2×D12C2×D412yes—
176C2×C4×D3—24no—
177C2×C4×D3—42no—
178C2×C4×D3—42no—
179C2×Dic6—28no—
180C2×Dic6—28no—
181C2×Dic6—22no—
182C2×Dic6—22no—
183C2×Dic6—21no—
184C2×Dic6—21no—
185C2×Dic6—22no—
186C2×Dic6—22no—
187C2×Dic6—21no—
188C2×Dic6—21no—
189C2×Dic6—24no—
190C2×Dic6—22no—
191C2×Dic6—21no—
192C2×Dic6—42no—
193C2×Dic6—42no—
194C2×Dic6—44no—
195C2×Dic6—44no—
196C2×Dic6C2×D414yes—
197C2×Dic6C2×D412yes—
198C3×M16—28no—
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C2×C3×C8C2×D414yes—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—42no—
207C3×C22⋊C4—42no—
208C3×C42—42no—
209(C2×D3)⋊C4—44no—
210(C2×D3)⋊C4—44no—
211C4×Dic3—22no—
212C4×Dic3—22no—
213C4×Dic3—22no—
214C4×Dic3—22no—
215#208—44no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219Dic12—216no—
220#204—216no—
221C2×#41—34no—
222C2×#41—62no—
223C2×#41—62no—
224C2×#41—62no—
225C2×#41—62no—
226C2×#41—62no—
227C22×Q8—34no—
228C22×Q8—32no—
229C22×Q8—31no—
230C22×Q8—62no—
231C22×Q8—61no—
232C22×Q8—62no—
233C22×Q8—62no—
234C22×Q8—62no—
235C22×Q8—62no—
236C22×D4—31no—
237C22×D4—61no—
238C23×C4—21no—
239C23×C4—32no—
240C23×C4—61no—
241C23×C4—61no—
242C23×C4C22×D311yescharacteristic
243#137—632no—
244C2×Q16—64no—
245C2×Q16—64no—
246C2×Q16—64no—
247C2×SD16—64no—
248C2×SD16—64no—
249C2×SD16—64no—
250C2×M16—22no—
251C2×M16—22no—
252C2×M16—62no—
253C2×M16—62no—
254C2×M16—62no—
255C2×M16D1214yes—
256C22×C8—32no—
257C22×C8C22×D312yes—
258#117—22no—
259#117—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4C22×D312yes—
264C2×C4⋊C4C22×D312yes—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—62no—
267C2×C22⋊C4—62no—
268C2×C42—21no—
269C2×C42—31no—
270C2×C42—31no—
271C2×C42—32no—
272C2×C42—62no—
273C2×C42—62no—
274C4⋊C8—68no—
275#103—64no—
276#103—68no—
277#102—64no—
278#95—64no—
279#95—64no—
280#95—62no—
281#95—62no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3#12112yes—
285C23×C3#12112yes—
286C23×C3#12112yes—
287C23×C3C22×D411yescharacteristic
288C22×D3—22no—
289C22×D3—41no—
290C22×D3—41no—
291C22×D3—42no—
292C22×D3—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—44no—
298C2×C3×C4—84no—
299C2×C3×C4#12014yes—
300C2×C3×C4#12112yes—
301C2×C3×C4#12112yes—
302C2×C3×C4#12112yes—
303C2×C3×C4#12314yes—
304C2×C3×C4#12314yes—
305C2×C3×C4#12414yes—
306C2×C3×C4C22×D411yescharacteristic
307C2×C3×C4C22×D411yescharacteristic
308C2×C3×C4C22×D411yescharacteristic
309C2×C3×C4C2×#4111yescharacteristic
310C2×C3×C4C2×#4111yescharacteristic
311C2×C3×C4C2×#4111yescharacteristic
312#67—48no—
313C2×Dic3—22no—
314C2×Dic3—22no—
315C2×Dic3—22no—
316C2×Dic3—24no—
317C2×Dic3—22no—
318C2×Dic3—22no—
319C2×Dic3—22no—
320C2×Dic3—48no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325C2×Dic3—48no—
326C2×Dic3#13714yescomplemented
327D12—22no—
328D12—22no—
329D12—41no—
330D12—41no—
331C4×D3—48no—
332Dic6—24no—
333Dic6—22no—
334Dic6—24no—
335Dic6—22no—
336Dic6—42no—
337Dic6—41no—
338Dic6—42no—
339Dic6—41no—
340Dic6—48no—
341C3×C8—24no—
342C3⋊C8—42no—
343C24—61no—
344C24D3×D411yescharacteristic
345#41—68no—
346#41—68no—
347#41—124no—
348#41—124no—
349C2×Q8—34no—
350C2×Q8—32no—
351C2×Q8—64no—
352C2×Q8—62no—
353C2×Q8—61no—
354C2×Q8—68no—
355C2×Q8—68no—
356C2×Q8—62no—
357C2×Q8—61no—
358C2×Q8—62no—
359C2×Q8—61no—
360C2×Q8—62no—
361C2×Q8—61no—
362C2×Q8—62no—
363C2×Q8—61no—
364C2×Q8—124no—
365C2×Q8—122no—
366C2×Q8—124no—
367C2×Q8—122no—
368C2×D4—32no—
369C2×D4—64no—
370C2×D4—61no—
371C2×D4—61no—
372C2×D4—61no—
373C2×D4—61no—
374C2×D4—61no—
375C2×D4—122no—
376C2×D4—122no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—32no—
380C22×C4—32no—
381C22×C4—44no—
382C22×C4—42no—
383C22×C4—64no—
384C22×C4—64no—
385C22×C4—64no—
386C22×C4—61no—
387C22×C4—62no—
388C22×C4—64no—
389C22×C4—122no—
390C22×C4—122no—
391C22×C4—122no—
392C22×C4—122no—
393C22×C4—122no—
394C22×C4C2×D1212yes—
395C22×C4#23512yes—
396C22×C4#23512yes—
397C22×C4D3×D412yes—
398C22×C4D3×D412yes—
399C22×C4D3×D411yescharacteristic
400C22×C4#23912yes—
401C22×C4C23×D311yescharacteristic
402Q16—616no—
403SD16—616no—
404M16—28no—
405M16—124no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—62no—
409C2×C8—62no—
410C2×C8—62no—
411C2×C8C2×D1214yes—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—42no—
417C22⋊C4—42no—
418C22⋊C4—124no—
419C22⋊C4—124no—
420C42—42no—
421C42—62no—
422C42—62no—
423C42—62no—
424C42—62no—
425C22×C3—24no—
426C22×C3—22no—
427C22×C3—22no—
428C22×C3—24no—
429C22×C3#39414yes—
430C22×C3C2×#12111yescharacteristic
431C22×C3C2×#12111yescharacteristic
432C22×C3C2×#12111yescharacteristic
433C22×C3C2×#13712yes—
434C22×C3#57512yes—
435C2×D3—42no—
436C2×D3—42no—
437C2×D3—42no—
438C2×D3—82no—
439C2×D3—82no—
440C3×C4—82no—
441C3×C4—82no—
442C3×C4#39414yes—
443C3×C4C2×#12011yescharacteristic
444C3×C4C2×#12311yescharacteristic
445C3×C4C2×#12311yescharacteristic
446C3×C4C2×#12411yescharacteristic
447Dic3—24no—
448Dic3—44no—
449Dic3—42no—
450Dic3—42no—
451C23—22no—
452C23—22no—
453C23—62no—
454C23—121no—
455C23—121no—
456C23—122no—
457C23—122no—
458C23#88412yes—
459C23#88412yes—
460C23#89512yes—
461C23C2×D3×D411yescharacteristic
462Q8—64no—
463Q8—62no—
464Q8—64no—
465Q8—62no—
466Q8—128no—
467Q8—122no—
468Q8—121no—
469Q8—122no—
470Q8—121no—
471D4—62no—
472D4—62no—
473D4—128no—
474D4—121no—
475D4—121no—
476C2×C4—34no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—62no—
483C2×C4—62no—
484C2×C4—62no—
485C2×C4—64no—
486C2×C4—62no—
487C2×C4—62no—
488C2×C4—62no—
489C2×C4—84no—
490C2×C4—128no—
491C2×C4—128no—
492C2×C4—121no—
493C2×C4—122no—
494C2×C4—121no—
495C2×C4—122no—
496C2×C4—128no—
497C2×C4#87514yes—
498C2×C4#88214yes—
499C2×C4#88412yes—
500C2×C4#88412yes—
501C2×C4#89414yes—
502C2×C4#89414yes—
503C2×C4#89512yes—
504C2×C4C22×D1211yescharacteristic
505C2×C4C2×#23511yescharacteristic
506C2×C4C2×#23511yescharacteristic
507C2×C4C2×D3×D411yescharacteristic
508C2×C4C2×D3×D411yescharacteristic
509C2×C4C2×#23911yescharacteristic
510C8—24no—
511C8—122no—
512C2×C3—24no—
513C2×C3C2×#39411yescharacteristic
514C2×C3#303712yes—
515C2×C3#303911yescharacteristic
516C2×C3#304011yescharacteristic
517C2×C3#305612yes—
518D3—82no—
519C22—24no—
520C22—22no—
521C22—24no—
522C22—22no—
523C22—122no—
524C22—122no—
525C22—122no—
526C22—242no—
527C22—242no—
528C22#467214yes—
529C22C2×#88411yescharacteristic
530C22C2×#88411yescharacteristic
531C22C2×#89511yescharacteristic
532C22#577312yes—
533C22D3×#13712yes—
534C4—64no—
535C4—82no—
536C4—82no—
537C4—122no—
538C4—122no—
539C4—124no—
540C4#467214yes—
541C4C2×#87511yescharacteristic
542C4C2×#88211yescharacteristic
543C4C2×#89411yescharacteristic
544C4C2×#89411yescharacteristic
545C3#2354711yescharacteristic, complemented, Sylow
546C2—24no—
547C2—242no—
548C2C2×#467211yescharacteristic
549C2#7891312yes—
550C2#7894311yescharacteristic
551C2#7897411yescharacteristic
552C2#7921212yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1736C2
socleC23×C3C22×D4
prime radicalC22×#235C22
hypercenter#1736D3
hypocenterC3#23547
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
2. C2×#5118C2semidirect
4. C2×#5004C2semidirect
5. C2×#4973C2semidirect
7. C3×#1736C2semidirect
8. #68864C2semidirect
9. #68863C2semidirect
48. #1736D3semidirect
326. C2×Dic3#137semidirect
545. C3#23547semidirect

Maximal subgroups

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

subgroupindex
C22×#9092
C2×#51182
#822282
C2×#50042
C2×#49732
C2×#49662
C3×#17362
#688642
#688632
#235473