Finite Group Databasebeta

← #10944139

Subgroups — #10944139

Order 768.

1372 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5516C211yesmaximal, characteristic, complemented
2#82287C211yesmaximal, characteristic
3C2×#4845C212yesmaximal, complemented
4C2×#4739C211yesmaximal, characteristic, complemented
5C2×#4738C211yesmaximal, characteristic
6C3×#1798C211yesmaximal, characteristic, complemented
7#70485C212yesmaximal, complemented
8#69933C211yesmaximal, characteristic
9#69931C211yesmaximal, characteristic, complemented
10#68686C212yesmaximal, complemented
11#68685C212yesmaximal, complemented
12#32918—31nomaximal, Sylow
13C23×Dic6C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C2×#930C2211yescharacteristic
16C2×#926C2211yescharacteristic
17C2×#924C2211yescharacteristic
18C22×Dic12C2212yes—
19C22×#204C2212yes—
20#5516—28no—
21C2×#887—22no—
22C2×#877—22no—
23C2×#875C2211yescharacteristic
24C2×C4×Dic6C2212yes—
25C2×C4×Dic6C2212yes—
26C2×C3×C4⋊C8C2212yes—
27C2×C3×C22⋊C8C2211yescharacteristic
28C2×C3×C22⋊C8C2211yescharacteristic
29C2×#821C2212yes—
30C2×#816C2212yes—
31C2×#816C2212yes—
32C2×#816C2212yes—
33#4994—22no—
34#4961—22no—
35#4939C2211yescharacteristic
36#4935C2211yescharacteristic
37#4929C2211yescharacteristic
38#4845—28no—
39#4739—24no—
40#4738—24no—
41C3×#336C2212yes—
42#4551C2212yes—
43#4551C2212yes—
44C2×#519—31no—
45C2×#464—32no—
46C2×#451—31no—
47C2×#448—31no—
48#2620—31no—
49#1880—32no—
50#1878—32no—
51#1875—31no—
52#1873—31no—
53#1803—32no—
54#1798D311yescharacteristic, complemented
55C23×C3×C4C2311yescharacteristic
56C22×#67—21no—
57C23×Dic3—21no—
58C22×D12C2311yescharacteristic
59C22×Dic6—21no—
60C22×Dic6—22no—
61C22×Dic6—24no—
62C22×Dic6—24no—
63C22×Dic6D414yes—
64C22×Dic6C2311yescharacteristic
65C22×Dic6C2312yes—
66C22×C3×C8C2312yes—
67C22×C3×C8C2312yes—
68C3×#117—24no—
69C2×C3×C4⋊C4D412yes—
70C2×C3×C22⋊C4D412yes—
71C2×C3×C42C2311yescharacteristic
72C2×C3×C42C2311yescharacteristic
73C2×#217—21no—
74C2×#217—21no—
75#930—22no—
76#930—22no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—21no—
79C2×(C2×D3)⋊C4—22no—
80#926—24no—
81C2×C4⋊Dic3—22no—
82C2×C4⋊Dic3C2311yescharacteristic
83C2×C4⋊Dic3C2311yescharacteristic
84C2×C4⋊Dic3C2312yes—
85#924—22no—
86#924—22no—
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4—21no—
90C2×C4×Dic3—22no—
91C2×C4×Dic3—22no—
92C2×C4×Dic3—21no—
93C2×Dic12—24no—
94C2×Dic12—24no—
95C2×Dic12—24no—
96C2×#204—24no—
97C2×#204—24no—
98C2×#204—24no—
99#887—216no—
100#877—216no—
101#875—24no—
102C4×Dic6—24no—
103C4×Dic6—24no—
104C4×Dic6—24no—
105C4×Dic6D418yescomplemented
106C3×C4⋊C8—24no—
107C3×C22⋊C8—22no—
108C3×C22⋊C8—22no—
109C3⋊#95—22no—
110#821—24no—
111#816—24no—
112#816—24no—
113#816—24no—
114C23×Q8—31no—
115C22×Q16—32no—
116C22×SD16—32no—
117C2×#124—31no—
118C2×C22⋊Q8—31no—
119C2×#121—31no—
120#519—68no—
121C2×C4×Q8—32no—
122C2×C4×Q8—32no—
123C2×C4×Q8—62no—
124C2×#117—31no—
125C2×#117—62no—
126C2×#117C2×D311yescharacteristic
127#464—68no—
128#451—64no—
129#448—64no—
130C2×C4⋊C8C2×D312yes—
131C2×#103—32no—
132C2×#103—32no—
133C2×#103—32no—
134C2×#102—32no—
135C2×C22⋊C8C2×D311yescharacteristic
136C2×C22⋊C8C2×D311yescharacteristic
137#396—62no—
138#394—62no—
139#391—31no—
140#390—31no—
141#384—31no—
142#340—32no—
143#340—32no—
144#336C2×D312yes—
145C24×C3C2×D411yescharacteristic
146C23×D3—21no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4C2×D412yes—
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D412yes—
153C22×C3×C4#4112yes—
154C22×C3×C4C2411yescharacteristic
155C2×#67—42no—
156C2×#67—82no—
157C22×Dic3—24no—
158C22×Dic3—24no—
159C22×Dic3—24no—
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3—21no—
163C22×Dic3—21no—
164C22×Dic3—21no—
165C2×D12—22no—
166C2×D12—21no—
167C2×D12—22no—
168C2×D12—21no—
169C2×Dic6—24no—
170C2×Dic6—24no—
171C2×Dic6—216no—
172C2×Dic6—24no—
173C2×Dic6—22no—
174C2×Dic6—22no—
175C2×Dic6—22no—
176C2×Dic6—216no—
177C2×Dic6—22no—
178C2×Dic6—21no—
179C2×Dic6—22no—
180C2×Dic6—21no—
181C2×Dic6C2×D414yes—
182C2×Dic6C2×D414yes—
183C2×Dic6#4114yes—
184C2×C3×C8—22no—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C2×C3×C8—22no—
188C2×C3×C8—22no—
189C2×C3×C8—22no—
190C3×C4⋊C4—24no—
191C3×C22⋊C4—24no—
192C3×C42—22no—
193C3×C42C2×D414yes—
194#217—24no—
195#217—42no—
196(C2×D3)⋊C4—44no—
197(C2×D3)⋊C4—42no—
198(C2×D3)⋊C4—82no—
199C4⋊Dic3—22no—
200C4⋊Dic3—21no—
201C4⋊Dic3—21no—
202C4⋊Dic3—21no—
203C4⋊Dic3—21no—
204C4⋊Dic3—82no—
205C4⋊Dic3C2×D414yes—
206Dic3⋊C4—28no—
207Dic3⋊C4—28no—
208Dic3⋊C4—42no—
209C4×Dic3—28no—
210C4×Dic3—24no—
211C4×Dic3—28no—
212Dic12—48no—
213#204—48no—
214C22×Q8—34no—
215C22×Q8—32no—
216C22×Q8—31no—
217C22×Q8—64no—
218C22×Q8—64no—
219C22×Q8—61no—
220C22×Q8—62no—
221C22×D4—31no—
222C22×D4—61no—
223C23×C4—61no—
224C23×C4C22×D311yescharacteristic
225C2×Q16—64no—
226C2×Q16—64no—
227C2×Q16—64no—
228C2×SD16—64no—
229C2×SD16—64no—
230C2×SD16—64no—
231C22×C8C22×D312yes—
232C22×C8C22×D312yes—
233#124—64no—
234C22⋊Q8—62no—
235C22⋊Q8—62no—
236#121—62no—
237#121—62no—
238C4×Q8—38no—
239C4×Q8—64no—
240C4×Q8—64no—
241C4×Q8—616no—
242C4×Q8—64no—
243#117—24no—
244#117—616no—
245#117—64no—
246C2×C4⋊C4—32no—
247C2×C4⋊C4—31no—
248C2×C4⋊C4—31no—
249C2×C4⋊C4—62no—
250C2×C4⋊C4—62no—
251C2×C4⋊C4—61no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4D1212yes—
254C2×C22⋊C4—61no—
255C2×C22⋊C4—61no—
256C2×C22⋊C4—61no—
257C2×C22⋊C4—62no—
258C2×C22⋊C4—62no—
259C2×C22⋊C4D1212yes—
260C2×C42—61no—
261C2×C42—62no—
262C2×C42—62no—
263C2×C42C22×D311yescharacteristic
264C2×C42C22×D311yescharacteristic
265C4⋊C8—24no—
266#103—64no—
267#103—64no—
268#103—64no—
269#102—64no—
270C22⋊C8—22no—
271C22⋊C8—22no—
272#95—62no—
273C23×C3—21no—
274C23×C3—21no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3#12112yes—
278C23×C3C22×D411yescharacteristic
279C22×D3—41no—
280C22×D3—41no—
281C22×D3—41no—
282C22×D3—42no—
283C22×D3—42no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—24no—
289C2×C3×C4—24no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4#12014yes—
294C2×C3×C4#12112yes—
295C2×C3×C4C22×D411yescharacteristic
296C2×C3×C4C22×D412yes—
297C2×C3×C4C2×#4111yescharacteristic
298C2×C3×C4C2×#4112yes—
299#67—84no—
300C2×Dic3—216no—
301C2×Dic3—24no—
302C2×Dic3—24no—
303C2×Dic3—24no—
304C2×Dic3—24no—
305C2×Dic3—42no—
306C2×Dic3—42no—
307C2×Dic3—41no—
308C2×Dic3—41no—
309C2×Dic3—41no—
310C2×Dic3—41no—
311C2×Dic3—41no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314D12—42no—
315D12—42no—
316Dic6—216no—
317Dic6—24no—
318Dic6—24no—
319Dic6—24no—
320Dic6—42no—
321Dic6—42no—
322Dic6#12118yes—
323C3×C8—42no—
324C3×C8—42no—
325C24—61no—
326C24C2×D1211yescharacteristic
327C2×Q8—34no—
328C2×Q8—34no—
329C2×Q8—34no—
330C2×Q8—616no—
331C2×Q8—64no—
332C2×Q8—64no—
333C2×Q8—616no—
334C2×Q8—64no—
335C2×Q8—62no—
336C2×Q8—62no—
337C2×Q8—62no—
338C2×Q8—62no—
339C2×Q8—61no—
340C2×Q8—61no—
341C2×Q8—62no—
342C2×D4—62no—
343C2×D4—61no—
344C2×D4—61no—
345C2×D4—62no—
346C2×D4—122no—
347C2×D4—242no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—64no—
351C22×C4—64no—
352C22×C4—64no—
353C22×C4—62no—
354C22×C4—62no—
355C22×C4—61no—
356C22×C4—61no—
357C22×C4—61no—
358C22×C4C2×D1212yes—
359C22×C4C2×D1212yes—
360C22×C4C2×D1211yescharacteristic
361C22×C4D3×D412yes—
362C22×C4#23912yes—
363C22×C4C23×D311yescharacteristic
364Q16—128no—
365SD16—128no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C4⋊C4—24no—
373C4⋊C4—34no—
374C4⋊C4—68no—
375C4⋊C4—68no—
376C4⋊C4—62no—
377C4⋊C4—61no—
378C4⋊C4—61no—
379C4⋊C4—61no—
380C4⋊C4—61no—
381C4⋊C4—122no—
382C4⋊C4—242no—
383C22⋊C4—24no—
384C22⋊C4—64no—
385C22⋊C4—122no—
386C22⋊C4—124no—
387C22⋊C4—122no—
388C22⋊C4—242no—
389C42—22no—
390C42—64no—
391C42—68no—
392C42—68no—
393C42D3×D414yes—
394C22×C3—22no—
395C22×C3—22no—
396C22×C3—22no—
397C22×C3—42no—
398C22×C3—42no—
399C22×C3C2×#12011yescharacteristic
400C22×C3C2×#12111yescharacteristic
401C22×C3#52511yescharacteristic
402C22×C3#57712yes—
403C22×C3#57812yes—
404C2×D3—81no—
405C2×D3—82no—
406C2×D3—82no—
407C2×D3—81no—
408C2×D3—81no—
409C3×C4—22no—
410C3×C4—24no—
411C3×C4—42no—
412C3×C4C2×#12112yes—
413C3×C4#52512yes—
414Dic3—28no—
415Dic3—44no—
416Dic3—81no—
417Dic3—81no—
418Dic3—81no—
419C23—21no—
420C23—22no—
421C23—22no—
422C23—21no—
423C23—121no—
424C23—122no—
425C23—122no—
426C23—121no—
427C23—121no—
428C23#89512yes—
429C23C22×D1211yescharacteristic
430Q8—38no—
431Q8—616no—
432Q8—64no—
433Q8—64no—
434Q8—64no—
435Q8—122no—
436Q8—122no—
437D4—122no—
438D4—122no—
439D4—244no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—21no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—616no—
450C2×C4—64no—
451C2×C4—64no—
452C2×C4—64no—
453C2×C4—64no—
454C2×C4—122no—
455C2×C4—122no—
456C2×C4—121no—
457C2×C4—122no—
458C2×C4—122no—
459C2×C4—121no—
460C2×C4—121no—
461C2×C4—121no—
462C2×C4—121no—
463C2×C4#88214yes—
464C2×C4#89512yes—
465C2×C4C2×D3×D411yescharacteristic
466C2×C4C2×D3×D412yes—
467C2×C4C2×#23911yescharacteristic
468C2×C4D3×#4112yes—
469C8—42no—
470C8—42no—
471C2×C3—42no—
472C2×C3#284711yescharacteristic
473C2×C3#301312yes—
474C2×C3#304311yescharacteristic
475C2×C3#304411yescharacteristic
476C2×C3#312112yes—
477D3—161no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—42no—
482C22—42no—
483C22—242no—
484C22—241no—
485C22—242no—
486C22—241no—
487C22—241no—
488C22C2×#88211yescharacteristic
489C22C2×#89511yescharacteristic
490C22#552411yescharacteristic
491C22#575112yes—
492C22#575312yes—
493C4—22no—
494C4—24no—
495C4—42no—
496C4—68no—
497C4—124no—
498C4—241no—
499C4—241no—
500C4—241no—
501C4D3×#12112yes—
502C4#562712yes—
503C3#3291811yescharacteristic, complemented, Sylow
504C2—42no—
505C2—481no—
506C2#7404811yescharacteristic
507C2#7868012yes—
508C2#7909911yescharacteristic
509C2#7910011yescharacteristic
510C2#7939412yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1798C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#1798D3
hypocenterC3#32918
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1798

Complements

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

NH ≅ G/Ntype
1. C2×#5516C2semidirect
3. C2×#4845C2semidirect
4. C2×#4739C2semidirect
6. C3×#1798C2semidirect
7. #70485C2semidirect
9. #69931C2semidirect
10. #68686C2semidirect
11. #68685C2semidirect
54. #1798D3semidirect
105. C4×Dic6D4semidirect
503. C3#32918semidirect

Maximal subgroups

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

subgroupindex
C2×#55162
#822872
C2×#48452
C2×#47392
C2×#47382
C3×#17982
#704852
#699332
#699312
#686862
#686852
#329183