Finite Group Databasebeta

← #10945035

Subgroups — #10945035

Order 768.

1292 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5523C211yesmaximal, characteristic
2#82263C211yesmaximal, characteristic, complemented
3C2×#4733C211yesmaximal, characteristic
4C2×#4722C211yesmaximal, characteristic, complemented
5C3×#1818C211yesmaximal, characteristic, complemented
6#69960C211yesmaximal, characteristic
7#69924C211yesmaximal, characteristic, complemented
8#68685C212yesmaximal, complemented
9#68683C212yesmaximal, complemented
10#68672C212yesmaximal, complemented
11#68670C212yesmaximal, complemented
12#33017—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C22×C4⋊Dic3C2211yescharacteristic
16C2×#924C2211yescharacteristic
17C22×Dic12C2211yescharacteristic
18C22×#204C2211yescharacteristic
19#5523—24no—
20#5523—24no—
21C2×#892—21no—
22C2×D3×C4⋊C4—21no—
23C2×#67⋊C4—22no—
24C2×C4×D12C2212yes—
25C2×C4×Dic6C2212yes—
26C2×C3×C22⋊C8C2211yescharacteristic
27C2×C3×C22⋊C8C2211yescharacteristic
28#5212C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2212yes—
31C2×#818C2212yes—
32C2×#816C2211yescharacteristic
33C2×#816C2212yes—
34C2×#816C2212yes—
35#4985—21no—
36#4983—21no—
37#4952—22no—
38#4929C2212yes—
39#4928C2212yes—
40#4733—24no—
41#4722—24no—
42C3×#336C2212yes—
43C3×#336C2212yes—
44#4551C2212yes—
45#4551C2212yes—
46C2×#520—31no—
47C2×#484—31no—
48C2×#452—31no—
49#2625—31no—
50#1878—32no—
51#1877—32no—
52#1863—32no—
53#1862—32no—
54#1823—31no—
55#1818D311yescharacteristic, complemented
56#1788—31no—
57C23×C3×C4C2311yescharacteristic
58C22×#67—21no—
59C23×Dic3—21no—
60C2×#235—22no—
61C2×#235—24no—
62C2×#235—24no—
63C2×#235D414yes—
64C22×D12C2311yescharacteristic
65C22×C4×D3—21no—
66C22×C4×D3—22no—
67C22×Dic6C2311yescharacteristic
68C22×Dic6C2311yescharacteristic
69C22×C3×C8C2311yescharacteristic
70C22×C3×C8C2311yescharacteristic
71C22×C3×C8C2312yes—
72C3×#117—22no—
73C3×#117—22no—
74C2×C3×C4⋊C4D411yescharacteristic
75C2×C3×C4⋊C4D411yescharacteristic
76C2×C3×C22⋊C4D412yes—
77C2×C3×C42C2312yes—
78C2×#217—21no—
79C2×#217—21no—
80C2×(C2×D3)⋊C4—22no—
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3—22no—
83C2×C4⋊Dic3D414yes—
84C2×C4⋊Dic3C2311yescharacteristic
85C2×C4⋊Dic3C2312yes—
86C2×C4⋊Dic3C2312yes—
87#924—22no—
88#924—22no—
89C2×Dic3⋊C4—22no—
90C2×Dic3⋊C4—21no—
91C2×Dic3⋊C4—22no—
92C2×C4×Dic3—22no—
93C2×Dic12—22no—
94C2×Dic12—24no—
95C2×#204—22no—
96C2×#204—24no—
97#892—28no—
98D3×C4⋊C4—28no—
99#67⋊C4—216no—
100C4×D12—24no—
101C4×D12—24no—
102C4×Dic6—24no—
103C4×Dic6—24no—
104C3×C22⋊C8—22no—
105C3×C22⋊C8—22no—
106C3⋊#95—22no—
107C3⋊#95—22no—
108C3⋊#95—22no—
109C3⋊#95—21no—
110#821—22no—
111#821—24no—
112#818—24no—
113#816—22no—
114#816—24no—
115#816—24no—
116C22×#41—31no—
117C22×Q16—31no—
118C22×SD16—31no—
119C2×C22⋊Q8—31no—
120#520—64no—
121#520—64no—
122C2×C4×Q8—32no—
123C2×C4×D4—32no—
124C2×C4×D4—62no—
125C2×#117—61no—
126C2×#117C2×D311yescharacteristic
127C22×C4⋊C4—31no—
128C22×C4⋊C4—61no—
129#484—64no—
130#452—64no—
131C2×#107—32no—
132C2×#103—32no—
133C2×#103—31no—
134C2×#103—32no—
135C2×#102—31no—
136C2×#102—32no—
137C2×C22⋊C8C2×D311yescharacteristic
138C2×C22⋊C8C2×D311yescharacteristic
139#400—61no—
140#398—61no—
141#397—62no—
142#391—32no—
143#389—32no—
144#380—31no—
145#340—32no—
146#340—32no—
147#336C2×D312yes—
148#336C2×D312yes—
149C24×C3C2×D411yescharacteristic
150C23×D3—21no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2411yescharacteristic
159C2×#67—28no—
160C2×#67—24no—
161C22×Dic3—21no—
162C22×Dic3—22no—
163C22×Dic3—24no—
164C22×Dic3—22no—
165C22×Dic3—21no—
166C22×Dic3—21no—
167C22×Dic3—81no—
168C22×Dic3—81no—
169#235—216no—
170#235—28no—
171#235—28no—
172C2×D12—21no—
173C2×D12—22no—
174C2×D12C2×D412yes—
175C2×D12#4114yes—
176C2×C4×D3—28no—
177C2×C4×D3—24no—
178C2×C4×D3—28no—
179C2×C4×D3—82no—
180C2×C4×D3—82no—
181C2×Dic6—21no—
182C2×Dic6—22no—
183C2×Dic6—21no—
184C2×Dic6—21no—
185C2×Dic6—22no—
186C2×Dic6—22no—
187C2×Dic6C2×D412yes—
188C2×Dic6#4114yes—
189C2×C3×C8—22no—
190C2×C3×C8—21no—
191C2×C3×C8—21no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C22⋊C4—24no—
199C3×C42—22no—
200C3×C42—22no—
201#217—42no—
202#217—81no—
203(C2×D3)⋊C4—28no—
204C4⋊Dic3—24no—
205C4⋊Dic3—22no—
206C4⋊Dic3—21no—
207C4⋊Dic3—21no—
208C4⋊Dic3—22no—
209C4⋊Dic3—22no—
210C4⋊Dic3C2×D414yes—
211Dic3⋊C4—28no—
212Dic3⋊C4—42no—
213Dic3⋊C4—82no—
214C4×Dic3—28no—
215Dic12—44no—
216#204—44no—
217C2×#41—34no—
218C2×#41—64no—
219C2×#41—64no—
220C2×#41—62no—
221C22×Q8—31no—
222C22×Q8—31no—
223C22×D4—31no—
224C22×D4—61no—
225C23×C4—61no—
226C23×C4—61no—
227C23×C4—62no—
228C23×C4C22×D311yescharacteristic
229C2×Q16—64no—
230C2×Q16—62no—
231C2×SD16—64no—
232C2×SD16—62no—
233C22×C8C22×D311yescharacteristic
234C22×C8C22×D311yescharacteristic
235C22×C8C22×D312yes—
236C22⋊Q8—62no—
237C22⋊Q8—62no—
238C4×Q8—64no—
239C4×Q8—64no—
240C4×D4—64no—
241C4×D4—64no—
242C4×D4—616no—
243#117—22no—
244#117—22no—
245#117—68no—
246C2×C4⋊C4—34no—
247C2×C4⋊C4—32no—
248C2×C4⋊C4—31no—
249C2×C4⋊C4—32no—
250C2×C4⋊C4—68no—
251C2×C4⋊C4—61no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4D1211yescharacteristic
257C2×C4⋊C4D1211yescharacteristic
258C2×C22⋊C4—62no—
259C2×C22⋊C4—61no—
260C2×C22⋊C4—61no—
261C2×C22⋊C4D1212yes—
262C2×C42—62no—
263C2×C42C22×D312yes—
264#107—64no—
265#103—64no—
266#103—64no—
267#103—62no—
268#102—64no—
269#102—62no—
270C22⋊C8—22no—
271C22⋊C8—22no—
272#95—61no—
273#95—62no—
274#95—62no—
275#95—62no—
276C23×C3—22no—
277C23×C3—21no—
278C23×C3—21no—
279C23×C3—22no—
280C23×C3#12012yes—
281C23×C3C22×D411yescharacteristic
282C22×D3—22no—
283C22×D3—24no—
284C22×D3—41no—
285C22×D3—41no—
286C22×D3—42no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—24no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—24no—
296C2×C3×C4#12012yes—
297C2×C3×C4C22×D411yescharacteristic
298C2×C3×C4C22×D411yescharacteristic
299C2×C3×C4C2×#4112yes—
300C2×C3×C4C2×#4112yes—
301#67—216no—
302C2×Dic3—22no—
303C2×Dic3—24no—
304C2×Dic3—24no—
305C2×Dic3—42no—
306C2×Dic3—42no—
307C2×Dic3—41no—
308C2×Dic3—41no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311C2×Dic3—41no—
312C2×Dic3—42no—
313C2×Dic3—41no—
314C2×Dic3—42no—
315C2×Dic3—84no—
316D12—22no—
317D12—24no—
318D12#12314yes—
319C4×D3—216no—
320C4×D3—88no—
321Dic6—22no—
322Dic6—24no—
323Dic6—41no—
324Dic6—41no—
325Dic6—42no—
326Dic6#12314yes—
327C3×C8—41no—
328C3×C8—41no—
329C3×C8—42no—
330C24—61no—
331C24C2×D1211yescharacteristic
332#41—68no—
333#41—68no—
334#41—616no—
335C2×Q8—32no—
336C2×Q8—34no—
337C2×Q8—61no—
338C2×Q8—62no—
339C2×Q8—61no—
340C2×Q8—61no—
341C2×Q8—62no—
342C2×Q8—62no—
343C2×D4—32no—
344C2×D4—34no—
345C2×D4—68no—
346C2×D4—64no—
347C2×D4—61no—
348C2×D4—62no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—68no—
352C22×C4—68no—
353C22×C4—61no—
354C22×C4—64no—
355C22×C4—62no—
356C22×C4—64no—
357C22×C4—62no—
358C22×C4—61no—
359C22×C4—61no—
360C22×C4—242no—
361C22×C4—242no—
362C22×C4—241no—
363C22×C4—241no—
364C22×C4C2×D1212yes—
365C22×C4C2×D1212yes—
366C22×C4C2×D1211yescharacteristic
367C22×C4D3×D412yes—
368C22×C4D3×D412yes—
369C22×C4C23×D311yescharacteristic
370Q16—124no—
371SD16—124no—
372C2×C8—21no—
373C2×C8—22no—
374C2×C8—21no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—34no—
382C4⋊C4—68no—
383C4⋊C4—64no—
384C4⋊C4—62no—
385C4⋊C4—61no—
386C4⋊C4—61no—
387C4⋊C4—62no—
388C4⋊C4—62no—
389C4⋊C4—122no—
390C4⋊C4—242no—
391C22⋊C4—24no—
392C22⋊C4—68no—
393C22⋊C4—122no—
394C22⋊C4—241no—
395C42—22no—
396C42—22no—
397C42—68no—
398C22×C3—22no—
399C22×C3—22no—
400C22×C3—22no—
401C22×C3—42no—
402C22×C3—42no—
403C22×C3C2×#12011yescharacteristic
404C22×C3#52512yes—
405C22×C3#57611yescharacteristic
406C22×C3#57711yescharacteristic
407C22×C3#57812yes—
408C2×D3—24no—
409C2×D3—44no—
410C2×D3—44no—
411C2×D3—42no—
412C2×D3—42no—
413C3×C4—22no—
414C3×C4—42no—
415C3×C4—42no—
416C3×C4C2×#12311yescharacteristic
417C3×C4C2×#12311yescharacteristic
418C3×C4#52512yes—
419Dic3—24no—
420Dic3—44no—
421Dic3—81no—
422Dic3—82no—
423Dic3—81no—
424C23—21no—
425C23—21no—
426C23—22no—
427C23—22no—
428C23—62no—
429C23—64no—
430C23—121no—
431C23—121no—
432C23—122no—
433C23#88212yes—
434C23C22×D1211yescharacteristic
435Q8—34no—
436Q8—62no—
437Q8—64no—
438Q8—121no—
439Q8—121no—
440Q8—122no—
441D4—34no—
442D4—616no—
443D4—62no—
444D4—64no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—616no—
455C2×C4—62no—
456C2×C4—64no—
457C2×C4—64no—
458C2×C4—122no—
459C2×C4—122no—
460C2×C4—121no—
461C2×C4—121no—
462C2×C4—121no—
463C2×C4—122no—
464C2×C4—121no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4—122no—
468C2×C4—248no—
469C2×C4—244no—
470C2×C4#88212yes—
471C2×C4C2×D3×D411yescharacteristic
472C2×C4C2×D3×D411yescharacteristic
473C2×C4D3×#4112yes—
474C2×C4D3×#4112yes—
475C8—41no—
476C8—41no—
477C8—42no—
478C2×C3—42no—
479C2×C3#284911yescharacteristic
480C2×C3#301111yescharacteristic
481C2×C3#301211yescharacteristic
482C2×C3#312012yes—
483C2×C3#312412yes—
484D3—44no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—42no—
489C22—42no—
490C22—64no—
491C22—124no—
492C22—124no—
493C22—122no—
494C22—122no—
495C22C2×#88211yescharacteristic
496C22#552412yes—
497C22#575111yescharacteristic
498C22#575211yescharacteristic
499C22#575312yes—
500C4—22no—
501C4—42no—
502C4—42no—
503C4—64no—
504C4—124no—
505C4—242no—
506C4—241no—
507C4—241no—
508C4D3×#12311yescharacteristic
509C4D3×#12311yescharacteristic
510C4#565612yes—
511C3#3301711yescharacteristic, complemented, Sylow
512C2—42no—
513C2—124no—
514C2#7414611yescharacteristic
515C2#7867211yescharacteristic
516C2#7867911yescharacteristic
517C2#7938512yes—
518C2#7938812yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1818C2
socleC23×C3C22×D4
prime radicalC22×#235C22
hypercenter#1818D3
hypocenterC3#33017
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1818

Complements

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

NH ≅ G/Ntype
2. #82263C2semidirect
4. C2×#4722C2semidirect
5. C3×#1818C2semidirect
7. #69924C2semidirect
8. #68685C2semidirect
9. #68683C2semidirect
10. #68672C2semidirect
11. #68670C2semidirect
55. #1818D3semidirect
511. C3#33017semidirect

Maximal subgroups

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

subgroupindex
C2×#55232
#822632
C2×#47332
C2×#47222
C3×#18182
#699602
#699242
#686852
#686832
#686722
#686702
#330173