Finite Group Databasebeta

← #10945028

Subgroups — #10945028

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×#5827C211yesmaximal, characteristic
2C2×C3×#452C211yesmaximal, characteristic
3#82787C211yesmaximal, characteristic, complemented
4C2×#5044C211yesmaximal, characteristic, complemented
5#72135C211yesmaximal, characteristic
6#72065C212yesmaximal, complemented
7#72047C212yesmaximal, complemented
8#70235C212yesmaximal, complemented
9#70167C212yesmaximal, complemented
10#69924C211yesmaximal, characteristic, complemented
11#69911C211yesmaximal, characteristic, complemented
12#33017—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C22×C3×Q16C2211yescharacteristic
15C22×C3×SD16C2211yescharacteristic
16C2×C3×C22⋊Q8C2211yescharacteristic
17#5827—24no—
18#5827—24no—
19C2×Q8×Dic3C2212yes—
20C2×D4×Dic3—22no—
21C2×D4×Dic3C2212yes—
22C2×#926—21no—
23C2×#926C2211yescharacteristic
24C22×C4⋊Dic3—21no—
25C22×C4⋊Dic3C2211yescharacteristic
26C3×#452—24no—
27C2×C3×#103C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29C2×C3×C22⋊C8C2211yescharacteristic
30#5229C2212yes—
31C2×#835C2212yes—
32C2×#835C2212yes—
33C2×#832C2212yes—
34#5212C2211yescharacteristic
35C2×#830C2211yescharacteristic
36#5044—24no—
37#4986—21no—
38#4981—21no—
39#4972C2212yes—
40C2×#807C2212yes—
41#4952—22no—
42#4551C2212yes—
43#4551C2212yes—
44#4550C2212yes—
45#4550C2212yes—
46C2×#520—31no—
47C2×#484—31no—
48C2×#452D311yescharacteristic
49#2625—31no—
50#1878—32no—
51#1877—32no—
52#1863—32no—
53#1862—32no—
54#1823—31no—
55#1818—31no—
56#1788—31no—
57C2×C3×#41—24no—
58C2×C3×#41—24no—
59C2×C3×#41—22no—
60C2×C3×#41D414yes—
61C22×C3×Q8C2311yescharacteristic
62C22×C3×Q8C2311yescharacteristic
63C22×C3×D4—21no—
64C22×C3×D4C2311yescharacteristic
65C23×C3×C4—21no—
66C23×C3×C4C2311yescharacteristic
67C23×Dic3—21no—
68C23×Dic3—22no—
69C2×C3×Q16—22no—
70C2×C3×Q16—24no—
71C2×C3×SD16—22no—
72C2×C3×SD16—24no—
73C22×C3×C8C2311yescharacteristic
74C22×C3×C8C2311yescharacteristic
75C3×C22⋊Q8—22no—
76C3×C22⋊Q8—22no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C22⋊C4—21no—
80C2×#217—22no—
81C2×#217—21no—
82C2×#217D412yes—
83Q8×Dic3—24no—
84Q8×Dic3—24no—
85D4×Dic3—216no—
86D4×Dic3—24no—
87D4×Dic3—24no—
88#926—28no—
89#926—22no—
90#926—22no—
91C2×C4⋊Dic3—28no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3—22no—
94C2×C4⋊Dic3—22no—
95C2×C4⋊Dic3D411yescharacteristic
96C2×C4⋊Dic3D411yescharacteristic
97C2×C4⋊Dic3D414yes—
98C2×C4⋊Dic3C2312yes—
99C2×C4⋊Dic3C2312yes—
100C2×Dic3⋊C4—22no—
101C2×C4×Dic3—22no—
102C2×C4×Dic3C2312yes—
103C22×C3⋊C8C2312yes—
104C3×#103—22no—
105C3×#102—22no—
106C3×C22⋊C8—22no—
107#835—24no—
108#835—24no—
109#832—24no—
110C3⋊#95—21no—
111C3⋊#95—22no—
112C3⋊#95—22no—
113C3⋊#95—22no—
114#830—22no—
115#807—24no—
116C22×#41C2×D311yescharacteristic
117C22×Q16C2×D311yescharacteristic
118C22×SD16C2×D311yescharacteristic
119C2×C22⋊Q8C2×D311yescharacteristic
120#520—64no—
121#520—64no—
122C2×C4×Q8—32no—
123C2×C4×D4—32no—
124C2×C4×D4—62no—
125C2×#117—31no—
126C2×#117—61no—
127C22×C4⋊C4—31no—
128C22×C4⋊C4—61no—
129#484—64no—
130#452—24no—
131C2×#107—32no—
132C2×#103—32no—
133C2×#103—32no—
134C2×#103C2×D311yescharacteristic
135C2×#102—32no—
136C2×#102C2×D311yescharacteristic
137C2×C22⋊C8—31no—
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#336—32no—
148#336—32no—
149C24×C3—21no—
150C24×C3C2×D411yescharacteristic
151C3×#41—216no—
152C3×#41—28no—
153C3×#41—28no—
154C2×C3×Q8—21no—
155C2×C3×Q8—22no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C2×C3×Q8C2×D412yes—
161C2×C3×Q8#4114yes—
162C2×C3×D4—28no—
163C2×C3×D4—24no—
164C2×C3×D4—21no—
165C2×C3×D4—22no—
166C2×C3×D4C2×D412yes—
167C2×C3×D4#4114yes—
168C22×C3×C4—28no—
169C22×C3×C4—21no—
170C22×C3×C4—24no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2411yescharacteristic
179C22×Dic3—22no—
180C22×Dic3—28no—
181C22×Dic3—24no—
182C22×Dic3—22no—
183C22×Dic3—82no—
184C22×Dic3—82no—
185C22×Dic3—81no—
186C22×Dic3—81no—
187C22×Dic3C2×D412yes—
188C22×Dic3C2×D412yes—
189C3×Q16—44no—
190C3×SD16—44no—
191C2×C3×C8—21no—
192C2×C3×C8—21no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—42no—
198C3×C22⋊C4—42no—
199#217—28no—
200#217—24no—
201#217—81no—
202C4⋊Dic3—28no—
203C4⋊Dic3—22no—
204C4⋊Dic3—24no—
205C4⋊Dic3—22no—
206C4⋊Dic3—22no—
207C4⋊Dic3—22no—
208C4⋊Dic3—22no—
209C4⋊Dic3C2×D414yes—
210Dic3⋊C4—82no—
211C4×Dic3—28no—
212C4×Dic3—22no—
213C4×Dic3—22no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C2×#41—24no—
218C2×#41—24no—
219C2×#41—22no—
220C2×#41#6714yes—
221C22×Q8C22×D311yescharacteristic
222C22×Q8C22×D311yescharacteristic
223C22×D4—21no—
224C22×D4C22×D311yescharacteristic
225C23×C4—21no—
226C23×C4—62no—
227C23×C4—61no—
228C23×C4C22×D311yescharacteristic
229C2×Q16—24no—
230C2×Q16—22no—
231C2×SD16—24no—
232C2×SD16—22no—
233C22×C8—32no—
234C22×C8C22×D311yescharacteristic
235C22×C8C22×D311yescharacteristic
236C22⋊Q8—22no—
237C22⋊Q8—22no—
238C4×Q8—64no—
239C4×Q8—64no—
240C4×D4—64no—
241C4×D4—616no—
242C4×D4—64no—
243#117—68no—
244#117—62no—
245#117—62no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—34no—
248C2×C4⋊C4—32no—
249C2×C4⋊C4—31no—
250C2×C4⋊C4—31no—
251C2×C4⋊C4—32no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—68no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4—62no—
257C2×C4⋊C4C22×D311yescharacteristic
258C2×C22⋊C4—21no—
259C2×C22⋊C4—32no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—61no—
262C2×C42—32no—
263C2×C42—62no—
264#107—64no—
265#103—22no—
266#103—64no—
267#103—64no—
268#102—22no—
269#102—64no—
270C22⋊C8—22no—
271C22⋊C8—62no—
272#95—62no—
273#95—62no—
274#95—62no—
275#95—61no—
276C23×C3—22no—
277C23×C3—22no—
278C23×C3—24no—
279C23×C3—21no—
280C23×C3—21no—
281C23×C3—22no—
282C23×C3—41no—
283C23×C3—41no—
284C23×C3—42no—
285C23×C3#12012yes—
286C23×C3C22×D411yescharacteristic
287C3×Q8—22no—
288C3×Q8—24no—
289C3×Q8—41no—
290C3×Q8—41no—
291C3×Q8—42no—
292C3×Q8#12314yes—
293C3×D4—216no—
294C3×D4—22no—
295C3×D4—24no—
296C3×D4#12314yes—
297C2×C3×C4—216no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—24no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—24no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—42no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4#12012yes—
310C2×C3×C4C22×D411yescharacteristic
311C2×C3×C4C22×D411yescharacteristic
312C2×C3×C4C2×#4112yes—
313C2×C3×C4C2×#4112yes—
314C2×Dic3—22no—
315C2×Dic3—22no—
316C2×Dic3—24no—
317C2×Dic3—22no—
318C2×Dic3—22no—
319C2×Dic3—24no—
320C2×Dic3—42no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—84no—
326C2×Dic3—88no—
327C3×C8—41no—
328C3×C8—41no—
329C3⋊C8—42no—
330C24—21no—
331C24D3×D411yescharacteristic
332#41—216no—
333#41—28no—
334#41—28no—
335C2×Q8—21no—
336C2×Q8—22no—
337C2×Q8—21no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—21no—
341C2×Q8#23714yes—
342C2×Q8C2×#6712yes—
343C2×D4—28no—
344C2×D4—24no—
345C2×D4—21no—
346C2×D4—22no—
347C2×D4#23714yes—
348C2×D4C2×#6712yes—
349C22×C4—28no—
350C22×C4—21no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—32no—
357C22×C4—32no—
358C22×C4—62no—
359C22×C4—68no—
360C22×C4—64no—
361C22×C4—62no—
362C22×C4—242no—
363C22×C4—242no—
364C22×C4—241no—
365C22×C4—241no—
366C22×C4D3×D411yescharacteristic
367C22×C4D3×D412yes—
368C22×C4C2×#6712yes—
369C22×C4C23×D311yescharacteristic
370Q16—44no—
371SD16—44no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—62no—
377C2×C8—62no—
378C2×C8—62no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—34no—
382C4⋊C4—42no—
383C4⋊C4—68no—
384C4⋊C4—64no—
385C4⋊C4—62no—
386C4⋊C4—62no—
387C4⋊C4—62no—
388C4⋊C4—62no—
389C4⋊C4—62no—
390C4⋊C4—242no—
391C22⋊C4—42no—
392C22⋊C4—68no—
393C22⋊C4—64no—
394C22⋊C4—241no—
395C42—68no—
396C42—62no—
397C42—62no—
398C22×C3—22no—
399C22×C3—24no—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3—42no—
403C22×C3—44no—
404C22×C3—42no—
405C22×C3—44no—
406C22×C3—42no—
407C22×C3—42no—
408C22×C3C2×#12011yescharacteristic
409C22×C3#52512yes—
410C22×C3#57611yescharacteristic
411C22×C3#57711yescharacteristic
412C22×C3#57812yes—
413C3×C4—22no—
414C3×C4—24no—
415C3×C4—81no—
416C3×C4—81no—
417C3×C4C2×#12311yescharacteristic
418C3×C4C2×#12311yescharacteristic
419C3×C4#52512yes—
420Dic3—42no—
421Dic3—42no—
422Dic3—44no—
423Dic3—82no—
424C23—21no—
425C23—21no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—24no—
430C23—41no—
431C23—41no—
432C23—42no—
433C23#93712yes—
434C23C2×D3×D411yescharacteristic
435Q8—22no—
436Q8—24no—
437Q8—41no—
438Q8—41no—
439Q8—42no—
440Q8#93514yes—
441D4—216no—
442D4—22no—
443D4—24no—
444D4#93514yes—
445C2×C4—216no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—62no—
458C2×C4—62no—
459C2×C4—64no—
460C2×C4—62no—
461C2×C4—64no—
462C2×C4—62no—
463C2×C4—122no—
464C2×C4—122no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4—122no—
468C2×C4—244no—
469C2×C4—248no—
470C2×C4#93712yes—
471C2×C4C2×D3×D411yescharacteristic
472C2×C4C2×#23712yes—
473C2×C4C2×#23712yes—
474C2×C4C22×#6711yescharacteristic
475C8—41no—
476C8—41no—
477C8—122no—
478C2×C3—42no—
479C2×C3—44no—
480C2×C3#284911yescharacteristic
481C2×C3#301111yescharacteristic
482C2×C3#301211yescharacteristic
483C2×C3#312012yes—
484C2×C3#312412yes—
485C22—22no—
486C22—22no—
487C22—24no—
488C22—22no—
489C22—42no—
490C22—42no—
491C22—44no—
492C22—42no—
493C22—44no—
494C22—42no—
495C22#558612yes—
496C22#576911yescharacteristic
497C22#577011yescharacteristic
498C22#577912yes—
499C22C2×#93711yescharacteristic
500C4—22no—
501C4—24no—
502C4—81no—
503C4—81no—
504C4—122no—
505C4—122no—
506C4—124no—
507C4—242no—
508C4#558612yes—
509C4C2×#93511yescharacteristic
510C4C2×#93511yescharacteristic
511C3#3301711yescharacteristic, complemented, Sylow
512C2—42no—
513C2—44no—
514C2#8021612yes—
515C2#8021812yes—
516C2#8260211yescharacteristic
517C2#8391211yescharacteristic
518C2#8398311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#452C2
socleC23×C3C22×D4
prime radicalC22×C3×#41C22
hypercenterC2×#452D3
hypocenterC3#33017
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#452

Complements

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

NH ≅ G/Ntype
3. #82787C2semidirect
4. C2×#5044C2semidirect
6. #72065C2semidirect
7. #72047C2semidirect
8. #70235C2semidirect
9. #70167C2semidirect
10. #69924C2semidirect
11. #69911C2semidirect
511. C3#33017semidirect

Maximal subgroups

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

subgroupindex
C2×#58272
C2×C3×#4522
#827872
C2×#50442
#721352
#720652
#720472
#702352
#701672
#699242
#699112
#330173