Finite Group Databasebeta

← #10829297

Subgroups — #10829297

Order 768.

1120 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5730C211yesmaximal, characteristic
2C2×#5094C211yesmaximal, characteristic, complemented
3C2×#4716C211yesmaximal, characteristic, complemented
4#74240C211yesmaximal, characteristic
5C3×#1993C211yesmaximal, characteristic, complemented
6#69837C211yesmaximal, characteristic
7#69559C211yesmaximal, characteristic, complemented
8#68888C212yesmaximal, complemented
9#68878C212yesmaximal, complemented
10#68875C212yesmaximal, complemented
11#68824C212yesmaximal, complemented
12#24134—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×#946—22no—
15C22×Dic12C2211yescharacteristic
16#5730—24no—
17#5730—24no—
18C2×#897C2212yes—
19C2×#897C2212yes—
20C2×#893—21no—
21C2×D3×C4⋊C4C2212yes—
22C2×#890—21no—
23C2×C4×D12C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C2×C3×#97C2211yescharacteristic
27C3×#382C2211yescharacteristic
28C2×(C2×D3)⋊C8C2212yes—
29C2×(C2×D3)⋊C8C2212yes—
30C2×#817C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×#816C2211yescharacteristic
33#5094—24no—
34#4987C2212yes—
35C2×#810C2212yes—
36C2×#810C2212yes—
37C2×#808C2212yes—
38C2×C4⋊(C3⋊C8)C2211yescharacteristic
39#4716—24no—
40#4672—21no—
41#4670—22no—
42#4661—21no—
43C3×#340C2212yes—
44C3×#340C2212yes—
45#4529C2212yes—
46C2×#554—31no—
47C2×#475—31no—
48C2×#441—31no—
49#2536—31no—
50#2035—32no—
51#2001—32no—
52#1993D311yescharacteristic, complemented
53#1955—31no—
54#1872—32no—
55#1853—32no—
56#1805—31no—
57C22×C3×Q8—21no—
58C22×D12D411yescharacteristic
59C22×C4×D3—22no—
60C22×C4×D3C2312yes—
61C22×Dic6C2311yescharacteristic
62C22×Dic6C2311yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C22×C3×C8C2311yescharacteristic
65C3×C4⋊Q8—22no—
66C3×C4⋊Q8D414yes—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4C2312yes—
69C2×C3×C4⋊C4C2312yes—
70C2×C3×C42C2311yescharacteristic
71#946—216no—
72C2×(C2×D3)⋊C4—22no—
73C2×(C2×D3)⋊C4—22no—
74C2×(C2×D3)⋊C4—22no—
75C2×(C2×D3)⋊C4D412yes—
76C2×C4⋊Dic3—22no—
77C2×C4⋊Dic3—22no—
78C2×C4⋊Dic3D411yescharacteristic
79C2×C4⋊Dic3C2311yescharacteristic
80C2×C4⋊Dic3C2311yescharacteristic
81C2×Dic3⋊C4—21no—
82C2×Dic3⋊C4—22no—
83C2×Dic3⋊C4—21no—
84C2×C4×Dic3—21no—
85C2×C4×Dic3—21no—
86C22×C3⋊C8C2312yes—
87C2×Dic12—22no—
88C2×Dic12—22no—
89C2×Dic12—22no—
90#897—24no—
91#897—24no—
92#897—24no—
93#897—42no—
94#893—28no—
95D3×C4⋊C4—24no—
96D3×C4⋊C4—24no—
97#890—28no—
98C4×D12—22no—
99C4×D12—22no—
100C4×Dic6—22no—
101C4×Dic6—22no—
102C4×Dic6—22no—
103C4×Dic6D414yes—
104C3×#97—22no—
105C3×#95—21no—
106C3×#95—21no—
107C3⋊#95—22no—
108(C2×D3)⋊C8—24no—
109(C2×D3)⋊C8—42no—
110#817—22no—
111#816—22no—
112#816—22no—
113#810—24no—
114#810—42no—
115#808—24no—
116C4⋊(C3⋊C8)—22no—
117C22×Q16—31no—
118#554—64no—
119#554—64no—
120C2×C4⋊Q8C2×D311yescharacteristic
121C2×#125—61no—
122C2×C22⋊Q8—32no—
123C2×C22⋊Q8—32no—
124C2×C22⋊Q8—62no—
125C2×C4×Q8—31no—
126C2×C4×Q8—31no—
127C2×C4×D4—31no—
128C2×C4×D4—61no—
129C22×C4⋊C4—32no—
130#475—64no—
131#441—64no—
132C2×#106—31no—
133C2×#106—32no—
134C2×C4⋊C8—31no—
135C2×#103—31no—
136C2×#103—31no—
137C2×#103—32no—
138C2×#103—32no—
139C2×C22⋊C8—32no—
140C2×C22⋊C8—32no—
141C2×#97C2×D311yescharacteristic
142#398—61no—
143#394—61no—
144#388—62no—
145#382C2×D311yescharacteristic
146#380—32no—
147#340—32no—
148#340C2×D312yes—
149#340C2×D312yes—
150C23×D3C2×D412yes—
151C2×C3×Q8—24no—
152C2×C3×Q8—82no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2411yescharacteristic
158C22×Dic3—21no—
159C22×Dic3—21no—
160C22×Dic3—21no—
161C22×Dic3—21no—
162C22×Dic3C2×D412yes—
163C2×D12—24no—
164C2×D12—22no—
165C2×C4×D3—22no—
166C2×C4×D3—22no—
167C2×C4×D3—22no—
168C2×C4×D3—22no—
169C2×C4×D3—22no—
170C2×C4×D3—22no—
171C2×C4×D3—28no—
172C2×C4×D3—82no—
173C2×C4×D3—82no—
174C2×Dic6—21no—
175C2×Dic6—21no—
176C2×Dic6—21no—
177C2×Dic6—21no—
178C2×Dic6—21no—
179C2×Dic6—21no—
180C2×Dic6—21no—
181C2×Dic6—21no—
182C2×Dic6—21no—
183C2×Dic6—21no—
184C2×Dic6C2×D412yes—
185C2×Dic6#4112yes—
186C2×C3×C8—21no—
187C2×C3×C8—21no—
188C2×C3×C8—21no—
189C2×C3×C8—21no—
190C2×C3×C8—21no—
191C2×C3×C8#4112yes—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4C2×D414yes—
197C3×C42C2×D412yes—
198C3×C42C2×D412yes—
199(C2×D3)⋊C4—28no—
200(C2×D3)⋊C4—24no—
201(C2×D3)⋊C4—44no—
202(C2×D3)⋊C4—44no—
203C4⋊Dic3—28no—
204C4⋊Dic3—22no—
205C4⋊Dic3—21no—
206C4⋊Dic3—21no—
207C4⋊Dic3—21no—
208C4⋊Dic3—44no—
209C4⋊Dic3C2×D412yes—
210Dic3⋊C4—28no—
211Dic3⋊C4—24no—
212Dic3⋊C4—42no—
213C4×Dic3—24no—
214C4×Dic3—42no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218Dic12—28no—
219C22×Q8—21no—
220C22×Q8—31no—
221C22×Q8—31no—
222C22×D4—31no—
223C23×C4—32no—
224C23×C4—62no—
225C2×Q16—62no—
226C2×Q16—62no—
227C2×Q16—62no—
228C22×C8—32no—
229C22×C8C22×D311yescharacteristic
230C22×C8C22×D311yescharacteristic
231C4⋊Q8—22no—
232C4⋊Q8D1214yes—
233#125—68no—
234C22⋊Q8—64no—
235C22⋊Q8—616no—
236C22⋊Q8—64no—
237C22⋊Q8—64no—
238C22⋊Q8—122no—
239C4×Q8—34no—
240C4×Q8—62no—
241C4×Q8—62no—
242C4×Q8—62no—
243C4×D4—68no—
244C4×D4—62no—
245C4×D4—62no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—31no—
248C2×C4⋊C4—31no—
249C2×C4⋊C4—31no—
250C2×C4⋊C4—64no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—61no—
254C2×C4⋊C4—64no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4—61no—
257C2×C4⋊C4C22×D312yes—
258C2×C4⋊C4C22×D312yes—
259C2×C22⋊C4—32no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—62no—
263C2×C42—61no—
264C2×C42—61no—
265C2×C42C22×D311yescharacteristic
266#106—62no—
267#106—64no—
268C4⋊C8—62no—
269#103—62no—
270#103—62no—
271#103—64no—
272#103—122no—
273C22⋊C8—64no—
274C22⋊C8—122no—
275#97—22no—
276#95—21no—
277#95—21no—
278#95—62no—
279C23×C3C22×D411yescharacteristic
280C22×D3—22no—
281C22×D3—22no—
282C22×D3—22no—
283C22×D3—22no—
284C22×D3—22no—
285C22×D3—22no—
286C22×D3#13714yes—
287C3×Q8—84no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—24no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4#12012yes—
297C2×C3×C4#12012yes—
298C2×C3×C4C22×D411yescharacteristic
299C2×C3×C4C22×D411yescharacteristic
300C2×C3×C4C2×#4111yescharacteristic
301C2×C3×C4C2×#4111yescharacteristic
302C2×C3×C4H(5,𝔽2)11yescharacteristic
303C2×C3×C42−1+411yescharacteristic
304C2×Dic3—22no—
305C2×Dic3—22no—
306C2×Dic3—22no—
307C2×Dic3—22no—
308C2×Dic3—22no—
309C2×Dic3—22no—
310C2×Dic3—41no—
311C2×Dic3—41no—
312C2×Dic3—41no—
313C2×Dic3—41no—
314C2×Dic3—42no—
315C2×Dic3—41no—
316C2×Dic3—42no—
317D12—28no—
318C4×D3—42no—
319C4×D3—42no—
320C4×D3—42no—
321C4×D3—42no—
322C4×D3—88no—
323Dic6—22no—
324Dic6—22no—
325Dic6—22no—
326Dic6—22no—
327Dic6—41no—
328Dic6—41no—
329Dic6—41no—
330Dic6—41no—
331C3×C8—22no—
332C3×C8—41no—
333C3⋊C8—42no—
334C24—32no—
335C2×Q8—24no—
336C2×Q8—32no—
337C2×Q8—32no—
338C2×Q8—61no—
339C2×Q8—61no—
340C2×Q8—61no—
341C2×Q8—61no—
342C2×Q8—61no—
343C2×Q8—61no—
344C2×Q8—61no—
345C2×Q8—61no—
346C2×Q8—61no—
347C2×Q8—61no—
348C2×Q8—82no—
349C2×D4—64no—
350C2×D4—62no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—32no—
354C22×C4—68no—
355C22×C4—61no—
356C22×C4—61no—
357C22×C4—61no—
358C22×C4—62no—
359C22×C4—62no—
360C22×C4—62no—
361C22×C4—62no—
362C22×C4—62no—
363C22×C4—62no—
364C22×C4—61no—
365C22×C4—242no—
366C22×C4—242no—
367C22×C4D3×D411yescharacteristic
368C22×C4D3×D411yescharacteristic
369C22×C4C23×D311yescharacteristic
370Q16—68no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—62no—
377C2×C8—62no—
378C2×C8—62no—
379C2×C8#23912yes—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—32no—
385C4⋊C4—68no—
386C4⋊C4—68no—
387C4⋊C4—62no—
388C4⋊C4—64no—
389C4⋊C4—61no—
390C4⋊C4—61no—
391C4⋊C4—61no—
392C4⋊C4—124no—
393C4⋊C4—122no—
394C4⋊C4C2×D1214yes—
395C22⋊C4—68no—
396C22⋊C4—64no—
397C22⋊C4—124no—
398C22⋊C4—124no—
399C42—64no—
400C42—122no—
401C42C2×D1212yes—
402C42D3×D412yes—
403C22×C3C2×#12011yescharacteristic
404C22×C3#54611yescharacteristic
405C22×C3#54711yescharacteristic
406C22×C3C2×#13712yes—
407C22×C3#57712yes—
408C2×D3—24no—
409C2×D3—24no—
410C2×D3—24no—
411C2×D3—42no—
412C2×D3—42no—
413C2×D3—42no—
414C2×D3—42no—
415C3×C4—22no—
416C3×C4—22no—
417C3×C4—44no—
418C3×C4—82no—
419C3×C4#52511yescharacteristic
420C3×C4#53711yescharacteristic
421C3×C4#54811yescharacteristic
422C3×C4#55011yescharacteristic
423Dic3—42no—
424Dic3—42no—
425Dic3—42no—
426Dic3—42no—
427Dic3—81no—
428C23—34no—
429C23—62no—
430C23—62no—
431C23—62no—
432C23—62no—
433C23—62no—
434C23—62no—
435C23C2×D3×D411yescharacteristic
436Q8—62no—
437Q8—62no—
438Q8—62no—
439Q8—62no—
440Q8—84no—
441Q8—121no—
442Q8—121no—
443Q8—121no—
444Q8—121no—
445D4—68no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—62no—
455C2×C4—62no—
456C2×C4—62no—
457C2×C4—62no—
458C2×C4—62no—
459C2×C4—62no—
460C2×C4—121no—
461C2×C4—121no—
462C2×C4—121no—
463C2×C4—121no—
464C2×C4—122no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4—122no—
468C2×C4—122no—
469C2×C4—121no—
470C2×C4—122no—
471C2×C4—248no—
472C2×C4#88212yes—
473C2×C4#88212yes—
474C2×C4C22×D1211yescharacteristic
475C2×C4C2×D3×D411yescharacteristic
476C2×C4C2×#23911yescharacteristic
477C2×C4#100711yescharacteristic
478C2×C4D3×#4111yescharacteristic
479C2×C4#100911yescharacteristic
480C8—22no—
481C8—41no—
482C8—122no—
483C2×C3#240711yescharacteristic
484C2×C3#300411yescharacteristic
485C2×C3#300911yescharacteristic
486C2×C3#331612yes—
487C2×C3#334412yes—
488D3—44no—
489C22—64no—
490C22—64no—
491C22—64no—
492C22—122no—
493C22—122no—
494C22—122no—
495C22—122no—
496C22C2×#88211yescharacteristic
497C22#567611yescharacteristic
498C22#572611yescharacteristic
499C22#577512yes—
500C22D3×#13712yes—
501C4—22no—
502C4—22no—
503C4—44no—
504C4—82no—
505C4—122no—
506C4—122no—
507C4—122no—
508C4—122no—
509C4—241no—
510C4#557611yescharacteristic
511C4#557711yescharacteristic
512C4#562711yescharacteristic
513C4#562911yescharacteristic
514C3#2413411yescharacteristic, complemented, Sylow
515C2—124no—
516C2#7661811yescharacteristic
517C2#7892911yescharacteristic
518C2#7895811yescharacteristic
519C2#8170112yes—
520C2#8199112yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1993C2
socleC23×C3C22×D4
prime radicalC22×D12D4
hypercenter#1993D3
hypocenterC3#24134
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
2. C2×#5094C2semidirect
3. C2×#4716C2semidirect
5. C3×#1993C2semidirect
7. #69559C2semidirect
8. #68888C2semidirect
9. #68878C2semidirect
10. #68875C2semidirect
11. #68824C2semidirect
52. #1993D3semidirect
514. C3#24134semidirect

Maximal subgroups

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

subgroupindex
C2×#57302
C2×#50942
C2×#47162
#742402
C3×#19932
#698372
#695592
#688882
#688782
#688752
#688242
#241343