Finite Group Databasebeta

← #10844686

Subgroups — #10844686

Order 768.

954 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4843C211yesmaximal, characteristic
2C2×#4830C211yesmaximal, characteristic, complemented
3#74369C211yesmaximal, characteristic, complemented
4#73712C211yesmaximal, characteristic, complemented
5C3×#1947C211yesmaximal, characteristic, complemented
6#70639C212yesmaximal, complemented
7#70628C212yesmaximal, complemented
8#69580C211yesmaximal, characteristic, complemented
9C4×#809C211yesmaximal, characteristic, complemented
10#68862C212yesmaximal, complemented
11#68853C212yesmaximal, complemented
12#25292—31nomaximal, Sylow
13C22×D24C2212yes—
14C2×#897—21no—
15C2×#875—21no—
16C2×#874C2211yescharacteristic
17C2×C4×D12C2211yescharacteristic
18C2×C4×Dic6C2211yescharacteristic
19C2×C3×#107C2212yes—
20C2×C3×C4⋊C8C2211yescharacteristic
21C3×#384C2211yescharacteristic
22C3×C4×C4⋊C4C2211yescharacteristic
23C2×#821C2212yes—
24C2×Dic3⋊C8C2212yes—
25C2×C8×Dic3C2212yes—
26#4994—22no—
27#4988C2212yes—
28#4988C2212yes—
29#4988C2212yes—
30#4987—21no—
31#4984C2212yes—
32C2×#809C2211yescharacteristic
33C2×#809C2211yescharacteristic
34C2×#809C2212yes—
35C2×#809C2212yes—
36#4939—21no—
37C2×C4×C3⋊C8C2211yescharacteristic
38#4843—24no—
39#4830—24no—
40#4672—22no—
41C3×#340C2212yes—
42#4529C2211yescharacteristic
43#4529C2211yescharacteristic
44C2×#459—31no—
45C2×C8×Q8—31no—
46#2548—31no—
47#2538—31no—
48#2067—32no—
49#2054—32no—
50#1970—32no—
51#1957—31no—
52#1947D311yescharacteristic, complemented
53#1934—32no—
54C4×#102—31no—
55C22×D12—21no—
56C22×D12—21no—
57C22×D12C2312yes—
58C22×D12C2311yescharacteristic
59C22×C4×D3—21no—
60C22×Dic6D411yescharacteristic
61C22×C3×C8C2312yes—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C4⋊C4C2311yescharacteristic
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C42—21no—
68C2×C3×C42C2311yescharacteristic
69C2×(C2×D3)⋊C4—21no—
70C2×(C2×D3)⋊C4—22no—
71C2×(C2×D3)⋊C4—22no—
72C2×(C2×D3)⋊C4—22no—
73C2×(C2×D3)⋊C4—22no—
74C2×(C2×D3)⋊C4—22no—
75C2×(C2×D3)⋊C4—22no—
76C2×C4⋊Dic3—22no—
77C2×C4⋊Dic3D411yescharacteristic
78C2×C4⋊Dic3C2311yescharacteristic
79C2×Dic3⋊C4D412yes—
80C2×C4×Dic3C2312yes—
81C22×C3⋊C8C2311yescharacteristic
82C22×C3⋊C8C2311yescharacteristic
83C2×D24—24no—
84C2×D24—24no—
85C2×D24—42no—
86#897—44no—
87#875—44no—
88#874—22no—
89#874—22no—
90C4×D12—22no—
91C4×D12—22no—
92C4×Dic6—22no—
93C4×Dic6—22no—
94C3×#107—42no—
95C3×C4⋊C8D414yes—
96C3×#95—21no—
97C3⋊#95—22no—
98#821—24no—
99Dic3⋊C8—24no—
100C8×Dic3—24no—
101#809—22no—
102#809—22no—
103#809—24no—
104#809—42no—
105C4×C3⋊C8—22no—
106C22×D8—32no—
107C2×#127—31no—
108C2×#124—61no—
109C2×C22⋊Q8—61no—
110C2×C4×Q8—31no—
111C2×C4×D4—31no—
112#459—64no—
113C8×Q8—64no—
114C2×#107C2×D312yes—
115C2×C4⋊C8—32no—
116C2×C4⋊C8C2×D311yescharacteristic
117C2×#102—32no—
118C2×#102—31no—
119C2×#102—32no—
120C2×#102—31no—
121C2×#102—32no—
122C2×C4×C8—32no—
123C2×C4×C8—31no—
124#396—62no—
125#394—62no—
126#390—32no—
127#390—32no—
128#390—32no—
129#390—61no—
130#389—32no—
131#384C2×D311yescharacteristic
132#380—61no—
133C4×C4⋊C4C2×D311yescharacteristic
134#340—31no—
135#340—31no—
136#340C2×D312yes—
137C23×D3—21no—
138C23×D3—22no—
139C22×C3×C4—22no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2411yescharacteristic
145C22×Dic3—21no—
146C22×Dic3C2×D412yes—
147C22×Dic3C2×D412yes—
148C2×D12—21no—
149C2×D12—21no—
150C2×D12—21no—
151C2×D12—21no—
152C2×D12—22no—
153C2×D12—22no—
154C2×D12—22no—
155C2×D12—22no—
156C2×D12—22no—
157C2×D12—21no—
158C2×D12—22no—
159C2×D12—42no—
160C2×D12—42no—
161C2×D12—42no—
162C2×D12—42no—
163C2×D12—81no—
164C2×D12—81no—
165C2×D12#4112yes—
166C2×C4×D3—42no—
167C2×C4×D3—42no—
168C2×C4×D3—42no—
169C2×Dic6—22no—
170C2×Dic6—22no—
171C2×Dic6D814yescomplemented
172C2×C3×C8—22no—
173C2×C3×C8—22no—
174C2×C3×C8C2×D414yes—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—42no—
182C3×C4⋊C4—81no—
183C3×C42—42no—
184C3×C42C2×D412yes—
185C3×C42#4112yes—
186(C2×D3)⋊C4—42no—
187(C2×D3)⋊C4—44no—
188(C2×D3)⋊C4—44no—
189(C2×D3)⋊C4—44no—
190(C2×D3)⋊C4—82no—
191(C2×D3)⋊C4—82no—
192(C2×D3)⋊C4—82no—
193C4⋊Dic3—22no—
194C4⋊Dic3—21no—
195C4⋊Dic3—21no—
196C4⋊Dic3—44no—
197Dic3⋊C4—24no—
198C4×Dic3—22no—
199C4×Dic3—22no—
200C2×C3⋊C8—21no—
201C2×C3⋊C8—21no—
202C2×C3⋊C8—21no—
203C2×C3⋊C8—21no—
204C2×C3⋊C8—21no—
205C2×C3⋊C8#4112yes—
206D24—48no—
207C22×Q8—31no—
208C22×D4—31no—
209C22×D4—32no—
210C22×D4—61no—
211C22×D4—61no—
212C23×C4—61no—
213C2×D8—64no—
214C2×D8—64no—
215C2×D8—122no—
216C22×C8—31no—
217C22×C8—31no—
218C22×C8C22×D312yes—
219#127—62no—
220#127—62no—
221#124—124no—
222C22⋊Q8—124no—
223C4×Q8—62no—
224C4×Q8—62no—
225C4×D4—62no—
226C4×D4—62no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—31no—
230C2×C4⋊C4—32no—
231C2×C4⋊C4—31no—
232C2×C4⋊C4—62no—
233C2×C4⋊C4C22×D311yescharacteristic
234C2×C4⋊C4C22×D311yescharacteristic
235C2×C4⋊C4C22×D312yes—
236C2×C22⋊C4—62no—
237C2×C22⋊C4—62no—
238C2×C22⋊C4—62no—
239C2×C22⋊C4—61no—
240C2×C22⋊C4—62no—
241C2×C22⋊C4—62no—
242C2×C22⋊C4—62no—
243C2×C42—21no—
244C2×C42—32no—
245C2×C42C22×D311yescharacteristic
246#107—42no—
247C4⋊C8—64no—
248C4⋊C8D1214yes—
249#102—62no—
250#102—64no—
251#102—62no—
252#102—64no—
253#102—122no—
254C4×C8—64no—
255C4×C8—62no—
256#95—21no—
257#95—62no—
258C23×C3C22×D411yescharacteristic
259C22×D3—22no—
260C22×D3—41no—
261C22×D3—41no—
262C22×D3—41no—
263C22×D3—41no—
264C22×D3—41no—
265C22×D3—41no—
266C22×D3—42no—
267C22×D3—42no—
268C22×D3—42no—
269C22×D3—42no—
270C22×D3—42no—
271C22×D3—42no—
272C22×D3—42no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—41no—
276C2×C3×C4—42no—
277C2×C3×C4—41no—
278C2×C3×C4—42no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—42no—
283C2×C3×C4—41no—
284C2×C3×C4#12112yes—
285C2×C3×C4#12112yes—
286C2×C3×C4C2×D812yes—
287C2×C3×C4#13512yes—
288C2×C3×C4C22×D411yescharacteristic
289C2×C3×C4C2×#4111yescharacteristic
290C2×C3×C4C2×#4111yescharacteristic
291C2×C3×C4C2×#4111yescharacteristic
292C2×C3×C4H(5,𝔽2)11yescharacteristic
293C2×C3×C42−1+411yescharacteristic
294C2×Dic3—22no—
295C2×Dic3—22no—
296C2×Dic3—22no—
297C2×Dic3—22no—
298C2×Dic3—22no—
299C2×Dic3—41no—
300C2×Dic3—41no—
301C2×Dic3—41no—
302C2×Dic3C2×D814yes—
303D12—22no—
304D12—22no—
305D12—42no—
306D12—41no—
307D12—42no—
308D12—42no—
309D12—41no—
310D12—42no—
311D12—84no—
312D12—84no—
313C4×D3—84no—
314Dic6—28no—
315C3×C8—24no—
316C3⋊C8—22no—
317C3⋊C8—41no—
318C24—61no—
319C24—62no—
320C2×Q8—34no—
321C2×Q8—62no—
322C2×Q8—62no—
323C2×D4—32no—
324C2×D4—61no—
325C2×D4—62no—
326C2×D4—61no—
327C2×D4—62no—
328C2×D4—61no—
329C2×D4—62no—
330C2×D4—61no—
331C2×D4—62no—
332C2×D4—61no—
333C2×D4—62no—
334C2×D4—62no—
335C2×D4—122no—
336C2×D4—122no—
337C2×D4—122no—
338C2×D4—122no—
339C2×D4—241no—
340C2×D4—241no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—32no—
345C22×C4—32no—
346C22×C4—61no—
347C22×C4—122no—
348C22×C4—122no—
349C22×C4—122no—
350C22×C4D3×D411yescharacteristic
351C22×C4D3×D411yescharacteristic
352C22×C4C23×D311yescharacteristic
353D8—128no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—32no—
357C2×C8—61no—
358C2×C8—61no—
359C2×C8—61no—
360C2×C8—61no—
361C2×C8—61no—
362C2×C8C2×D1214yes—
363C4⋊C4—22no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—22no—
369C4⋊C4—42no—
370C4⋊C4—64no—
371C4⋊C4—62no—
372C4⋊C4—61no—
373C4⋊C4—61no—
374C4⋊C4—81no—
375C4⋊C4—124no—
376C22⋊C4—124no—
377C22⋊C4—124no—
378C22⋊C4—124no—
379C22⋊C4—122no—
380C22⋊C4—242no—
381C22⋊C4—242no—
382C22⋊C4—242no—
383C42—42no—
384C42—62no—
385C42—62no—
386C42C2×D1212yes—
387C42#23912yes—
388C22×C3C2×#12111yescharacteristic
389C22×C3#54811yescharacteristic
390C22×C3#55011yescharacteristic
391C22×C3C22×D811yescharacteristic
392C22×C3C2×#13511yescharacteristic
393C22×C3#57611yescharacteristic
394C22×C3#57711yescharacteristic
395C2×D3—42no—
396C2×D3—42no—
397C2×D3—42no—
398C2×D3—81no—
399C2×D3—81no—
400C2×D3—81no—
401C2×D3—81no—
402C2×D3—82no—
403C2×D3—82no—
404C2×D3—82no—
405C2×D3—82no—
406C2×D3—82no—
407C2×D3—82no—
408C2×D3—82no—
409C3×C4—22no—
410C3×C4—81no—
411C3×C4—82no—
412C3×C4—81no—
413C3×C4#46614yes—
414C3×C4#52911yescharacteristic
415C3×C4#54211yescharacteristic
416C3×C4#54611yescharacteristic
417C3×C4#54811yescharacteristic
418Dic3—24no—
419Dic3—42no—
420Dic3—81no—
421C23—62no—
422C23—121no—
423C23—121no—
424C23—121no—
425C23—121no—
426C23—121no—
427C23—121no—
428C23—122no—
429C23—122no—
430C23—122no—
431C23—122no—
432C23—122no—
433C23—122no—
434C23—122no—
435C23C2×D3×D411yescharacteristic
436Q8—68no—
437D4—62no—
438D4—62no—
439D4—122no—
440D4—121no—
441D4—122no—
442D4—122no—
443D4—121no—
444D4—122no—
445D4—244no—
446D4—244no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—34no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—62no—
460C2×C4—62no—
461C2×C4—62no—
462C2×C4—62no—
463C2×C4—62no—
464C2×C4—121no—
465C2×C4—121no—
466C2×C4—121no—
467C2×C4—244no—
468C2×C4#89512yes—
469C2×C4#89512yes—
470C2×C4D3×D812yes—
471C2×C4#91912yes—
472C2×C4C22×D1211yescharacteristic
473C2×C4#100411yescharacteristic
474C2×C4C2×#23911yescharacteristic
475C2×C4#100711yescharacteristic
476C2×C4D3×#4111yescharacteristic
477C2×C4D3×#4111yescharacteristic
478C8—24no—
479C8—62no—
480C8—121no—
481C2×C3#242611yescharacteristic
482C2×C3C2×#46611yescharacteristic
483C2×C3#306211yescharacteristic
484C2×C3#335811yescharacteristic
485C2×C3#336111yescharacteristic
486C2×C3#339111yescharacteristic
487C2×C3#339411yescharacteristic
488D3—82no—
489D3—162no—
490C22—122no—
491C22—122no—
492C22—122no—
493C22—241no—
494C22—241no—
495C22—241no—
496C22—241no—
497C22—242no—
498C22—242no—
499C22—242no—
500C22—242no—
501C22—242no—
502C22—242no—
503C22—242no—
504C22C2×#89511yescharacteristic
505C22#559911yescharacteristic
506C22#563311yescharacteristic
507C22C2×D3×D811yescharacteristic
508C22C2×#91911yescharacteristic
509C22#577411yescharacteristic
510C22#577811yescharacteristic
511C4—22no—
512C4—64no—
513C4—81no—
514C4—81no—
515C4—82no—
516C4—122no—
517C4—241no—
518C4#488314yes—
519C4#555411yescharacteristic
520C4#557611yescharacteristic
521C4#569111yescharacteristic
522C4#573311yescharacteristic
523C3#2529211yescharacteristic, complemented, Sylow
524C2—242no—
525C2—482no—
526C2#7621811yescharacteristic
527C2C2×#488311yescharacteristic
528C2#7921311yescharacteristic
529C2#8032411yescharacteristic
530C2#8047211yescharacteristic
531C2#8099211yescharacteristic
532C2#8113411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1947C2
socleC23×C3C22×D4
prime radicalC2×#4843C2
hypercenter#1947D3
hypocenterC3#25292
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
2. C2×#4830C2semidirect
3. #74369C2semidirect
4. #73712C2semidirect
5. C3×#1947C2semidirect
6. #70639C2semidirect
7. #70628C2semidirect
8. #69580C2semidirect
9. C4×#809C2semidirect
10. #68862C2semidirect
11. #68853C2semidirect
52. #1947D3semidirect
171. C2×Dic6D8semidirect
523. C3#25292semidirect

Maximal subgroups

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

subgroupindex
C2×#48432
C2×#48302
#743692
#737122
C3×#19472
#706392
#706282
#695802
C4×#8092
#688622
#688532
#252923