Finite Group Databasebeta

← #10844690

Subgroups — #10844690

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
1C3×#2538C211yesmaximal, characteristic, complemented
2#82476C211yesmaximal, characteristic, complemented
3C2×Q8×C3⋊C8C211yesmaximal, characteristic, complemented
4C2×#5015C211yesmaximal, characteristic
5C4×#832C211yesmaximal, characteristic, complemented
6#69650C212yesmaximal, complemented
7#69647C212yesmaximal, complemented
8#69630C212yesmaximal, complemented
9#69615C212yesmaximal, complemented
10#69580C211yesmaximal, characteristic, complemented
11#69567C211yesmaximal, characteristic, complemented
12#25292—31nomaximal, Sylow
13C2×C3×C22⋊Q8—21no—
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×#947—21no—
17C22×#213C2212yes—
18C2×#874C2211yescharacteristic
19C3×#396—22no—
20C3×#390C2212yes—
21C3×#389C2212yes—
22C3×#384C2211yescharacteristic
23C3×#380—21no—
24C2×#832C2211yescharacteristic
25C2×#832C2211yescharacteristic
26Q8×C3⋊C8—24no—
27#5015—24no—
28#4991—22no—
29#4988—21no—
30C2×#809C2212yes—
31C2×#809C2212yes—
32C2×#809C2212yes—
33C2×#807C2212yes—
34#4939C2212yes—
35#4939C2212yes—
36C4×C4⋊Dic3C2211yescharacteristic
37C2×C4⋊(C3⋊C8)C2212yes—
38C2×C4⋊(C3⋊C8)C2211yescharacteristic
39C2×C4×C3⋊C8C2212yes—
40C2×C4×C3⋊C8C2211yescharacteristic
41#4529C2212yes—
42#4529C2211yescharacteristic
43#4529C2211yescharacteristic
44C2×#459—31no—
45C2×C8×Q8—31no—
46#2548—31no—
47#2538D311yescharacteristic, complemented
48#2067—32no—
49#2054—32no—
50#1970—32no—
51#1957—31no—
52#1947—31no—
53#1934—32no—
54C4×#102—31no—
55C22×C3×Q8D411yescharacteristic
56C22×C3×D4C2311yescharacteristic
57C23×C3×C4—21no—
58C22×D12—21no—
59C22×D12—21no—
60C22×D12C2312yes—
61C3×C22⋊Q8—44no—
62C3×C4×Q8—22no—
63C3×C4×Q8—22no—
64C3×C4×D4—22no—
65C3×C4×D4—22no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4D411yescharacteristic
70C2×C3×C4⋊C4D412yes—
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C42C2312yes—
77C2×C3×C42C2311yescharacteristic
78#947—44no—
79C2×#213—24no—
80C2×#213—24no—
81C2×#213—42no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—22no—
86C2×C4⋊Dic3C2311yescharacteristic
87C2×C4⋊Dic3C2311yescharacteristic
88C2×C4×Dic3—21no—
89C22×C3⋊C8C2311yescharacteristic
90C22×C3⋊C8C2312yes—
91C22×C3⋊C8C2311yescharacteristic
92#874—22no—
93#874—22no—
94C3×#95—22no—
95#832—22no—
96#832—22no—
97C3⋊#95—21no—
98#809—24no—
99#809—24no—
100#809—42no—
101#807—42no—
102C4⋊(C3⋊C8)—24no—
103C4⋊(C3⋊C8)D414yes—
104C4×C3⋊C8—22no—
105C4×C3⋊C8—24no—
106C22×D8—32no—
107C2×#127—31no—
108C2×#124—61no—
109C2×C22⋊Q8—21no—
110C2×C4×Q8C2×D311yescharacteristic
111C2×C4×D4C2×D311yescharacteristic
112#459—64no—
113C8×Q8—64no—
114C2×#107—32no—
115C2×C4⋊C8—31no—
116C2×C4⋊C8—32no—
117C2×#102—31no—
118C2×#102—31no—
119C2×#102—32no—
120C2×#102—32no—
121C2×#102—32no—
122C2×C4×C8—32no—
123C2×C4×C8—31no—
124#396—22no—
125#394—62no—
126#390—32no—
127#390—32no—
128#390—61no—
129#390C2×D312yes—
130#389C2×D312yes—
131#384C2×D311yescharacteristic
132#380—21no—
133C4×C4⋊C4—31no—
134#340—32no—
135#340—31no—
136#340—31no—
137C24×C3—21no—
138C23×D3—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8—22no—
141C2×C3×Q8D814yescomplemented
142C2×C3×D4—21no—
143C2×C3×D4—21no—
144C2×C3×D4—21no—
145C2×C3×D4—21no—
146C2×C3×D4—21no—
147C2×C3×D4#4112yes—
148C22×C3×C4—21no—
149C22×C3×C4—22no—
150C22×C3×C4—42no—
151C22×C3×C4—42no—
152C22×C3×C4—42no—
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D412yes—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2411yescharacteristic
158C22×Dic3—21no—
159C22×Dic3—21no—
160C2×D12—22no—
161C2×D12—22no—
162C2×D12—22no—
163C2×D12—22no—
164C2×D12—22no—
165C2×D12—22no—
166C2×D12—42no—
167C2×D12—42no—
168C2×D12—42no—
169C2×D12—42no—
170C2×D12—81no—
171C2×D12—81no—
172C3×C4⋊C4—24no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—42no—
179C3×C4⋊C4—44no—
180C3×C4⋊C4—81no—
181C3×C22⋊C4—42no—
182C3×C22⋊C4—44no—
183C3×C22⋊C4—82no—
184C3×C42—22no—
185C3×C42—22no—
186C3×C42C2×D412yes—
187C3×C42#4112yes—
188#213—48no—
189(C2×D3)⋊C4—44no—
190(C2×D3)⋊C4—44no—
191(C2×D3)⋊C4—82no—
192(C2×D3)⋊C4—82no—
193C4⋊Dic3—21no—
194C4⋊Dic3—21no—
195C4⋊Dic3—21no—
196C4⋊Dic3—21no—
197C4×Dic3—42no—
198C2×C3⋊C8—21no—
199C2×C3⋊C8—22no—
200C2×C3⋊C8—22no—
201C2×C3⋊C8—21no—
202C2×C3⋊C8—21no—
203C2×C3⋊C8—21no—
204C2×C3⋊C8—21no—
205C2×C3⋊C8C2×D414yes—
206C2×C3⋊C8#4112yes—
207C22×Q8#6711yescharacteristic
208C22×D4—32no—
209C22×D4—61no—
210C22×D4—61no—
211C22×D4C22×D311yescharacteristic
212C23×C4—21no—
213C2×D8—64no—
214C2×D8—64no—
215C2×D8—122no—
216C22×C8—32no—
217C22×C8—31no—
218C22×C8—31no—
219#127—62no—
220#127—62no—
221#124—124no—
222C22⋊Q8—44no—
223C4×Q8—22no—
224C4×Q8—22no—
225C4×D4—22no—
226C4×D4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—31no—
231C2×C4⋊C4—31no—
232C2×C4⋊C4#6711yescharacteristic
233C2×C4⋊C4#6712yes—
234C2×C4⋊C4C22×D311yescharacteristic
235C2×C4⋊C4C22×D312yes—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—62no—
240C2×C22⋊C4—62no—
241C2×C22⋊C4—62no—
242C2×C22⋊C4—62no—
243C2×C42—61no—
244C2×C42C22×D312yes—
245C2×C42C22×D311yescharacteristic
246#107—122no—
247C4⋊C8—34no—
248C4⋊C8—64no—
249#102—64no—
250#102—64no—
251#102—62no—
252#102—62no—
253#102—122no—
254C4×C8—64no—
255C4×C8—62no—
256#95—22no—
257#95—61no—
258C23×C3—22no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3—41no—
262C23×C3—41no—
263C23×C3—41no—
264C23×C3—41no—
265C23×C3C22×D411yescharacteristic
266C22×D3—42no—
267C22×D3—42no—
268C22×D3—42no—
269C22×D3—42no—
270C22×D3—42no—
271C22×D3—42no—
272C22×D3—42no—
273C3×Q8—28no—
274C3×D4—22no—
275C3×D4—22no—
276C3×D4—41no—
277C3×D4—41no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—41no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—42no—
291C2×C3×C4—84no—
292C2×C3×C4#12112yes—
293C2×C3×C4#12112yes—
294C2×C3×C4C2×D814yes—
295C2×C3×C4C2×D812yes—
296C2×C3×C4#13512yes—
297C2×C3×C4C22×D411yescharacteristic
298C2×C3×C4C2×#4111yescharacteristic
299C2×C3×C4C2×#4111yescharacteristic
300C2×C3×C4C2×#4111yescharacteristic
301C2×C3×C4H(5,𝔽2)11yescharacteristic
302C2×C3×C42−1+411yescharacteristic
303C2×Dic3—41no—
304C2×Dic3—41no—
305C2×Dic3—41no—
306C2×Dic3—41no—
307C2×Dic3—41no—
308C2×Dic3—41no—
309D12—42no—
310D12—42no—
311D12—42no—
312D12—42no—
313D12—84no—
314D12—84no—
315C3⋊C8—24no—
316C3⋊C8—22no—
317C3⋊C8—41no—
318C24—21no—
319C24—62no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8#21314yescomplemented
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—62no—
329C2×D4—62no—
330C2×D4—62no—
331C2×D4—62no—
332C2×D4—62no—
333C2×D4—62no—
334C2×D4—122no—
335C2×D4—122no—
336C2×D4—122no—
337C2×D4—122no—
338C2×D4—241no—
339C2×D4—241no—
340C2×D4#23912yes—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—61no—
347C22×C4—61no—
348C22×C4C2×#6712yes—
349C22×C4C2×#6712yes—
350C22×C4C2×#6711yescharacteristic
351C22×C4C2×#6711yescharacteristic
352C22×C4C23×D311yescharacteristic
353D8—128no—
354C2×C8—34no—
355C2×C8—32no—
356C2×C8—61no—
357C2×C8—62no—
358C2×C8—61no—
359C2×C8—62no—
360C2×C8—61no—
361C2×C8—61no—
362C2×C8—61no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—42no—
370C4⋊C4—44no—
371C4⋊C4—61no—
372C4⋊C4—61no—
373C4⋊C4—61no—
374C4⋊C4—61no—
375C4⋊C4—81no—
376C22⋊C4—44no—
377C22⋊C4—42no—
378C22⋊C4—82no—
379C22⋊C4—124no—
380C22⋊C4—124no—
381C22⋊C4—242no—
382C22⋊C4—242no—
383C42—22no—
384C42—22no—
385C42—122no—
386C42D3×D412yes—
387C42#23712yes—
388C22×C3—42no—
389C22×C3—42no—
390C22×C3—42no—
391C22×C3—81no—
392C22×C3—81no—
393C22×C3—81no—
394C22×C3—81no—
395C22×C3C2×#12111yescharacteristic
396C22×C3#54811yescharacteristic
397C22×C3#55011yescharacteristic
398C22×C3C22×D811yescharacteristic
399C22×C3C2×#13511yescharacteristic
400C22×C3#57611yescharacteristic
401C22×C3#57711yescharacteristic
402C2×D3—82no—
403C2×D3—82no—
404C2×D3—82no—
405C2×D3—82no—
406C2×D3—82no—
407C2×D3—82no—
408C2×D3—82no—
409C3×C4—24no—
410C3×C4—22no—
411C3×C4—42no—
412C3×C4—82no—
413C3×C4—81no—
414C3×C4#46614yes—
415C3×C4#52911yescharacteristic
416C3×C4#54211yescharacteristic
417C3×C4#54611yescharacteristic
418C3×C4#54811yescharacteristic
419Dic3—81no—
420Dic3—81no—
421C23—22no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—122no—
429C23—122no—
430C23—122no—
431C23—122no—
432C23—122no—
433C23—122no—
434C23—122no—
435C23C22×#6711yescharacteristic
436Q8—28no—
437D4—22no—
438D4—22no—
439D4—41no—
440D4—41no—
441D4—122no—
442D4—122no—
443D4—122no—
444D4—122no—
445D4—244no—
446D4—244no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—84no—
461C2×C4—121no—
462C2×C4—121no—
463C2×C4—121no—
464C2×C4—121no—
465C2×C4—121no—
466C2×C4—121no—
467C2×C4C2×#21314yes—
468C2×C4C2×#21312yes—
469C2×C4#93912yes—
470C2×C4#93912yes—
471C2×C4#95012yes—
472C2×C4C2×D3×D411yescharacteristic
473C2×C4C2×#23711yescharacteristic
474C2×C4C2×#23911yescharacteristic
475C2×C4#100711yescharacteristic
476C2×C4D3×#4111yescharacteristic
477C2×C4#100911yescharacteristic
478C8—62no—
479C8—64no—
480C8—121no—
481C2×C3—82no—
482C2×C3#242611yescharacteristic
483C2×C3C2×#46611yescharacteristic
484C2×C3#306211yescharacteristic
485C2×C3#335811yescharacteristic
486C2×C3#336111yescharacteristic
487C2×C3#339111yescharacteristic
488C2×C3#339411yescharacteristic
489D3—162no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—81no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22—242no—
498C22—242no—
499C22—242no—
500C22—242no—
501C22—242no—
502C22—242no—
503C22—242no—
504C22C22×#21311yescharacteristic
505C22C2×#93911yescharacteristic
506C22#581711yescharacteristic
507C22C2×#95011yescharacteristic
508C22#583411yescharacteristic
509C22#583511yescharacteristic
510C22#583711yescharacteristic
511C4—24no—
512C4—22no—
513C4—42no—
514C4—82no—
515C4—81no—
516C4—241no—
517C4—241no—
518C4#503814yes—
519C4#562411yescharacteristic
520C4#563011yescharacteristic
521C4#563311yescharacteristic
522C4#564311yescharacteristic
523C3#2529211yescharacteristic, complemented, Sylow
524C2—82no—
525C2—482no—
526C2C2×#503811yescharacteristic
527C2#8275511yescharacteristic
528C2#8342211yescharacteristic
529C2#8356211yescharacteristic
530C2#8358211yescharacteristic
531C2#8364311yescharacteristic
532C2#8367911yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2538C2
socleC23×C3C22×D4
prime radicalC2×#5015C2
hypercenter#2538D3
hypocenterC3#25292
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2538

Complements

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

NH ≅ G/Ntype
1. C3×#2538C2semidirect
2. #82476C2semidirect
3. C2×Q8×C3⋊C8C2semidirect
5. C4×#832C2semidirect
6. #69650C2semidirect
7. #69647C2semidirect
8. #69630C2semidirect
9. #69615C2semidirect
10. #69580C2semidirect
11. #69567C2semidirect
47. #2538D3semidirect
141. C2×C3×Q8D8semidirect
322. C2×Q8#213semidirect
523. C3#25292semidirect

Maximal subgroups

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

subgroupindex
C3×#25382
#824762
C2×Q8×C3⋊C82
C2×#50152
C4×#8322
#696502
#696472
#696302
#696152
#695802
#695672
#252923