Finite Group Databasebeta

← #10845933

Subgroups — #10845933

Order 768.

1018 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2542C211yesmaximal, characteristic
2C2×#5090C211yesmaximal, characteristic, complemented
3C2×#5051C212yesmaximal, complemented
4#82376C211yesmaximal, characteristic
5C2×D4×C3⋊C8C211yesmaximal, characteristic
6#71785C211yesmaximal, characteristic, complemented
7#71700C212yesmaximal, complemented
8#71696C212yesmaximal, complemented
9#71695C212yesmaximal, complemented
10C4×#810C211yesmaximal, characteristic, complemented
11#69466C211yesmaximal, characteristic, complemented
12#25386—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C22⋊Q8C2212yes—
15C2×C3×C22⋊Q8C2212yes—
16C2×C3×C4×D4C2211yescharacteristic
17C22×#216C2212yes—
18C2×#924C2212yes—
19C23×C3⋊C8C2212yes—
20C2×C4×Dic6C2211yescharacteristic
21C3×#399—22no—
22C3×#390—21no—
23C3×#384—21no—
24C3×C4×C4⋊C4C2211yescharacteristic
25C2×#835C2212yes—
26C2×#835C2212yes—
27C2×#835C2212yes—
28#5221—21no—
29#5212C2212yes—
30C2×#830C2212yes—
31#5090—24no—
32#5051—28no—
33D4×C3⋊C8—24no—
34#4972—21no—
35C2×#810C2211yescharacteristic
36C2×#810C2211yescharacteristic
37C2×#807C2212yes—
38#4948—22no—
39#4935C2211yescharacteristic
40C2×C4⋊(C3⋊C8)C2211yescharacteristic
41C2×C4×C3⋊C8C2211yescharacteristic
42#4529C2211yescharacteristic
43#4529C2211yescharacteristic
44#4529C2212yes—
45C2×#467—32no—
46C2×#462—31no—
47C2×C8×D4—31no—
48#2542D311yescharacteristic
49#2538—31no—
50#2036—32no—
51#2001—32no—
52#1956—31no—
53#1947—31no—
54#1892—32no—
55C4×#103—31no—
56C22×C3×Q8C2312yes—
57C22×C3×D4D411yescharacteristic
58C23×C3×C4C2312yes—
59C22×Dic6C2311yescharacteristic
60C3×C4⋊Q8—22no—
61C3×C4⋊Q8—22no—
62C3×C22⋊Q8—24no—
63C3×C22⋊Q8—24no—
64C3×C22⋊Q8—42no—
65C3×C22⋊Q8—42no—
66C3×C4×D4—22no—
67C3×C4×D4—22no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4D411yescharacteristic
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4D412yes—
78C2×C3×C42—21no—
79C2×C3×C42C2311yescharacteristic
80C2×#217—22no—
81C2×#217—22no—
82C2×#216—24no—
83C2×#216—24no—
84C2×#216—42no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3C2312yes—
88C2×C4⋊Dic3C2311yescharacteristic
89#924—24no—
90#924—42no—
91C2×Dic3⋊C4—21no—
92C2×Dic3⋊C4—22no—
93C2×C4×Dic3—21no—
94C22×C3⋊C8—24no—
95C22×C3⋊C8—24no—
96C22×C3⋊C8—42no—
97C22×C3⋊C8C2311yescharacteristic
98C22×C3⋊C8C2311yescharacteristic
99C22×C3⋊C8C2312yes—
100C4×Dic6—22no—
101C4×Dic6—22no—
102C3×#95—21no—
103#835—24no—
104#835—24no—
105#835—42no—
106C3⋊#95—22no—
107#830—24no—
108#810—22no—
109#810D414yescomplemented
110#807—24no—
111C4⋊(C3⋊C8)D414yes—
112C4×C3⋊C8—22no—
113C22×Q16—32no—
114C23×C8—32no—
115C2×C4⋊Q8C2×D311yescharacteristic
116C2×C22⋊Q8—32no—
117C2×C22⋊Q8C2×D312yes—
118C2×C22⋊Q8C2×D312yes—
119C2×C4×Q8—31no—
120C2×C4×D4C2×D311yescharacteristic
121#467—68no—
122#462—64no—
123C8×D4—64no—
124C2×#107—32no—
125C2×C4⋊C8—31no—
126C2×#103—31no—
127C2×#103—32no—
128C2×#103—32no—
129C2×#103—31no—
130C2×#103—32no—
131C2×C22⋊C8—32no—
132C2×C4×C8—31no—
133#399—22no—
134#396—62no—
135#390—21no—
136#390—61no—
137#389—61no—
138#384—21no—
139#384—31no—
140#380—32no—
141C4×C4⋊C4C2×D311yescharacteristic
142#340—31no—
143#340—32no—
144#340—31no—
145C24×C3C2×D412yes—
146C2×C3×Q8—22no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×Q8—22no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4Q1614yescomplemented
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2411yescharacteristic
168C22×Dic3—21no—
169C22×Dic3—21no—
170C22×Dic3—22no—
171C2×Dic6—21no—
172C2×Dic6—21no—
173C2×Dic6—21no—
174C2×Dic6—21no—
175C2×Dic6—21no—
176C2×Dic6C2×D412yes—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—21no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—44no—
181C3×C4⋊C4—44no—
182C3×C4⋊C4—42no—
183C3×C4⋊C4—42no—
184C3×C4⋊C4C2×D412yes—
185C3×C4⋊C4#4112yes—
186C3×C22⋊C4—24no—
187C3×C22⋊C4—44no—
188C3×C22⋊C4—44no—
189C3×C42—42no—
190C3×C42C2×D412yes—
191C3×C42#4112yes—
192#217—44no—
193#217—82no—
194#216—48no—
195C4⋊Dic3—22no—
196C4⋊Dic3—21no—
197C4⋊Dic3—21no—
198C4⋊Dic3—22no—
199C4⋊Dic3—42no—
200C4⋊Dic3—81no—
201Dic3⋊C4—44no—
202Dic3⋊C4—42no—
203C4×Dic3—42no—
204C2×C3⋊C8—21no—
205C2×C3⋊C8—22no—
206C2×C3⋊C8—21no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8—21no—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8—48no—
212C2×C3⋊C8C2×D414yes—
213C2×C3⋊C8C2×D412yes—
214C22×Q8—31no—
215C22×Q8C22×D312yes—
216C22×D4#6711yescharacteristic
217C23×C4C22×D312yes—
218C2×Q16—64no—
219C2×Q16—64no—
220C2×Q16—122no—
221C22×C8—31no—
222C22×C8—32no—
223C22×C8—31no—
224C22×C8—64no—
225C22×C8—64no—
226C22×C8—122no—
227C4⋊Q8—22no—
228C4⋊Q8—22no—
229C22⋊Q8—24no—
230C22⋊Q8—24no—
231C22⋊Q8—42no—
232C22⋊Q8—42no—
233C22⋊Q8—64no—
234C22⋊Q8—122no—
235C4×Q8—62no—
236C4×Q8—62no—
237C4×D4—22no—
238C4×D4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—31no—
244C2×C4⋊C4—32no—
245C2×C4⋊C4—61no—
246C2×C4⋊C4—62no—
247C2×C4⋊C4—61no—
248C2×C4⋊C4—61no—
249C2×C4⋊C4#6711yescharacteristic
250C2×C4⋊C4C22×D311yescharacteristic
251C2×C4⋊C4C22×D311yescharacteristic
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—62no—
255C2×C22⋊C4—62no—
256C2×C22⋊C4#6712yes—
257C2×C42—21no—
258C2×C42—61no—
259C2×C42C22×D311yescharacteristic
260#107—64no—
261C4⋊C8—34no—
262#103—34no—
263#103—64no—
264#103—64no—
265#103—62no—
266#103—122no—
267C22⋊C8—64no—
268C4×C8—62no—
269#95—21no—
270#95—62no—
271C23×C3—22no—
272C23×C3—22no—
273C23×C3—22no—
274C23×C3—22no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3C2×Q1614yes—
278C23×C3C22×D411yescharacteristic
279C3×Q8—42no—
280C3×Q8—42no—
281C3×Q8—42no—
282C3×Q8—42no—
283C3×D4—28no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4#12112yes—
303C2×C3×C4#12112yes—
304C2×C3×C4C2×Q1612yes—
305C2×C3×C4#13512yes—
306C2×C3×C4C22×D411yescharacteristic
307C2×C3×C4C22×D411yescharacteristic
308C2×C3×C4C2×#4111yescharacteristic
309C2×C3×C4C2×#4111yescharacteristic
310C2×C3×C4H(5,𝔽2)11yescharacteristic
311C2×C3×C4H(5,𝔽2)11yescharacteristic
312C2×Dic3—22no—
313C2×Dic3—41no—
314C2×Dic3—41no—
315C2×Dic3—41no—
316C2×Dic3—42no—
317C2×Dic3—42no—
318C2×Dic3—42no—
319C2×Dic3—41no—
320C2×Dic3—41no—
321Dic6—22no—
322Dic6—22no—
323Dic6—41no—
324Dic6—41no—
325C3⋊C8—22no—
326C3⋊C8—24no—
327C3⋊C8—41no—
328C24C2×#6712yes—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—32no—
336C2×Q8—61no—
337C2×Q8—61no—
338C2×Q8—61no—
339C2×Q8—61no—
340C2×Q8—61no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4#21614yescomplemented
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—61no—
354C22×C4—61no—
355C22×C4—62no—
356C22×C4C2×#6712yes—
357C22×C4C2×#6711yescharacteristic
358C22×C4C2×#6711yescharacteristic
359C22×C4C23×D311yescharacteristic
360Q16—128no—
361C2×C8—34no—
362C2×C8—32no—
363C2×C8—61no—
364C2×C8—62no—
365C2×C8—61no—
366C2×C8—62no—
367C2×C8—61no—
368C2×C8—61no—
369C2×C8—61no—
370C2×C8—128no—
371C4⋊C4—22no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—61no—
379C4⋊C4—62no—
380C4⋊C4—61no—
381C4⋊C4—62no—
382C4⋊C4—124no—
383C4⋊C4—122no—
384C4⋊C4—122no—
385C4⋊C4—241no—
386C4⋊C4D3×D412yes—
387C4⋊C4#23712yes—
388C22⋊C4—24no—
389C22⋊C4—44no—
390C22⋊C4—44no—
391C22⋊C4—124no—
392C22⋊C4—242no—
393C42—42no—
394C42—122no—
395C42D3×D412yes—
396C42#23712yes—
397C22×C3—24no—
398C22×C3—24no—
399C22×C3—24no—
400C22×C3—42no—
401C22×C3—42no—
402C22×C3—42no—
403C22×C3—42no—
404C22×C3C2×#12111yescharacteristic
405C22×C3D4211yescharacteristic
406C22×C3#54611yescharacteristic
407C22×C3C22×Q1611yescharacteristic
408C22×C3C2×#13511yescharacteristic
409C22×C3#57711yescharacteristic
410C22×C3#57811yescharacteristic
411C3×C4—22no—
412C3×C4—42no—
413C3×C4—42no—
414C3×C4—42no—
415C3×C4—82no—
416C3×C4#46714yescomplemented
417C3×C4#53211yescharacteristic
418C3×C4#54011yescharacteristic
419C3×C4D4211yescharacteristic
420C3×C4#55011yescharacteristic
421Dic3—42no—
422Dic3—82no—
423Dic3—81no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23C2×#21614yes—
431C23C22×#6711yescharacteristic
432Q8—42no—
433Q8—42no—
434Q8—42no—
435Q8—42no—
436Q8—62no—
437Q8—62no—
438Q8—121no—
439Q8—121no—
440D4—28no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—62no—
460C2×C4—121no—
461C2×C4—121no—
462C2×C4—122no—
463C2×C4—121no—
464C2×C4—122no—
465C2×C4—121no—
466C2×C4—122no—
467C2×C4—121no—
468C2×C4#93912yes—
469C2×C4#93912yes—
470C2×C4C2×#21612yes—
471C2×C4#95012yes—
472C2×C4C2×D3×D411yescharacteristic
473C2×C4C2×D3×D411yescharacteristic
474C2×C4C2×#23711yescharacteristic
475C2×C4C2×#23711yescharacteristic
476C2×C4#100411yescharacteristic
477C2×C4#100411yescharacteristic
478C8—62no—
479C8—64no—
480C8—121no—
481C2×C3—44no—
482C2×C3#242811yescharacteristic
483C2×C3C2×#46711yescharacteristic
484C2×C3#306311yescharacteristic
485C2×C3#328411yescharacteristic
486C2×C3D4×Q1611yescharacteristic
487C2×C3#329911yescharacteristic
488C2×C3#330211yescharacteristic
489C22—24no—
490C22—24no—
491C22—24no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22D4×#6711yescharacteristic
497C22C2×#93911yescharacteristic
498C22#580611yescharacteristic
499C22C22×#21611yescharacteristic
500C22C2×#95011yescharacteristic
501C22#583711yescharacteristic
502C22#583811yescharacteristic
503C4—22no—
504C4—42no—
505C4—42no—
506C4—42no—
507C4—82no—
508C4—122no—
509C4—242no—
510C4—241no—
511C4#505114yescomplemented
512C4#571611yescharacteristic
513C4#571711yescharacteristic
514C4#571811yescharacteristic
515C4#571911yescharacteristic
516C3#2538611yescharacteristic, complemented, Sylow
517C2—44no—
518C2C2×#505111yescharacteristic
519C2#8318611yescharacteristic
520C2#8340011yescharacteristic
521C2#8351011yescharacteristic
522C2#8353011yescharacteristic
523C2#8355111yescharacteristic
524C2D4×#21611yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2542C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
hypercenter#2542D3
hypocenterC3#25386
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2542

Complements

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

NH ≅ G/Ntype
2. C2×#5090C2semidirect
3. C2×#5051C2semidirect
6. #71785C2semidirect
7. #71700C2semidirect
8. #71696C2semidirect
9. #71695C2semidirect
10. C4×#810C2semidirect
11. #69466C2semidirect
109. #810D4semidirect
154. C2×C3×D4Q16semidirect
343. C2×D4#216semidirect
416. C3×C4#467semidirect
511. C4#5051semidirect
516. C3#25386semidirect

Maximal subgroups

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

subgroupindex
C3×#25422
C2×#50902
C2×#50512
#823762
C2×D4×C3⋊C82
#717852
#717002
#716962
#716952
C4×#8102
#694662
#253863