Finite Group Databasebeta

← #10723350

Subgroups — #10723350

Order 768.

1670 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊Q8C211yesmaximal, characteristic
2C2×#5229C212yesmaximal
3C2×#4980C212yesmaximal
4#72134C212yesmaximal
5#71569C212yesmaximal
6#71565C212yesmaximal
7#71564C212yesmaximal
8#71534C211yesmaximal, characteristic
9#71534C211yesmaximal, characteristic
10#14420—31nomaximal, Sylow
11C23×C3×Q8C2212yes—
12C2×C3×C4⋊Q8—28no—
13C2×C3×C4⋊Q8C418yescomplemented
14C22×C3×C4⋊C4C2212yes—
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C42C2211yescharacteristic
18C22×C4⋊Dic3C2212yes—
19C22×C4×Dic3C2212yes—
20#5229—28no—
21C2×C3⋊#95C2212yes—
22C2×C3⋊#95C2212yes—
23C2×C3⋊#95C2212yes—
24C2×C3⋊#95C2212yes—
25C2×C3⋊#95C2211yescharacteristic
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2211yescharacteristic
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2211yescharacteristic
34C2×C3⋊#95C2211yescharacteristic
35#4980—28no—
36C22×C4⋊Q8D311yescharacteristic
37C2×#391—32no—
38C2×#387—32no—
39#1446—32no—
40#1445—32no—
41#1441—32no—
42#1436—32no—
43#1434—31no—
44#1434—31no—
45C22×C3×Q8—24no—
46C22×C3×Q8—24no—
47C22×C3×Q8—24no—
48C22×C3×Q8—24no—
49C22×C3×Q8—42no—
50C22×C3×Q8—42no—
51C22×C3×Q8C2×C418yes—
52C23×C3×C4C2312yes—
53C23×C3×C4C2312yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×Dic3C2312yes—
58C23×Dic3C2312yes—
59C23×Dic3C2312yes—
60C23×Dic3C2312yes—
61C3×C4⋊Q8—232no—
62C2×C3×C4⋊C4—24no—
63C2×C3×C4⋊C4—24no—
64C2×C3×C4⋊C4—24no—
65C2×C3×C4⋊C4C2×C414yes—
66C2×C3×C4⋊C4C2×C418yes—
67C2×C3×C4⋊C4C2×C414yes—
68C2×C3×C4⋊C4D418yes—
69C2×C3×C42—24no—
70C2×C3×C42C2×C414yes—
71C2×C4⋊Dic3—24no—
72C2×C4⋊Dic3—24no—
73C2×C4⋊Dic3—42no—
74C2×C4×Dic3—24no—
75C2×C4×Dic3—24no—
76C2×C4×Dic3—42no—
77C3⋊#95—24no—
78C3⋊#95—24no—
79C3⋊#95—24no—
80C3⋊#95—24no—
81C3⋊#95—24no—
82C3⋊#95—22no—
83C3⋊#95—24no—
84C3⋊#95—22no—
85C3⋊#95—24no—
86C3⋊#95—24no—
87C3⋊#95—42no—
88C3⋊#95—42no—
89C3⋊#95D414yes—
90C3⋊#95Q814yescomplemented
91C23×Q8C2×D312yes—
92C2×C4⋊Q8—28no—
93C2×C4⋊Q8Dic318yescomplemented
94C22×C4⋊C4—32no—
95C22×C4⋊C4C2×D312yes—
96C22×C4⋊C4C2×D311yescharacteristic
97C22×C4⋊C4C2×D311yescharacteristic
98C22×C42—32no—
99C22×C42C2×D311yescharacteristic
100#391—68no—
101#387—68no—
102C2×#95—31no—
103C2×#95—31no—
104C2×#95—32no—
105C2×#95—31no—
106C2×#95—32no—
107C2×#95—32no—
108C2×#95—31no—
109C2×#95—32no—
110C2×#95—32no—
111C2×#95—32no—
112C2×#95—32no—
113C2×#95—32no—
114C2×#95—32no—
115C2×#95—32no—
116C24×C3C2411yescharacteristic
117C2×C3×Q8—216no—
118C2×C3×Q8—216no—
119C2×C3×Q8—48no—
120C2×C3×Q8—48no—
121C2×C3×Q8—48no—
122C2×C3×Q8—48no—
123C2×C3×Q8—48no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4C22×C414yes—
142C22×C3×C4C22×C412yes—
143C22×C3×C4C22×C414yes—
144C22×C3×C4C22×C412yes—
145C22×C3×C4C22×C412yes—
146C22×C3×C4C2×D412yes—
147C22×C3×C4C2×D414yes—
148C22×C3×C4C2×D414yes—
149C22×C3×C4C2×D414yes—
150C22×C3×C4C2×Q812yes—
151C22×C3×C4#4112yes—
152C22×C3×C4#4112yes—
153C22×Dic3—22no—
154C22×Dic3—22no—
155C22×Dic3—22no—
156C22×Dic3—22no—
157C22×Dic3—22no—
158C22×Dic3—22no—
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3—22no—
165C22×Dic3—22no—
166C22×Dic3—22no—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3C2×D414yes—
180C22×Dic3C2×Q814yes—
181C3×C4⋊C4—216no—
182C3×C4⋊C4—28no—
183C3×C4⋊C4C22⋊C4116yes—
184C3×C42—28no—
185C4⋊Dic3—48no—
186C4×Dic3—48no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—42no—
192C22×Q8—42no—
193C22×Q8C2×Dic318yes—
194C23×C4—32no—
195C23×C4—32no—
196C23×C4—32no—
197C23×C4—32no—
198C23×C4C22×D312yes—
199C23×C4C22×D311yescharacteristic
200C23×C4C22×D311yescharacteristic
201C23×C4C22×D312yes—
202C23×C4C22×D311yescharacteristic
203C4⋊Q8—232no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—64no—
208C2×C4⋊C4—64no—
209C2×C4⋊C4—122no—
210C2×C4⋊C4C2×Dic314yes—
211C2×C4⋊C4C2×Dic318yes—
212C2×C4⋊C4C2×Dic314yes—
213C2×C4⋊C4#6718yes—
214C2×C42—24no—
215C2×C42—64no—
216C2×C42—64no—
217C2×C42—122no—
218C2×C42C2×Dic314yes—
219#95—34no—
220#95—34no—
221#95—64no—
222#95—64no—
223#95—64no—
224#95—62no—
225#95—64no—
226#95—62no—
227#95—64no—
228#95—64no—
229#95—64no—
230#95—64no—
231#95—122no—
232#95—122no—
233C23×C3C23×C411yescharacteristic
234C23×C3C22×D411yescharacteristic
235C23×C3C22×D412yes—
236C23×C3C22×Q811yescharacteristic
237C23×C3C2×#4111yescharacteristic
238C23×C3C2×#4111yescharacteristic
239C23×C3H(5,𝔽2)12yes—
240C23×C32−1+412yes—
241C23×C32−1+411yescharacteristic
242C23×C32−1+411yescharacteristic
243C23×C32−1+412yes—
244C3×Q8—432no—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4—24no—
248C2×C3×C4—24no—
249C2×C3×C4—22no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—24no—
254C2×C3×C4—24no—
255C2×C3×C4—24no—
256C2×C3×C4—24no—
257C2×C3×C4—24no—
258C2×C3×C4—24no—
259C2×C3×C4—24no—
260C2×C3×C4—24no—
261C2×C3×C4—24no—
262C2×C3×C4—24no—
263C2×C3×C4—24no—
264C2×C3×C4—24no—
265C2×C3×C4—24no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4C2×C22⋊C414yes—
271C2×C3×C4C2×C22⋊C418yes—
272C2×C3×C4C4×D414yes—
273C2×C3×C4C4×Q814yes—
274C2×Dic3—24no—
275C2×Dic3—24no—
276C2×Dic3—24no—
277C2×Dic3—24no—
278C2×Dic3—24no—
279C2×Dic3—24no—
280C2×Dic3—42no—
281C2×Dic3—42no—
282C2×Dic3—42no—
283C2×Dic3—42no—
284C2×Dic3—42no—
285C2×Dic3—42no—
286C2×Dic3—42no—
287C2×Dic3—42no—
288C2×Dic3—42no—
289C2×Dic3—42no—
290C2×Dic3—42no—
291C2×Dic3—42no—
292C2×Dic3—42no—
293C2×Dic3—42no—
294C2×Dic3—42no—
295C2×Dic3—42no—
296C2×Dic3—42no—
297C2×Dic3—42no—
298C2×Dic3—42no—
299C2×Dic3—42no—
300C2×Dic3—42no—
301C2×Dic3—42no—
302C24C23×D311yescharacteristic
303C2×Q8—216no—
304C2×Q8—216no—
305C2×Q8—48no—
306C2×Q8—48no—
307C2×Q8—48no—
308C2×Q8—48no—
309C2×Q8—48no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—34no—
328C22×C4—34no—
329C22×C4—62no—
330C22×C4—62no—
331C22×C4—62no—
332C22×C4—62no—
333C22×C4—62no—
334C22×C4—62no—
335C22×C4—62no—
336C22×C4—62no—
337C22×C4—62no—
338C22×C4—62no—
339C22×C4—62no—
340C22×C4—62no—
341C22×C4—62no—
342C22×C4—62no—
343C22×C4—62no—
344C22×C4—62no—
345C22×C4—62no—
346C22×C4—62no—
347C22×C4—62no—
348C22×C4—62no—
349C22×C4—62no—
350C22×C4—62no—
351C22×C4—62no—
352C22×C4—62no—
353C22×C4—62no—
354C22×C4—62no—
355C22×C4D3×D412yes—
356C22×C4#23712yes—
357C22×C4D3×Q812yes—
358C22×C4#23912yes—
359C22×C4C22×Dic314yes—
360C22×C4C22×Dic312yes—
361C22×C4C22×Dic312yes—
362C22×C4C22×Dic312yes—
363C22×C4C22×Dic314yes—
364C22×C4C2×#6714yes—
365C22×C4C2×#6714yes—
366C22×C4C2×#6714yes—
367C4⋊C4—216no—
368C4⋊C4—28no—
369C4⋊C4—128no—
370C4⋊C4#217116yes—
371C42—28no—
372C42—128no—
373C22×C3C22×C22⋊C411yescharacteristic
374C22×C3C2×C4×D411yescharacteristic
375C22×C3C2×C4×Q811yescharacteristic
376C22×C3#51811yescharacteristic
377C22×C3#51911yescharacteristic
378C22×C3#51911yescharacteristic
379C22×C3#51911yescharacteristic
380C22×C3#53612yes—
381C22×C3#53612yes—
382C22×C3#53612yes—
383C22×C3#54112yes—
384C22×C3#54212yes—
385C22×C3#54412yes—
386C22×C3D4212yes—
387C22×C3#54612yes—
388C22×C3#54711yescharacteristic
389C22×C3#54812yes—
390C22×C3D4×Q811yescharacteristic
391C22×C3D4×Q812yes—
392C22×C3#55611yescharacteristic
393C22×C3#55711yescharacteristic
394C22×C3#56312yes—
395C22×C3#56312yes—
396C3×C4—24no—
397C3×C4—28no—
398C3×C4—24no—
399C3×C4—24no—
400C3×C4—44no—
401Dic3—44no—
402Dic3—44no—
403Dic3—82no—
404Dic3—82no—
405C23C2×D3×D411yescharacteristic
406C23C2×#23711yescharacteristic
407C23#100412yes—
408C23C2×D3×Q811yescharacteristic
409C23C2×#23911yescharacteristic
410C23#100711yescharacteristic
411C23#100712yes—
412C23#100712yes—
413C23#100711yescharacteristic
414C23C23×Dic311yescharacteristic
415C23C22×#6712yes—
416Q8—432no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—64no—
443C2×C4—64no—
444C2×C4—64no—
445C2×C4—64no—
446C2×C4—64no—
447C2×C4—64no—
448C2×C4—122no—
449C2×C4—122no—
450C2×C4—122no—
451C2×C4—122no—
452C2×C4—122no—
453C2×C4—122no—
454C2×C4—122no—
455C2×C4—122no—
456C2×C4—122no—
457C2×C4—122no—
458C2×C4—122no—
459C2×C4—122no—
460C2×C4—122no—
461C2×C4—122no—
462C2×C4—122no—
463C2×C4—122no—
464C2×C4—122no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4—122no—
468C2×C4—122no—
469C2×C4—122no—
470C2×C4D4×Dic314yes—
471C2×C4Q8×Dic314yes—
472C2×C4C2×#21714yes—
473C2×C4C2×#21718yes—
474C2×C3#231011yescharacteristic
475C2×C3#232711yescharacteristic
476C2×C3#233111yescharacteristic
477C2×C3D4×C22⋊C411yescharacteristic
478C2×C3Q8×C22⋊C411yescharacteristic
479C2×C3#236511yescharacteristic
480C2×C3#238111yescharacteristic
481C2×C3#267412yes—
482C2×C3#267712yes—
483C2×C3#271312yes—
484C2×C3#271612yes—
485C22#572111yescharacteristic
486C22#572412yes—
487C22#572611yescharacteristic
488C22#572711yescharacteristic
489C22#572912yes—
490C22#573112yes—
491C22#573211yescharacteristic
492C22#573312yes—
493C22#573512yes—
494C22#573512yes—
495C22C2×D4×Dic311yescharacteristic
496C22#579811yescharacteristic
497C22D4×#6712yes—
498C22#580612yes—
499C22C2×Q8×Dic311yescharacteristic
500C22#581211yescharacteristic
501C22#581211yescharacteristic
502C22#581211yescharacteristic
503C22Q8×#6712yes—
504C22#581612yes—
505C22#581612yes—
506C22#581712yes—
507C22C22×#21711yescharacteristic
508C4—24no—
509C4—28no—
510C4—24no—
511C4—24no—
512C4—44no—
513C4—124no—
514C4—124no—
515C4—242no—
516C4—242no—
517C3#1442011yescharacteristic, complemented, Sylow
518C2#8322711yescharacteristic
519C2#8322811yescharacteristic
520C2#8322911yescharacteristic
521C2#8323111yescharacteristic
522C2#8324712yes—
523C2#8324812yes—
524C2#8326512yes—
525C2#8326612yes—
526C2D4×#21711yescharacteristic
527C2#8406211yescharacteristic
528C2Q8×#21711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC22×C3×C4⋊Q8C2
socle, prime radical, cosocleC24×C3C24
hypercenterC22×C4⋊Q8D3
hypocenterC3#14420
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×C4⋊Q8

Complements

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

NH ≅ G/Ntype
13. C2×C3×C4⋊Q8C4semidirect
90. C3⋊#95Q8semidirect
93. C2×C4⋊Q8Dic3semidirect
517. C3#14420semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊Q82
C2×#52292
C2×#49802
#721342
#715692
#715652
#715642
#715342
#144203