Finite Group Databasebeta

← #10845334

Subgroups — #10845334

Order 768.

894 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2547C211yesmaximal, characteristic
2C2×#5094C211yesmaximal, characteristic, complemented
3#82376C211yesmaximal, characteristic
4C2×D4×C3⋊C8C211yesmaximal, characteristic
5#71787C211yesmaximal, characteristic, complemented
6#69526C212yesmaximal, complemented
7#69520C212yesmaximal, complemented
8#69514C212yesmaximal, complemented
9#69511C212yesmaximal, complemented
10C4×#810C211yesmaximal, characteristic, complemented
11#69467C211yesmaximal, characteristic, complemented
12#25341—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×#124—21no—
15C2×C3×C4×D4C2211yescharacteristic
16C2×#924C2212yes—
17C23×C3⋊C8C2212yes—
18C2×C4×Dic6C2211yescharacteristic
19C3×#397—22no—
20C3×#391—21no—
21C3×#380C2212yes—
22C3×#380C2212yes—
23C3×C4×C4⋊C4C2211yescharacteristic
24#5221—21no—
25#5212C2212yes—
26C2×#830C2212yes—
27#5094—24no—
28D4×C3⋊C8—24no—
29#4972—21no—
30C2×#810C2212yes—
31C2×#810C2211yescharacteristic
32C2×#810C2211yescharacteristic
33C2×#808C2212yes—
34C2×#808C2212yes—
35#4948—22no—
36#4935C2211yescharacteristic
37C2×C4⋊(C3⋊C8)C2211yescharacteristic
38C2×C4×C3⋊C8C2211yescharacteristic
39#4529C2212yes—
40#4529C2211yescharacteristic
41#4529C2211yescharacteristic
42#4529C2212yes—
43#4529C2212yes—
44C2×#475—31no—
45C2×C8×D4—31no—
46#2547D311yescharacteristic
47#2538—31no—
48#2077—32no—
49#2035—32no—
50#1990—31no—
51#1948—31no—
52#1892—32no—
53#1807—32no—
54C4×#103—31no—
55C22×C3×Q8—21no—
56C22×C3×D4D411yescharacteristic
57C23×C3×C4C2312yes—
58C22×Dic6C2311yescharacteristic
59C3×C4⋊Q8—22no—
60C3×C4⋊Q8—22no—
61C3×#124—44no—
62C3×C4×D4—22no—
63C3×C4×D4—22no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4D411yescharacteristic
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C22⋊C4—22no—
70C2×C3×C22⋊C4—22no—
71C2×C3×C22⋊C4D412yes—
72C2×C3×C42—21no—
73C2×C3×C42C2311yescharacteristic
74C2×#217—22no—
75C2×#217—22no—
76C2×C4⋊Dic3—21no—
77C2×C4⋊Dic3—21no—
78C2×C4⋊Dic3C2312yes—
79C2×C4⋊Dic3C2311yescharacteristic
80#924—24no—
81#924—42no—
82C2×Dic3⋊C4—21no—
83C2×Dic3⋊C4—22no—
84C2×C4×Dic3—21no—
85C22×C3⋊C8—24no—
86C22×C3⋊C8—24no—
87C22×C3⋊C8—42no—
88C22×C3⋊C8C2311yescharacteristic
89C22×C3⋊C8C2312yes—
90C22×C3⋊C8C2311yescharacteristic
91C4×Dic6—22no—
92C4×Dic6—22no—
93C3×#95—21no—
94C3×#95—22no—
95C3×#95—22no—
96C3⋊#95—22no—
97#830—24no—
98#810—22no—
99#810—24no—
100#810—22no—
101#808—24no—
102#808—42no—
103C4⋊(C3⋊C8)D414yes—
104C4×C3⋊C8—22no—
105C23×C8—32no—
106C2×C4⋊Q8C2×D311yescharacteristic
107C2×#124—21no—
108C2×C22⋊Q8—32no—
109C2×C4×Q8—31no—
110C2×C4×D4C2×D311yescharacteristic
111#475—64no—
112C8×D4—64no—
113C2×#106—32no—
114C2×#106—32no—
115C2×C4⋊C8—31no—
116C2×#103—32no—
117C2×#103—31no—
118C2×#103—31no—
119C2×C22⋊C8—32no—
120C2×C4×C8—31no—
121#397—22no—
122#396—62no—
123#391—21no—
124#390—61no—
125#389—61no—
126#384—31no—
127#380—32no—
128#380C2×D312yes—
129#380C2×D312yes—
130C4×C4⋊C4C2×D311yescharacteristic
131#340—31no—
132#340—32no—
133#340—31no—
134#340—32no—
135#340—32no—
136C24×C3C2×D412yes—
137C2×C3×Q8—42no—
138C2×C3×Q8—42no—
139C2×C3×Q8—42no—
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4SD1614yes—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2411yescharacteristic
156C22×Dic3—21no—
157C22×Dic3—21no—
158C22×Dic3—22no—
159C2×Dic6—21no—
160C2×Dic6—21no—
161C2×Dic6—21no—
162C2×Dic6—21no—
163C2×Dic6—21no—
164C2×Dic6#4112yes—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—42no—
173C3×C22⋊C4—24no—
174C3×C22⋊C4—44no—
175C3×C22⋊C4—44no—
176C3×C42—42no—
177C3×C42C2×D412yes—
178C3×C42#4112yes—
179#217—44no—
180#217—82no—
181C4⋊Dic3—21no—
182C4⋊Dic3—21no—
183C4⋊Dic3—22no—
184C4⋊Dic3—22no—
185C4⋊Dic3—42no—
186C4⋊Dic3—81no—
187Dic3⋊C4—44no—
188Dic3⋊C4—42no—
189C4×Dic3—42no—
190C2×C3⋊C8—21no—
191C2×C3⋊C8—22no—
192C2×C3⋊C8—22no—
193C2×C3⋊C8—21no—
194C2×C3⋊C8—21no—
195C2×C3⋊C8—21no—
196C2×C3⋊C8—21no—
197C2×C3⋊C8—48no—
198C2×C3⋊C8C2×D414yes—
199C2×C3⋊C8#4112yes—
200C22×Q8—21no—
201C22×Q8—31no—
202C22×D4#6711yescharacteristic
203C23×C4C22×D312yes—
204C22×C8—31no—
205C22×C8—32no—
206C22×C8—31no—
207C22×C8—64no—
208C22×C8—64no—
209C22×C8—122no—
210C4⋊Q8—22no—
211C4⋊Q8—22no—
212#124—44no—
213C22⋊Q8—64no—
214C22⋊Q8—122no—
215C4×Q8—62no—
216C4×Q8—62no—
217C4×D4—22no—
218C4×D4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—31no—
221C2×C4⋊C4—32no—
222C2×C4⋊C4—61no—
223C2×C4⋊C4—62no—
224C2×C4⋊C4—61no—
225C2×C4⋊C4—61no—
226C2×C4⋊C4#6711yescharacteristic
227C2×C4⋊C4C22×D312yes—
228C2×C4⋊C4C22×D311yescharacteristic
229C2×C4⋊C4C22×D311yescharacteristic
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—62no—
233C2×C22⋊C4—62no—
234C2×C22⋊C4#6712yes—
235C2×C42—21no—
236C2×C42—61no—
237C2×C42C22×D311yescharacteristic
238#106—64no—
239#106—122no—
240C4⋊C8—34no—
241#103—62no—
242#103—64no—
243#103—62no—
244C22⋊C8—64no—
245C4×C8—62no—
246#95—21no—
247#95—22no—
248#95—22no—
249#95—62no—
250C23×C3—22no—
251C23×C3—22no—
252C23×C3—22no—
253C23×C3—22no—
254C23×C3—22no—
255C23×C3—22no—
256C23×C3C2×SD1614yes—
257C23×C3C22×D411yescharacteristic
258C3×Q8—84no—
259C3×D4—28no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4#12112yes—
279C2×C3×C4#12112yes—
280C2×C3×C4C2×SD1612yes—
281C2×C3×C4#13512yes—
282C2×C3×C4C22×D411yescharacteristic
283C2×C3×C4C2×#4111yescharacteristic
284C2×C3×C4C2×#4111yescharacteristic
285C2×C3×C4C2×#4111yescharacteristic
286C2×C3×C4H(5,𝔽2)11yescharacteristic
287C2×C3×C42−1+411yescharacteristic
288C2×Dic3—22no—
289C2×Dic3—41no—
290C2×Dic3—41no—
291C2×Dic3—41no—
292C2×Dic3—42no—
293C2×Dic3—42no—
294C2×Dic3—42no—
295C2×Dic3—41no—
296C2×Dic3—41no—
297Dic6—22no—
298Dic6—22no—
299Dic6—41no—
300Dic6—41no—
301C3⋊C8—22no—
302C3⋊C8—24no—
303C3⋊C8—41no—
304C24C2×#6712yes—
305C2×Q8—32no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—61no—
310C2×Q8—61no—
311C2×Q8—61no—
312C2×Q8—61no—
313C2×Q8—61no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4#21414yes—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—61no—
327C22×C4—61no—
328C22×C4—62no—
329C22×C4C2×#6712yes—
330C22×C4C2×#6711yescharacteristic
331C22×C4C2×#6711yescharacteristic
332C22×C4C23×D311yescharacteristic
333C2×C8—34no—
334C2×C8—32no—
335C2×C8—61no—
336C2×C8—62no—
337C2×C8—62no—
338C2×C8—61no—
339C2×C8—61no—
340C2×C8—61no—
341C2×C8—61no—
342C2×C8—128no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—42no—
351C4⋊C4—61no—
352C4⋊C4—61no—
353C4⋊C4—62no—
354C4⋊C4—62no—
355C4⋊C4—124no—
356C4⋊C4—122no—
357C4⋊C4—122no—
358C4⋊C4—241no—
359C22⋊C4—24no—
360C22⋊C4—44no—
361C22⋊C4—44no—
362C22⋊C4—124no—
363C22⋊C4—242no—
364C42—42no—
365C42—122no—
366C42D3×D412yes—
367C42#23712yes—
368C22×C3—24no—
369C22×C3—24no—
370C22×C3—24no—
371C22×C3—42no—
372C22×C3—42no—
373C22×C3—42no—
374C22×C3—42no—
375C22×C3C2×#12111yescharacteristic
376C22×C3#54611yescharacteristic
377C22×C3#54711yescharacteristic
378C22×C3C22×SD1611yescharacteristic
379C22×C3C2×#13511yescharacteristic
380C22×C3#57711yescharacteristic
381C22×C3#57811yescharacteristic
382C3×C4—22no—
383C3×C4—42no—
384C3×C4—81no—
385C3×C4—81no—
386C3×C4—82no—
387C3×C4#46514yes—
388C3×C4#52911yescharacteristic
389C3×C4#54211yescharacteristic
390C3×C4#54611yescharacteristic
391C3×C4#54811yescharacteristic
392Dic3—42no—
393Dic3—82no—
394Dic3—81no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23C2×#21414yes—
402C23C22×#6711yescharacteristic
403Q8—62no—
404Q8—62no—
405Q8—84no—
406Q8—121no—
407Q8—121no—
408D4—28no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—62no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4—121no—
431C2×C4—122no—
432C2×C4—122no—
433C2×C4—121no—
434C2×C4—122no—
435C2×C4—121no—
436C2×C4C2×#21412yes—
437C2×C4#93912yes—
438C2×C4#93912yes—
439C2×C4#95012yes—
440C2×C4C2×D3×D411yescharacteristic
441C2×C4C2×#23711yescharacteristic
442C2×C4D3×#4111yescharacteristic
443C2×C4D3×#4111yescharacteristic
444C2×C4#100911yescharacteristic
445C2×C4#101011yescharacteristic
446C8—62no—
447C8—64no—
448C8—121no—
449C2×C3—44no—
450C2×C3#242711yescharacteristic
451C2×C3C2×#46511yescharacteristic
452C2×C3#306311yescharacteristic
453C2×C3#329811yescharacteristic
454C2×C3#330411yescharacteristic
455C2×C3#333511yescharacteristic
456C2×C3#334211yescharacteristic
457C22—24no—
458C22—24no—
459C22—24no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22C22×#21411yescharacteristic
465C22C2×#93911yescharacteristic
466C22C2×#95011yescharacteristic
467C22#582411yescharacteristic
468C22#582911yescharacteristic
469C22#583711yescharacteristic
470C22#583811yescharacteristic
471C4—22no—
472C4—42no—
473C4—81no—
474C4—81no—
475C4—82no—
476C4—122no—
477C4—241no—
478C4—242no—
479C4#504114yes—
480C4#567611yescharacteristic
481C4#567711yescharacteristic
482C4#568011yescharacteristic
483C4#568211yescharacteristic
484C3#2534111yescharacteristic, complemented, Sylow
485C2—44no—
486C2C2×#504111yescharacteristic
487C2#8299611yescharacteristic
488C2#8340011yescharacteristic
489C2#8351711yescharacteristic
490C2#8353111yescharacteristic
491C2#8365911yescharacteristic
492C2#8367311yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2547C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
hypercenter#2547D3
hypocenterC3#25341
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2547

Complements

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

NH ≅ G/Ntype
2. C2×#5094C2semidirect
5. #71787C2semidirect
6. #69526C2semidirect
7. #69520C2semidirect
8. #69514C2semidirect
9. #69511C2semidirect
10. C4×#810C2semidirect
11. #69467C2semidirect
484. C3#25341semidirect

Maximal subgroups

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

subgroupindex
C3×#25472
C2×#50942
#823762
C2×D4×C3⋊C82
#717872
#695262
#695202
#695142
#695112
C4×#8102
#694672
#253413