Finite Group Databasebeta

← #10944183

Subgroups — #10944183

Order 768.

1334 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#1007C211yesmaximal, characteristic
2C3×#2546C211yesmaximal, characteristic, complemented
3C2×#5095C212yesmaximal
4#83209C212yesmaximal, complemented
5#71784C212yesmaximal, complemented
6#71637C211yesmaximal, characteristic, complemented
7#69693C214yesmaximal, complemented
8#69494C212yesmaximal, complemented
9#32923—31nomaximal, Sylow
10C2×#1007C2211yescharacteristic
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×C4⋊Q8C2212yes—
13C2×C3×C4×Q8C2211yescharacteristic
14#5812C2211yescharacteristic
15C2×#942C2214yes—
16#5574—22no—
17C4×#239—22no—
18#5572—22no—
19C4×D3×Q8—22no—
20#5504—22no—
21#5504C2212yes—
22C4×#235—22no—
23C4×#235C2212yes—
24C3×#398—22no—
25C3×#391C2212yes—
26C3×#389—22no—
27C3×C4×C4⋊C4C2211yescharacteristic
28C2×#836C2212yes—
29#5225C2214yes—
30#5095—28no—
31#5090—22no—
32#5090—22no—
33#5083—22no—
34#5083—22no—
35C2×#805C2214yes—
36#4924C2212yes—
37#4530C2212yes—
38#4523C2214yes—
39C4×2−1+4—31no—
40#3039—32no—
41C2×#456—32no—
42#2546D311yescharacteristic, complemented
43#1886—32no—
44#1882—34no—
45#1806—32no—
46#1801—31no—
47C22×C3×Q8C2312yes—
48C22×C3×Q8C2311yescharacteristic
49#1007—22no—
50#1007—24no—
51#1007D414yes—
52C2×#239—22no—
53C2×D3×Q8—22no—
54C2×#235—22no—
55C2×#235C2312yes—
56C3×C4⋊Q8—24no—
57C3×C4⋊Q8—24no—
58C3×C4⋊Q8—22no—
59C3×C4⋊Q8—22no—
60C3×C4×Q8—22no—
61C3×C4×Q8—22no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—24no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4D412yes—
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C42D412yes—
69C2×C3×C42C2312yes—
70C2×C3×C42C2311yescharacteristic
71Q8×Dic3—22no—
72Q8×Dic3—22no—
73C2×#216—24no—
74#942—28no—
75#942—28no—
76C2×#215—24no—
77C4×#67—22no—
78C4×#67—24no—
79C4×#67—22no—
80#926—22no—
81#926C2312yes—
82C2×#208C2314yes—
83#893—24no—
84#893—22no—
85#892—24no—
86#892—22no—
87D3×C4⋊C4—24no—
88D3×C4⋊C4—22no—
89#887—24no—
90#887—22no—
91C4×D12—22no—
92C4×D12—22no—
93C4×D12D412yes—
94C4×D12D412yes—
95#872—24no—
96#872—22no—
97C42×D3—24no—
98C42×D3—22no—
99C4×Dic6—22no—
100C4×Dic6—22no—
101C4×Dic6D412yes—
102C4×Dic6D412yes—
103C3×#95—21no—
104C3×#95—24no—
105#836—24no—
106#835—24no—
107#835—24no—
108#805—28no—
109C4⋊(C3⋊C8)—22no—
110C4⋊(C3⋊C8)—22no—
111C2×2−1+4—31no—
112C2×#137—34no—
113C2×C4⋊Q8C2×D312yes—
114C2×C4⋊Q8C2×D311yescharacteristic
115#520—32no—
116#520—62no—
117#520—62no—
118#519—31no—
119#519—62no—
120C4×#41—32no—
121C4×#41—62no—
122C4×#41—62no—
123C2×C4×Q8—62no—
124C2×C4×Q8C2×D311yescharacteristic
125#462—62no—
126#462—62no—
127#461—62no—
128#461—62no—
129#456—68no—
130#423—32no—
131C2×#104—34no—
132#419—34no—
133C2×#101—32no—
134#398—22no—
135#391C2×D312yes—
136#389—22no—
137C4×C4⋊C4C2×D311yescharacteristic
138#343—32no—
139#339—34no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8—24no—
143C2×C3×Q8—22no—
144C2×C3×Q8—24no—
145C2×C3×Q8—22no—
146C2×C3×Q8C2×D412yes—
147C2×C3×Q8#4112yes—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2411yescharacteristic
155C2×#67—22no—
156#239—28no—
157#239—28no—
158D3×Q8—28no—
159D3×Q8—28no—
160#235—28no—
161#235—28no—
162#235—22no—
163#235—22no—
164#235C2×D414yes—
165#235#4114yes—
166C2×D12—22no—
167C2×D12C2×D412yes—
168C2×C4×D3—24no—
169C2×C4×D3—22no—
170C2×C4×D3—22no—
171C2×C4×D3—24no—
172C2×C4×D3—22no—
173C2×Dic6—22no—
174C2×Dic6C2×D412yes—
175C3×C4⋊C4—24no—
176C3×C4⋊C4—24no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—42no—
179C3×C4⋊C4—41no—
180C3×C4⋊C4—41no—
181C3×C4⋊C4—48no—
182C3×C42—22no—
183C3×C42—22no—
184C3×C42—22no—
185C3×C42—22no—
186C3×C42C2×D412yes—
187C3×C42C2×D412yes—
188#217—22no—
189#216—48no—
190#215—48no—
191(C2×D3)⋊C4—24no—
192(C2×D3)⋊C4—22no—
193(C2×D3)⋊C4—22no—
194C4⋊Dic3—22no—
195C4⋊Dic3—22no—
196C4⋊Dic3C2×D412yes—
197C4⋊Dic3C2×D412yes—
198Dic3⋊C4—24no—
199Dic3⋊C4—22no—
200Dic3⋊C4—22no—
201C4×Dic3—24no—
202C4×Dic3—24no—
203C4×Dic3—22no—
204C4×Dic3—22no—
205C4×Dic3—22no—
206#208—24no—
207#208—24no—
208C2×C3⋊C8—24no—
2092−1+4—34no—
2102−1+4—64no—
2112−1+4—62no—
212C2×#41—32no—
213C2×#41—62no—
214C2×#41—62no—
215C22×Q8—62no—
216C22×Q8C22×D312yes—
217C22×Q8C22×D311yescharacteristic
218#137—68no—
219#137—68no—
220C2×Q16—64no—
221C2×SD16—64no—
222C2×M16—34no—
223C4⋊Q8—24no—
224C4⋊Q8—24no—
225C4⋊Q8—22no—
226C4⋊Q8—22no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×Q8—32no—
230C4×Q8—32no—
231C4×Q8—62no—
232C4×Q8—62no—
233C4×Q8—62no—
234C4×Q8—64no—
235C4×Q8—62no—
236C4×Q8—62no—
237C4×D4—32no—
238C4×D4—32no—
239C4×D4—62no—
240C4×D4—62no—
241C4×D4—62no—
242C4×D4—64no—
243C4×D4—62no—
244C4×D4—62no—
245C4×D4—64no—
246#117—32no—
247#117—64no—
248#117—62no—
249#117—64no—
250#117—62no—
251#117—62no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—62no—
257C2×C4⋊C4—64no—
258C2×C4⋊C4#6712yes—
259C2×C4⋊C4C22×D311yescharacteristic
260C2×C42—64no—
261C2×C42—62no—
262C2×C42#6712yes—
263C2×C42C22×D312yes—
264C2×C42C22×D311yescharacteristic
265C4⋊C8—62no—
266C4⋊C8—62no—
267#104—68no—
268#103—64no—
269#103—64no—
270#101—64no—
271#95—21no—
272#95—24no—
273C23×C3C22×D411yescharacteristic
274C22×D3—22no—
275C3×Q8—24no—
276C3×Q8—24no—
277C3×Q8—44no—
278C3×Q8—44no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—24no—
283C2×C3×C4—22no—
284C2×C3×C4—41no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4#12012yes—
292C2×C3×C4#12011yescharacteristic
293C2×C3×C4#12011yescharacteristic
294C2×C3×C4#12114yes—
295C2×C3×C4C22×D411yescharacteristic
296C2×C3×C4C22×D412yes—
297C2×C3×C4C2×#4111yescharacteristic
298C2×C3×C4C2×#4112yes—
299#67—24no—
300#67—24no—
301C2×Dic3—22no—
302C2×Dic3—22no—
303C2×Dic3—22no—
304D12—28no—
305D12—22no—
306D12#12114yes—
307C4×D3—216no—
308C4×D3—24no—
309C4×D3—24no—
310C4×D3—44no—
311C4×D3—82no—
312C4×D3—84no—
313Dic6—28no—
314Dic6—22no—
315Dic6#12114yes—
316C3⋊C8—44no—
317#41—34no—
318#41—34no—
319#41—68no—
320#41—68no—
321#41—68no—
322#41—68no—
323#41—62no—
324#41—62no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—24no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—24no—
331C2×Q8—32no—
332C2×Q8—62no—
333C2×Q8—68no—
334C2×Q8—68no—
335C2×Q8D3×D412yes—
336C2×Q8#23712yes—
337C2×D4—32no—
338C2×D4—62no—
339C2×D4—62no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—64no—
343C22×C4—64no—
344C22×C4—62no—
345C22×C4—62no—
346C22×C4—62no—
347C22×C4C2×#6711yescharacteristic
348C22×C4C2×#6712yes—
349C22×C4C2×#6712yes—
350C22×C4C2×#6711yescharacteristic
351C22×C4C23×D311yescharacteristic
352Q16—128no—
353SD16—128no—
354M16—64no—
355M16—64no—
356C2×C8—64no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—32no—
361C4⋊C4—32no—
362C4⋊C4—41no—
363C4⋊C4—42no—
364C4⋊C4—41no—
365C4⋊C4—48no—
366C4⋊C4—64no—
367C4⋊C4—62no—
368C4⋊C4—62no—
369C4⋊C4—62no—
370C4⋊C4—62no—
371C22⋊C4—64no—
372C22⋊C4—62no—
373C22⋊C4—62no—
374C22⋊C4—62no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—64no—
380C42—64no—
381C42—62no—
382C42—62no—
383C42—62no—
384C42D3×D412yes—
385C42D3×D412yes—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3C2×#12011yescharacteristic
389C22×C3C2×#12111yescharacteristic
390C22×C3#52511yescharacteristic
391C2×D3—24no—
392C2×D3—42no—
393C2×D3—42no—
394C3×C4—24no—
395C3×C4—21no—
396C3×C4—22no—
397C3×C4—44no—
398C3×C4—42no—
399C3×C4—82no—
400C3×C4#45612yescomplemented
401C3×C4C2×#12112yes—
402C3×C4#52512yes—
403Dic3—24no—
404Dic3—42no—
405Dic3—42no—
406C23—62no—
407C23C22×#6711yescharacteristic
408Q8—24no—
409Q8—24no—
410Q8—34no—
411Q8—44no—
412Q8—44no—
413Q8—68no—
414Q8—62no—
415D4—34no—
416D4—68no—
417D4—64no—
418D4—64no—
419D4—62no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—616no—
433C2×C4—62no—
434C2×C4—64no—
435C2×C4—64no—
436C2×C4—62no—
437C2×C4—62no—
438C2×C4—124no—
439C2×C4—242no—
440C2×C4—244no—
441C2×C4#93711yescharacteristic
442C2×C4#93712yes—
443C2×C4#93711yescharacteristic
444C2×C4#93814yes—
445C2×C4C2×D3×D411yescharacteristic
446C2×C4C2×D3×D412yes—
447C2×C4C2×#23711yescharacteristic
448C2×C4D3×#4112yes—
449C8—124no—
450C2×C3—21no—
451C2×C3—21no—
452C2×C3#284711yescharacteristic
453C2×C3C2×#45611yescharacteristic
454C2×C3#302211yescharacteristic
455D3—44no—
456C22—21no—
457C22—21no—
458C22—64no—
459C22—122no—
460C22—122no—
461C22C2×#93711yescharacteristic
462C22C2×#93811yescharacteristic
463C22#582811yescharacteristic
464C4—21no—
465C4—24no—
466C4—22no—
467C4—44no—
468C4—42no—
469C4—64no—
470C4—82no—
471C4—122no—
472C4—122no—
473C4#509512yescomplemented
474C4D3×#12112yes—
475C4#560612yes—
476C3#3292311yescharacteristic, complemented, Sylow
477C2—21no—
478C2—21no—
479C2—124no—
480C2#8272311yescharacteristic
481C2C2×#509511yescharacteristic
482C2#8341611yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2546C2
socleC22×C3C2×#120
prime radicalC2×#1007C22
hypercenter#2546D3
hypocenterC3#32923
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2546

Complements

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

NH ≅ G/Ntype
2. C3×#2546C2semidirect
4. #83209C2semidirect
5. #71784C2semidirect
6. #71637C2semidirect
7. #69693C2semidirect
8. #69494C2semidirect
42. #2546D3semidirect
400. C3×C4#456semidirect
473. C4#5095semidirect
476. C3#32923semidirect

Maximal subgroups

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

subgroupindex
C4×#10072
C3×#25462
C2×#50952
#832092
#717842
#716372
#696932
#694942
#329233