Finite Group Databasebeta

← #10945056

Subgroups — #10945056

Order 768.

1216 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5812C211yesmaximal, characteristic
2C2×#5228C211yesmaximal, characteristic
3C2×#4723C211yesmaximal, characteristic
4#73951C211yesmaximal, characteristic
5C3×#1787C211yesmaximal, characteristic
6#70239C212yesmaximal, complemented
7#70239C212yesmaximal, complemented
8#70238C212yesmaximal, complemented
9#70236C212yesmaximal, complemented
10#69911C211yesmaximal, characteristic
11#69503C211yesmaximal, characteristic
12#33019—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14#5812—24no—
15#5812—24no—
16C2×Q8×Dic3—21no—
17C2×Q8×Dic3—21no—
18C2×Q8×Dic3C2212yes—
19C2×Q8×Dic3C2212yes—
20C22×#216C2212yes—
21C2×#926—22no—
22C2×#926C2211yescharacteristic
23C2×#926C2211yescharacteristic
24C2×#924C2211yescharacteristic
25C2×C3×#103C2212yes—
26C2×C3×#103C2212yes—
27C2×C3×C22⋊C8C2211yescharacteristic
28C3×#380C2211yescharacteristic
29#5228—24no—
30C2×#835C2212yes—
31C2×#830C2211yescharacteristic
32C2×#818C2211yescharacteristic
33C2×#817C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×#816C2211yescharacteristic
36#4984C2212yes—
37#4973C2212yes—
38#4953—22no—
39#4723—24no—
40#4661—21no—
41#4659—21no—
42#4550C2212yes—
43#4550C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C2×#519—31no—
47C2×#485—31no—
48C2×#451—31no—
49#2625—31no—
50#1878—32no—
51#1878—32no—
52#1865—32no—
53#1863—32no—
54#1825—31no—
55#1818—31no—
56#1787D311yescharacteristic
57C22×C3×Q8—24no—
58C22×C3×Q8—24no—
59C22×C3×Q8—21no—
60C22×C3×Q8—22no—
61C22×C3×Q8D414yes—
62C22×C3×Q8C2312yes—
63C23×C3×C4—21no—
64C23×C3×C4C2311yescharacteristic
65C22×Dic6C2311yescharacteristic
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2311yescharacteristic
68C2×C3×C4⋊C4C2312yes—
69C2×C3×C22⋊C4—21no—
70C2×#217—21no—
71C2×#217—21no—
72C2×#217D412yes—
73Q8×Dic3—28no—
74Q8×Dic3—28no—
75Q8×Dic3—24no—
76Q8×Dic3—24no—
77Q8×Dic3—24no—
78Q8×Dic3—24no—
79C2×#216—24no—
80C2×#216—24no—
81C2×#216—24no—
82#926—216no—
83#926—22no—
84#926—22no—
85#926—22no—
86#926D414yes—
87C2×C4⋊Dic3—22no—
88C2×C4⋊Dic3—22no—
89C2×C4⋊Dic3—22no—
90C2×C4⋊Dic3D411yescharacteristic
91C2×C4⋊Dic3D411yescharacteristic
92C2×C4⋊Dic3C2311yescharacteristic
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95#924—22no—
96#924—22no—
97C2×Dic3⋊C4—21no—
98C2×C4×Dic3—21no—
99C2×C4×Dic3—22no—
100C2×C4×Dic3—22no—
101C2×C4×Dic3C2312yes—
102C22×C3⋊C8C2312yes—
103C3×#103—24no—
104C3×#103—24no—
105C3×C22⋊C8—22no—
106C3×#95—21no—
107#835—24no—
108C3⋊#95—22no—
109C3⋊#95—22no—
110C3⋊#95—22no—
111#830—22no—
112#818—22no—
113#817—22no—
114#816—22no—
115#816—22no—
116C23×Q8C2×D311yescharacteristic
117C22×Q16—32no—
118C2×C22⋊Q8—31no—
119#519—64no—
120#519—64no—
121C2×C4×Q8—32no—
122C2×C4×Q8—32no—
123C2×C4×Q8—61no—
124C2×C4×Q8—61no—
125C2×#117—31no—
126C2×#117—31no—
127C2×#117—62no—
128#485—64no—
129#451—64no—
130C2×#107—31no—
131C2×#106—31no—
132C2×#103—32no—
133C2×#103—31no—
134C2×#103—31no—
135C2×#103C2×D312yes—
136C2×#103C2×D312yes—
137C2×C22⋊C8—31no—
138C2×C22⋊C8C2×D311yescharacteristic
139#400—61no—
140#398—61no—
141#397—62no—
142#391—32no—
143#389—32no—
144#380C2×D311yescharacteristic
145#340—32no—
146#340—32no—
147#336—32no—
148#336—32no—
149C24×C3C2×D411yescharacteristic
150C2×C3×Q8—28no—
151C2×C3×Q8—28no—
152C2×C3×Q8—28no—
153C2×C3×Q8—216no—
154C2×C3×Q8—24no—
155C2×C3×Q8—22no—
156C2×C3×Q8—22no—
157C2×C3×Q8—22no—
158C2×C3×Q8C2×D414yes—
159C2×C3×Q8#4114yes—
160C2×C3×Q8#4114yes—
161C22×C3×C4—28no—
162C22×C3×C4—24no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2411yescharacteristic
171C22×Dic3—21no—
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3—21no—
175C22×Dic3C2×D412yes—
176C22×Dic3C2×D412yes—
177C2×Dic6—21no—
178C2×Dic6—21no—
179C2×Dic6—22no—
180C2×Dic6—22no—
181C2×C3×C8—21no—
182C2×C3×C8—21no—
183C2×C3×C8—22no—
184C2×C3×C8—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—22no—
187C3×C22⋊C4—81no—
188#217—24no—
189#217—24no—
190#217—42no—
191#216—48no—
192C4⋊Dic3—28no—
193C4⋊Dic3—28no—
194C4⋊Dic3—22no—
195C4⋊Dic3—22no—
196C4⋊Dic3—21no—
197C4⋊Dic3—21no—
198C4⋊Dic3—21no—
199C4⋊Dic3—21no—
200C4⋊Dic3—82no—
201C4⋊Dic3C2×D412yes—
202C4⋊Dic3C2×D412yes—
203Dic3⋊C4—42no—
204C4×Dic3—28no—
205C4×Dic3—28no—
206C4×Dic3—24no—
207C4×Dic3—22no—
208C4×Dic3—22no—
209C2×C3⋊C8—22no—
210C2×C3⋊C8—22no—
211C2×C3⋊C8—22no—
212C22×Q8—24no—
213C22×Q8—24no—
214C22×Q8—21no—
215C22×Q8—22no—
216C22×Q8—31no—
217C22×Q8D1214yes—
218C22×Q8C22×D312yes—
219C23×C4—21no—
220C23×C4C22×D311yescharacteristic
221C2×Q16—64no—
222C2×Q16—64no—
223C2×Q16—64no—
224C22×C8—32no—
225C22×C8C22×D311yescharacteristic
226C22×C8C22×D311yescharacteristic
227C22⋊Q8—62no—
228C22⋊Q8—62no—
229C4×Q8—64no—
230C4×Q8—64no—
231C4×Q8—68no—
232C4×Q8—68no—
233C4×Q8—64no—
234C4×Q8—64no—
235#117—34no—
236#117—616no—
237#117—62no—
238#117—62no—
239#117—62no—
240C2×C4⋊C4—31no—
241C2×C4⋊C4—31no—
242C2×C4⋊C4—31no—
243C2×C4⋊C4—31no—
244C2×C4⋊C4—31no—
245C2×C4⋊C4—62no—
246C2×C4⋊C4—62no—
247C2×C4⋊C4—61no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4C22×D312yes—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—32no—
252C2×C22⋊C4—61no—
253C2×C22⋊C4—61no—
254C2×C42—32no—
255C2×C42—62no—
256C2×C42—62no—
257C2×C42—61no—
258#107—62no—
259#106—62no—
260#103—24no—
261#103—24no—
262#103—62no—
263#103—62no—
264#103—64no—
265C22⋊C8—22no—
266C22⋊C8—62no—
267#95—21no—
268#95—62no—
269#95—62no—
270#95—62no—
271C23×C3—22no—
272C23×C3—21no—
273C23×C3—21no—
274C23×C3—22no—
275C23×C3#12012yes—
276C23×C3C22×D411yescharacteristic
277C3×Q8—216no—
278C3×Q8—24no—
279C3×Q8—24no—
280C3×Q8—24no—
281C3×Q8#12318yes—
282C2×C3×C4—216no—
283C2×C3×C4—24no—
284C2×C3×C4—24no—
285C2×C3×C4—24no—
286C2×C3×C4—22no—
287C2×C3×C4—24no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4#12012yes—
294C2×C3×C4C22×D411yescharacteristic
295C2×C3×C4C22×D411yescharacteristic
296C2×C3×C4C2×#4112yes—
297C2×C3×C4C2×#4112yes—
298C2×Dic3—22no—
299C2×Dic3—22no—
300C2×Dic3—22no—
301C2×Dic3—22no—
302C2×Dic3—22no—
303C2×Dic3—22no—
304C2×Dic3—24no—
305C2×Dic3—41no—
306C2×Dic3—41no—
307C2×Dic3—42no—
308C2×Dic3—41no—
309C2×Dic3—41no—
310C2×Dic3—42no—
311C2×Dic3—41no—
312C2×Dic3—41no—
313Dic6—42no—
314Dic6—41no—
315Dic6—41no—
316C3×C8—41no—
317C3×C8—41no—
318C3⋊C8—42no—
319C24D3×D411yescharacteristic
320C2×Q8—28no—
321C2×Q8—28no—
322C2×Q8—28no—
323C2×Q8—216no—
324C2×Q8—24no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—61no—
329C2×Q8—61no—
330C2×Q8—62no—
331C2×Q8—62no—
332C2×Q8C2×D1214yes—
333C2×Q8#23714yes—
334C2×Q8#23714yes—
335C22×C4—28no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—32no—
342C22×C4—32no—
343C22×C4—61no—
344C22×C4—61no—
345C22×C4—61no—
346C22×C4—61no—
347C22×C4C2×D1212yes—
348C22×C4D3×D411yescharacteristic
349C22×C4D3×D412yes—
350C22×C4C23×D311yescharacteristic
351Q16—128no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—62no—
357C2×C8—62no—
358C2×C8—62no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—32no—
362C4⋊C4—32no—
363C4⋊C4—68no—
364C4⋊C4—68no—
365C4⋊C4—62no—
366C4⋊C4—62no—
367C4⋊C4—61no—
368C4⋊C4—61no—
369C4⋊C4—61no—
370C4⋊C4—61no—
371C4⋊C4—122no—
372C4⋊C4—242no—
373C22⋊C4—64no—
374C22⋊C4—64no—
375C22⋊C4—81no—
376C22⋊C4—122no—
377C42—68no—
378C42—68no—
379C42—64no—
380C42—62no—
381C42—62no—
382C22×C3—22no—
383C22×C3—22no—
384C22×C3—22no—
385C22×C3—42no—
386C22×C3—42no—
387C22×C3C2×#12011yescharacteristic
388C22×C3#52512yes—
389C22×C3#57711yescharacteristic
390C22×C3#57811yescharacteristic
391C22×C3#57812yes—
392C3×C4—28no—
393C3×C4—22no—
394C3×C4—82no—
395C3×C4C2×#12312yes—
396C3×C4#52511yescharacteristic
397C3×C4#52511yescharacteristic
398Dic3—42no—
399Dic3—42no—
400Dic3—42no—
401Dic3—42no—
402Dic3—81no—
403Dic3—81no—
404C23—21no—
405C23—21no—
406C23—22no—
407C23—22no—
408C23#88212yes—
409C23C2×D3×D411yescharacteristic
410Q8—216no—
411Q8—24no—
412Q8—24no—
413Q8—24no—
414Q8—122no—
415Q8—121no—
416Q8—121no—
417Q8#88618yes—
418C2×C4—216no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—62no—
430C2×C4—62no—
431C2×C4—62no—
432C2×C4—62no—
433C2×C4—62no—
434C2×C4—62no—
435C2×C4—64no—
436C2×C4—121no—
437C2×C4—121no—
438C2×C4—122no—
439C2×C4—121no—
440C2×C4—121no—
441C2×C4—121no—
442C2×C4—121no—
443C2×C4—122no—
444C2×C4#88212yes—
445C2×C4C22×D1211yescharacteristic
446C2×C4C2×D3×D411yescharacteristic
447C2×C4C2×#23712yes—
448C2×C4C2×#23712yes—
449C8—41no—
450C8—41no—
451C8—122no—
452C2×C3—42no—
453C2×C3#284911yescharacteristic
454C2×C3#301111yescharacteristic
455C2×C3#301311yescharacteristic
456C2×C3#312112yes—
457C2×C3#312712yes—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—42no—
462C22—42no—
463C22C2×#88211yescharacteristic
464C22#558612yes—
465C22#576211yescharacteristic
466C22#576611yescharacteristic
467C22#577912yes—
468C4—28no—
469C4—22no—
470C4—82no—
471C4—122no—
472C4—122no—
473C4—122no—
474C4—122no—
475C4—241no—
476C4—241no—
477C4C2×#88612yes—
478C4#558611yescharacteristic
479C4#558611yescharacteristic
480C3#3301911yescharacteristic, complemented, Sylow
481C2—42no—
482C2#7384711yescharacteristic
483C2#7879511yescharacteristic
484C2#7896111yescharacteristic
485C2#8003212yes—
486C2#8017712yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1787C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1787D3
hypocenterC3#33019
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
6. #70239C2semidirect
7. #70239C2semidirect
8. #70238C2semidirect
9. #70236C2semidirect
480. C3#33019semidirect

Maximal subgroups

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

subgroupindex
C2×#58122
C2×#52282
C2×#47232
#739512
C3×#17872
#702392
#702382
#702362
#699112
#695032
#330193