Finite Group Databasebeta

← #10897679

Subgroups — #10897679

Order 768.

774 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Dic3×C4⋊Q8C211yesmaximal, characteristic, complemented
2#76289C211yesmaximal, characteristic
3C3×#2023C211yesmaximal, characteristic
4#71782C211yesmaximal, characteristic, complemented
5#70173C212yesmaximal
6#70168C212yesmaximal
7#69640C211yesmaximal, characteristic
8#68815C212yesmaximal, complemented
9#68814C212yesmaximal, complemented
10#68639C211yesmaximal, characteristic, complemented
11#68607C211yesmaximal, characteristic, complemented
12#29388—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×Q8×Dic3—21no—
15C2×D4×Dic3C2211yescharacteristic
16C2×#877—21no—
17C2×C3×#102C2212yes—
18C3×#390C2211yescharacteristic
19#5229—21no—
20#5221C2211yescharacteristic
21C2×#832C2212yes—
22C2×#818C2212yes—
23#4984—21no—
24#4984C2212yes—
25Dic3×C4⋊C4—21no—
26Dic3×C4⋊C4C2211yescharacteristic
27Dic3×C4⋊C4C2212yes—
28#4972C2211yescharacteristic
29C2×#807C2212yes—
30#4945—21no—
31#4944—21no—
32Dic3×C22⋊C4—21no—
33#4935C2211yescharacteristic
34C42×Dic3C2211yescharacteristic
35#4928C2211yescharacteristic
36C3×#340C2212yes—
37C3×#336C2211yescharacteristic
38#4551C2212yes—
39#4550C2212yes—
40#4550C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43#4524C2211yescharacteristic
44C4×C4⋊Q8—31no—
45#2293—31no—
46#2083—31no—
47#2071—31no—
48#2057—31no—
49#2023D311yescharacteristic
50#1925—32no—
51#1918—32no—
52#1918—32no—
53#1862—32no—
54#1839—31no—
55C22×C3×Q8—21no—
56C22×C3×D4C2311yescharacteristic
57C23×Dic3—21no—
58C22×C3×C8C2312yes—
59C3×C4⋊Q8—24no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4C2311yescharacteristic
62C2×C3×C4⋊C4C2312yes—
63C2×C3×C22⋊C4—21no—
64C2×C3×C22⋊C4—21no—
65C2×C3×C42C2311yescharacteristic
66C2×#217—21no—
67C2×#217—21no—
68C2×#217—21no—
69Q8×Dic3—44no—
70D4×Dic3—22no—
71D4×Dic3—22no—
72C2×C4⋊Dic3—21no—
73C2×C4⋊Dic3—21no—
74C2×C4⋊Dic3—22no—
75C2×C4⋊Dic3C2311yescharacteristic
76C2×C4⋊Dic3C2312yes—
77C2×C4⋊Dic3C2311yescharacteristic
78C2×Dic3⋊C4—21no—
79C2×Dic3⋊C4—21no—
80C2×C4×Dic3—21no—
81C2×C4×Dic3—22no—
82C2×C4×Dic3D411yescharacteristic
83C2×C4×Dic3D411yescharacteristic
84C2×C4×Dic3D411yescharacteristic
85C2×C4×Dic3D411yescharacteristic
86C2×C4×Dic3C2311yescharacteristic
87C2×C4×Dic3C2311yescharacteristic
88C22×C3⋊C8C2312yes—
89#877—44no—
90C3×#102—24no—
91#832—24no—
92C3⋊#95—21no—
93C3⋊#95—21no—
94C3⋊#95—21no—
95C3⋊#95—22no—
96C3⋊#95—22no—
97#818—24no—
98#807—24no—
99C2×C4⋊Q8C2×D311yescharacteristic
100C2×C4×Q8—61no—
101C2×C4×D4—31no—
102C2×#117—61no—
103C2×#107—32no—
104C2×#107—32no—
105C2×#102—32no—
106C2×#102C2×D312yes—
107#391—61no—
108#390—31no—
109#390C2×D311yescharacteristic
110#389—32no—
111#389—31no—
112#389—31no—
113#389—61no—
114#388—61no—
115#388—61no—
116#384—31no—
117C4×C4⋊C4—32no—
118C4×C4⋊C4—31no—
119C4×C4⋊C4—61no—
120C4×C22⋊C4—61no—
121C43—31no—
122#340—32no—
123#340—32no—
124#340—32no—
125#340C2×D312yes—
126#336—32no—
127#336—32no—
128#336—31no—
129#336C2×D311yescharacteristic
130C24×C3—21no—
131C2×C3×Q8—24no—
132C2×C3×Q8—42no—
133C2×C3×Q8—42no—
134C2×C3×D4—22no—
135C2×C3×D4—21no—
136C2×C3×D4—22no—
137C2×C3×D4#4112yes—
138C22×C3×C4—22no—
139C22×C3×C4—21no—
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2411yescharacteristic
143C22×Dic3—24no—
144C22×Dic3—21no—
145C22×Dic3—21no—
146C22×Dic3—22no—
147C22×Dic3—42no—
148C22×Dic3—42no—
149C22×Dic3C2×D411yescharacteristic
150C22×Dic3C2×D411yescharacteristic
151C22×Dic3C2×D411yescharacteristic
152C22×Dic3C2×D411yescharacteristic
153C2×C3×C8—22no—
154C2×C3×C8—22no—
155C2×C3×C8—22no—
156C3×C4⋊C4—24no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4#4114yes—
160C3×C22⋊C4—42no—
161C3×C22⋊C4—42no—
162C3×C42—22no—
163#217—24no—
164#217—42no—
165#217—42no—
166C4⋊Dic3—21no—
167C4⋊Dic3—22no—
168C4⋊Dic3—22no—
169C4⋊Dic3—22no—
170C4⋊Dic3—42no—
171C4⋊Dic3—42no—
172C4⋊Dic3—44no—
173C4⋊Dic3#4112yes—
174Dic3⋊C4—42no—
175Dic3⋊C4—42no—
176C4×Dic3—22no—
177C4×Dic3—21no—
178C4×Dic3—21no—
179C4×Dic3—22no—
180C4×Dic3—41no—
181C4×Dic3—41no—
182C4×Dic3—42no—
183C4×Dic3—44no—
184C4×Dic3D814yes—
185C2×C3⋊C8—22no—
186C2×C3⋊C8—22no—
187C2×C3⋊C8—22no—
188C22×Q8—21no—
189C22×D4C22×D311yescharacteristic
190C23×C4—61no—
191C22×C8—32no—
192C22×C8C22×D312yes—
193C4⋊Q8—24no—
194C4×Q8—124no—
195C4×D4—62no—
196C4×D4—62no—
197#117—124no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—31no—
200C2×C4⋊C4—32no—
201C2×C4⋊C4—31no—
202C2×C4⋊C4—61no—
203C2×C4⋊C4—62no—
204C2×C4⋊C4—61no—
205C2×C4⋊C4—61no—
206C2×C4⋊C4—61no—
207C2×C4⋊C4C22×D311yescharacteristic
208C2×C4⋊C4C22×D312yes—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—61no—
212C2×C22⋊C4—61no—
213C2×C22⋊C4—61no—
214C2×C42—31no—
215C2×C42—31no—
216C2×C42—31no—
217C2×C42—31no—
218C2×C42—31no—
219C2×C42—31no—
220C2×C42—61no—
221C2×C42—62no—
222C2×C42C22×D311yescharacteristic
223#107—64no—
224#107—64no—
225#102—24no—
226#102—64no—
227#95—61no—
228#95—61no—
229#95—62no—
230#95—62no—
231#95—61no—
232C23×C3—22no—
233C23×C3—41no—
234C23×C3—41no—
235C23×C3—41no—
236C23×C3—41no—
237C23×C3—41no—
238C23×C3—41no—
239C23×C3C22×D411yescharacteristic
240C3×Q8—48no—
241C3×D4—24no—
242C3×D4—42no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—24no—
248C2×C3×C4—42no—
249C2×C3×C4—42no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4C2×D812yes—
254C2×C3×C4#13712yes—
255C2×C3×C4C2×#4112yes—
256C2×C3×C4C2×#4111yescharacteristic
257C2×C3×C4C2×#4111yescharacteristic
258C2×C3×C42−1+412yes—
259C2×Dic3—21no—
260C2×Dic3—21no—
261C2×Dic3—21no—
262C2×Dic3—21no—
263C2×Dic3—21no—
264C2×Dic3—22no—
265C2×Dic3—21no—
266C2×Dic3—21no—
267C2×Dic3—21no—
268C2×Dic3—21no—
269C2×Dic3—21no—
270C2×Dic3—48no—
271C2×Dic3—41no—
272C2×Dic3—41no—
273C2×Dic3—41no—
274C2×Dic3—42no—
275C2×Dic3—41no—
276C2×Dic3—42no—
277C2×Dic3—42no—
278C2×Dic3—41no—
279C2×Dic3C2×D812yes—
280C2×Dic3C2×D812yes—
281C3×C8—42no—
282C3⋊C8—42no—
283C24—21no—
284C2×Q8—24no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×D4—22no—
288C2×D4—21no—
289C2×D4—22no—
290C2×D4#23912yes—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—31no—
294C22×C4—31no—
295C22×C4—31no—
296C22×C4—31no—
297C22×C4—64no—
298C22×C4—61no—
299C22×C4—61no—
300C22×C4—62no—
301C22×C4—122no—
302C22×C4—122no—
303C22×C4D3×D411yescharacteristic
304C22×C4D3×D411yescharacteristic
305C22×C4C23×D311yescharacteristic
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—62no—
310C2×C8—62no—
311C2×C8—62no—
312C4⋊C4—24no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—32no—
316C4⋊C4—61no—
317C4⋊C4—62no—
318C4⋊C4—62no—
319C4⋊C4—62no—
320C4⋊C4—122no—
321C4⋊C4—122no—
322C4⋊C4—124no—
323C4⋊C4—122no—
324C4⋊C4—122no—
325C4⋊C4#23714yes—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—64no—
329C22⋊C4—122no—
330C22⋊C4—122no—
331C42—22no—
332C42—34no—
333C42—62no—
334C42—61no—
335C42—61no—
336C42—62no—
337C42—121no—
338C42—121no—
339C42—122no—
340C42—124no—
341C22×C3—42no—
342C22×C3—42no—
343C22×C3—42no—
344C22×C3—81no—
345C22×C3—81no—
346C22×C3—81no—
347C22×C3—81no—
348C22×C3#52511yescharacteristic
349C22×C3#52511yescharacteristic
350C22×C3#53611yescharacteristic
351C22×C3C22×D811yescharacteristic
352C22×C3C2×#13711yescharacteristic
353C22×C3#57511yescharacteristic
354C22×C3#57511yescharacteristic
355C3×C4—22no—
356C3×C4—22no—
357C3×C4—44no—
358C3×C4—81no—
359C3×C4#55212yes—
360C3×C4#55612yes—
361Dic3—22no—
362Dic3—22no—
363Dic3—42no—
364Dic3—41no—
365Dic3—41no—
366Dic3—82no—
367Dic3—81no—
368C23—22no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23C2×D3×D411yescharacteristic
376Q8—48no—
377D4—24no—
378D4—42no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—24no—
384C2×C4—32no—
385C2×C4—32no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—61no—
392C2×C4—61no—
393C2×C4—61no—
394C2×C4—61no—
395C2×C4—61no—
396C2×C4—61no—
397C2×C4—61no—
398C2×C4—61no—
399C2×C4—61no—
400C2×C4—61no—
401C2×C4—62no—
402C2×C4—128no—
403C2×C4—121no—
404C2×C4—121no—
405C2×C4—121no—
406C2×C4—122no—
407C2×C4—121no—
408C2×C4—122no—
409C2×C4—121no—
410C2×C4—122no—
411C2×C4D3×D812yes—
412C2×C4#91512yes—
413C2×C4C2×#23712yes—
414C2×C4C2×#23911yescharacteristic
415C2×C4D3×#4111yescharacteristic
416C2×C4#101012yes—
417C8—42no—
418C8—122no—
419C2×C3—82no—
420C2×C3#259911yescharacteristic
421C2×C3#310011yescharacteristic
422C2×C3#310011yescharacteristic
423C2×C3#311611yescharacteristic
424C2×C3#311711yescharacteristic
425C2×C3#322611yescharacteristic
426C2×C3#324011yescharacteristic
427C22—42no—
428C22—42no—
429C22—42no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22#558611yescharacteristic
435C22#562711yescharacteristic
436C22#567511yescharacteristic
437C22C2×D3×D811yescharacteristic
438C22C2×#91511yescharacteristic
439C22#577311yescharacteristic
440C22#577711yescharacteristic
441C4—22no—
442C4—22no—
443C4—44no—
444C4—62no—
445C4—62no—
446C4—81no—
447C4—121no—
448C4—122no—
449C4—121no—
450C4—242no—
451C4—241no—
452C4#564212yes—
453C4#567112yes—
454C3#2938811yescharacteristic, complemented, Sylow
455C2—82no—
456C2#7680511yescharacteristic
457C2#7998811yescharacteristic
458C2#8003111yescharacteristic
459C2#8094811yescharacteristic
460C2#8104711yescharacteristic
461C2#8157111yescharacteristic
462C2#8169311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2023C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#2023D3
hypocenterC3#29388
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2023

Complements

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

NH ≅ G/Ntype
1. Dic3×C4⋊Q8C2semidirect
4. #71782C2semidirect
8. #68815C2semidirect
9. #68814C2semidirect
10. #68639C2semidirect
11. #68607C2semidirect
454. C3#29388semidirect

Maximal subgroups

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

subgroupindex
Dic3×C4⋊Q82
#762892
C3×#20232
#717822
#701732
#701682
#696402
#688152
#688142
#686392
#686072
#293883