Finite Group Databasebeta

← #10694698

Subgroups — #10694698

Order 768.

564 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
1C2×C3×#465C211yesmaximal, characteristic
2#71970C211yesmaximal, characteristic, complemented
3C8×#217C211yesmaximal, characteristic, complemented
4#71701C211yesmaximal, characteristic, complemented
5#71690C211yesmaximal, characteristic, complemented
6#71686C211yesmaximal, characteristic
7#71675C211yesmaximal, characteristic
8#9781—31nomaximal, Sylow
9C22×C3×SD16C411yescharacteristic, complemented
10C23×C3×C8C2211yescharacteristic
11C2×C3×C22⋊Q8C2211yescharacteristic
12C2×C3×#121C2211yescharacteristic
13C3×#465—21no—
14C3×#465—21no—
15C3×#465—21no—
16C3×#465—21no—
17C2×C3×#106C411yescharacteristic, complemented
18C2×C3×#103C411yescharacteristic, complemented
19C2×C3×#102C411yescharacteristic, complemented
20C2×#834—22no—
21#5212C2211yescharacteristic
22#5212C2211yescharacteristic
23C4×#217C2211yescharacteristic
24C2×#830C2211yescharacteristic
25C2×#817—21no—
26C2×C8×Dic3—21no—
27#4551—21no—
28#4550—21no—
29#4538—22no—
30#4537—22no—
31#4526—22no—
32C2×#465D311yescharacteristic
33#1996—31no—
34#1964—31no—
35#1807—31no—
36C8×C22⋊C4—31no—
37#1520—31no—
38#1519—31no—
39C22×C3×Q8C2×C411yescharacteristic
40C22×C3×D4—21no—
41C22×C3×D4C2×C411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C2×C3×SD16—21no—
44C2×C3×SD16—21no—
45C2×C3×SD16—42no—
46C22×C3×C8—42no—
47C22×C3×C8C2×C411yescharacteristic
48C22×C3×C8C2×C411yescharacteristic
49C22×C3×C8D411yescharacteristic
50C22×C3×C8D411yescharacteristic
51C22×C3×C8D411yescharacteristic
52C22×C3×C8D411yescharacteristic
53C3×C22⋊Q8—21no—
54C3×C22⋊Q8—21no—
55C3×C22⋊Q8—21no—
56C3×C22⋊Q8—21no—
57C3×#121—21no—
58C3×#121—21no—
59C3×#121—21no—
60C3×#121—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4C2×C411yescharacteristic
63C2×C3×C4⋊C4C2×C411yescharacteristic
64C2×C3×C22⋊C4—21no—
65C2×C3×C22⋊C4—21no—
66C2×#217—21no—
67C2×#217—21no—
68C2×#217D412yes—
69C2×C4⋊Dic3—21no—
70C2×C4⋊Dic3—21no—
71C2×C4×Dic3—21no—
72C22×C3⋊C8—21no—
73C3×#106—21no—
74C3×#106—21no—
75C3×#103—21no—
76C3×#103—21no—
77C3×#102—21no—
78C3×#102—21no—
79#834—44no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82C3⋊#95—21no—
83#830—21no—
84#830D412yes—
85#817—42no—
86C8×Dic3—42no—
87C22×SD16Dic311yescharacteristic, complemented
88C23×C8C2×D311yescharacteristic
89C2×C22⋊Q8C2×D311yescharacteristic
90C2×#121C2×D311yescharacteristic
91#465—21no—
92#465—21no—
93#465—21no—
94#465—21no—
95C2×#106—61no—
96C2×#106Dic311yescharacteristic, complemented
97C2×#103Dic311yescharacteristic, complemented
98C2×#102Dic311yescharacteristic, complemented
99C2×#99—62no—
100C2×C22⋊C8—31no—
101C2×C4×C8—61no—
102#380—31no—
103#380—31no—
104C4×C22⋊C4—31no—
105#342—62no—
106#340—61no—
107#336—61no—
108#328—62no—
109#327—62no—
110C24×C3—21no—
111C24×C3C2×D411yescharacteristic
112C2×C3×Q8—21no—
113C2×C3×Q8—21no—
114C2×C3×Q8—41no—
115C2×C3×Q8—41no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4—42no—
123C2×C3×D4—42no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—41no—
128C22×C3×C4—41no—
129C22×C3×C4C22×C411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4#4111yescharacteristic
134C22×C3×C4#4111yescharacteristic
135C22×Dic3—22no—
136C22×Dic3—41no—
137C22×Dic3—41no—
138C3×SD16—42no—
139C3×SD16—42no—
140C2×C3×C8—22no—
141C2×C3×C8—22no—
142C2×C3×C8—22no—
143C2×C3×C8—41no—
144C2×C3×C8—41no—
145C2×C3×C8—41no—
146C2×C3×C8C22⋊C411yescharacteristic
147C2×C3×C8C22⋊C411yescharacteristic
148C2×C3×C8C22⋊C411yescharacteristic
149C2×C3×C8C22⋊C411yescharacteristic
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—41no—
155C3×C4⋊C4—41no—
156C3×C22⋊C4—41no—
157C3×C22⋊C4—41no—
158C3×C22⋊C4—41no—
159C3×C22⋊C4—41no—
160#217—22no—
161#217—42no—
162#217—42no—
163#217SD1614yescomplemented
164C4⋊Dic3—41no—
165C4⋊Dic3—41no—
166C4⋊Dic3—41no—
167C4⋊Dic3—41no—
168C4×Dic3—41no—
169C4×Dic3—41no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8—41no—
172C2×C3⋊C8—41no—
173C22×Q8C2×Dic311yescharacteristic
174C22×D4—21no—
175C22×D4C2×Dic311yescharacteristic
176C23×C4C22×D311yescharacteristic
177C2×SD16—21no—
178C2×SD16—21no—
179C2×SD16—42no—
180C22×C8—42no—
181C22×C8—61no—
182C22×C8C2×Dic311yescharacteristic
183C22×C8C2×Dic311yescharacteristic
184C22×C8#6711yescharacteristic
185C22×C8#6711yescharacteristic
186C22×C8#6711yescharacteristic
187C22×C8#6711yescharacteristic
188C22⋊Q8—21no—
189C22⋊Q8—21no—
190C22⋊Q8—21no—
191C22⋊Q8—21no—
192#121—21no—
193#121—21no—
194#121—21no—
195#121—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—61no—
198C2×C4⋊C4—61no—
199C2×C4⋊C4C2×Dic311yescharacteristic
200C2×C4⋊C4C2×Dic311yescharacteristic
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—32no—
204C2×C22⋊C4—61no—
205C2×C22⋊C4—61no—
206C2×C42—61no—
207#106—21no—
208#106—21no—
209#106—122no—
210#103—21no—
211#103—21no—
212#102—21no—
213#102—21no—
214#99—124no—
215C22⋊C8—32no—
216C22⋊C8—61no—
217C4×C8—122no—
218#95—61no—
219#95—61no—
220#95—61no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3—42no—
225C23×C3—81no—
226C23×C3—81no—
227C23×C3C4×D411yescharacteristic
228C23×C3#12111yescharacteristic
229C23×C3#12711yescharacteristic
230C23×C3C2×SD1612yes—
231C23×C3#13512yes—
232C3×Q8—41no—
233C3×Q8—41no—
234C3×Q8—81no—
235C3×D4—41no—
236C3×D4—41no—
237C3×D4—81no—
238C3×D4—82no—
239C3×D4—81no—
240C3×D4—81no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—42no—
254C2×C3×C4—81no—
255C2×C3×C4—81no—
256C2×C3×C4C2×C22⋊C411yescharacteristic
257C2×C3×C4C4×D411yescharacteristic
258C2×C3×C4#12111yescharacteristic
259C2×C3×C4#12411yescharacteristic
260C2×Dic3—24no—
261C2×Dic3—42no—
262C2×Dic3—42no—
263C2×Dic3—81no—
264C2×Dic3—81no—
265C2×Dic3—81no—
266C2×Dic3—81no—
267C2×Dic3—81no—
268C2×Dic3—81no—
269C3×C8—21no—
270C3×C8—21no—
271C3×C8—21no—
272C3×C8#9912yescomplemented
273C3⋊C8—42no—
274C24—21no—
275C24D3×D411yescharacteristic
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—41no—
287C2×D4—41no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—62no—
294C22×C4—121no—
295C22×C4—121no—
296C22×C4D3×D411yescharacteristic
297C22×C4#23711yescharacteristic
298C22×C4#23711yescharacteristic
299C22×C4C22×Dic311yescharacteristic
300C22×C4C2×#6711yescharacteristic
301C22×C4C2×#6711yescharacteristic
302SD16—42no—
303SD16—42no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—62no—
311C2×C8—121no—
312C2×C8—121no—
313C2×C8#21711yescharacteristic
314C2×C8#21711yescharacteristic
315C2×C8#21711yescharacteristic
316C2×C8#21711yescharacteristic
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—121no—
324C4⋊C4—121no—
325C4⋊C4—121no—
326C4⋊C4—121no—
327C22⋊C4—34no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—62no—
333C22⋊C4—122no—
334C22⋊C4—122no—
335C42—121no—
336C42—121no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—42no—
344C22×C3—42no—
345C22×C3—81no—
346C22×C3—81no—
347C22×C3—81no—
348C22×C3—162no—
349C22×C3#39011yescharacteristic
350C22×C3C4×SD1612yes—
351C22×C3#46512yes—
352C22×C3#49211yescharacteristic
353C22×C3#49511yescharacteristic
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—81no—
358C3×C4—81no—
359C3×C4—81no—
360C3×C4C2×#9911yescharacteristic
361C3×C4#41011yescharacteristic
362Dic3—44no—
363Dic3—161no—
364Dic3—161no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—42no—
369C23—81no—
370C23—81no—
371C23D3×SD1612yes—
372C23#91612yes—
373C23D4×Dic311yescharacteristic
374C23#93811yescharacteristic
375C23#94011yescharacteristic
376Q8—41no—
377Q8—41no—
378Q8—81no—
379D4—41no—
380D4—41no—
381D4—81no—
382D4—81no—
383D4—82no—
384D4—81no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—64no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—122no—
402C2×C4—122no—
403C2×C4—241no—
404C2×C4—241no—
405C2×C4—241no—
406C2×C4—241no—
407C2×C4—241no—
408C2×C4—241no—
409C2×C4D4×Dic311yescharacteristic
410C2×C4#93611yescharacteristic
411C2×C4#93811yescharacteristic
412C2×C4C2×#21711yescharacteristic
413C8—21no—
414C8—21no—
415C8—21no—
416C8—122no—
417C8#83412yescomplemented
418C2×C3—22no—
419C2×C3—21no—
420C2×C3—21no—
421C2×C3—22no—
422C2×C3—161no—
423C2×C3#196411yescharacteristic
424C2×C3#196811yescharacteristic
425C2×C3#197611yescharacteristic
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—42no—
433C22—42no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22—162no—
438C22Dic3×SD1612yes—
439C22#516811yescharacteristic
440C22#517112yes—
441C22#517511yescharacteristic
442C22#522111yescharacteristic
443C4—21no—
444C4—21no—
445C4—21no—
446C4—81no—
447C4—81no—
448C4—81no—
449C4—124no—
450C4—481no—
451C4—481no—
452C4C2×#83411yescharacteristic
453C4#524011yescharacteristic
454C3#978111yescharacteristic, complemented, Sylow
455C2—22no—
456C2—21no—
457C2—21no—
458C2—22no—
459C2—161no—
460C2#7168611yescharacteristic
461C2#7204911yescharacteristic
462C2#7208611yescharacteristic
subgroupG/H
centerC22#5221
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#465C2
socleC22×C3#390
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenterC2×#465D3
hypocenterC3#9781
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#465

Complements

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

NH ≅ G/Ntype
2. #71970C2semidirect
3. C8×#217C2semidirect
4. #71701C2semidirect
5. #71690C2semidirect
9. C22×C3×SD16C4semidirect
17. C2×C3×#106C4semidirect
18. C2×C3×#103C4semidirect
19. C2×C3×#102C4semidirect
87. C22×SD16Dic3semidirect
96. C2×#106Dic3semidirect
97. C2×#103Dic3semidirect
98. C2×#102Dic3semidirect
163. #217SD16semidirect
272. C3×C8#99semidirect
417. C8#834semidirect
454. C3#9781semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4652
#719702
C8×#2172
#717012
#716902
#716862
#716752
#97813