Finite Group Databasebeta

← #10694890

Subgroups — #10694890

Order 768.

522 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#465C211yesmaximal, characteristic
2#71977C211yesmaximal, characteristic, complemented
3C8×#217C211yesmaximal, characteristic, complemented
4#71691C211yesmaximal, characteristic, complemented
5#71688C211yesmaximal, characteristic, complemented
6#71676C211yesmaximal, characteristic
7#71673C211yesmaximal, characteristic
8#9813—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
20C4×#217C2211yescharacteristic
21#5207C2211yescharacteristic
22#5207C2211yescharacteristic
23C2×#830C2211yescharacteristic
24C2×#819—21no—
25C2×C8×Dic3—21no—
26#4552—21no—
27#4550—21no—
28#4538—22no—
29#4537—22no—
30#4528—22no—
31#4525—22no—
32C2×#465D311yescharacteristic
33#1810—31no—
34C8×C22⋊C4—31no—
35#1508—31no—
36#1507—31no—
37#1472—31no—
38#1471—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×C4×Dic3—21no—
68C2×#208—21no—
69C2×#208—21no—
70C22×C3⋊C8—21no—
71C3×#106—21no—
72C3×#106—21no—
73C3×#103—21no—
74C3×#103—21no—
75C3×#102—21no—
76C3×#102—21no—
77C3⋊#95D412yes—
78#830—21no—
79#830—21no—
80#830—21no—
81#830—21no—
82#830—21no—
83#830D412yes—
84#819—42no—
85C8×Dic3—42no—
86C22×SD16Dic311yescharacteristic, complemented
87C23×C8C2×D311yescharacteristic
88C2×C22⋊Q8C2×D311yescharacteristic
89C2×#121C2×D311yescharacteristic
90#465—21no—
91#465—21no—
92#465—21no—
93#465—21no—
94C2×#108—61no—
95C2×#106Dic311yescharacteristic, complemented
96C2×#103Dic311yescharacteristic, complemented
97C2×#102Dic311yescharacteristic, complemented
98#407—31no—
99#407—31no—
100C2×C22⋊C8—31no—
101C2×C4×C8—61no—
102C4×C22⋊C4—31no—
103#341—61no—
104#336—61no—
105#328—62no—
106#327—62no—
107#324—62no—
108#323—62no—
109C24×C3—21no—
110C24×C3#4111yescharacteristic
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—41no—
114C2×C3×Q8—41no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—82no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—41no—
126C22×C3×C4—41no—
127C22×C3×C4C22×C411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4#4111yescharacteristic
133C22×Dic3—22no—
134C3×SD16—42no—
135C3×SD16—42no—
136C2×C3×C8—22no—
137C2×C3×C8—22no—
138C2×C3×C8—22no—
139C2×C3×C8—41no—
140C2×C3×C8—41no—
141C2×C3×C8—41no—
142C2×C3×C8C22⋊C411yescharacteristic
143C2×C3×C8C22⋊C411yescharacteristic
144C2×C3×C8C22⋊C411yescharacteristic
145C2×C3×C8C22⋊C411yescharacteristic
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—41no—
151C3×C4⋊C4—41no—
152C3×C22⋊C4—41no—
153C3×C22⋊C4—41no—
154C3×C22⋊C4—41no—
155C3×C22⋊C4—41no—
156#217—24no—
157C4×Dic3—41no—
158C4×Dic3—41no—
159#208—41no—
160#208—41no—
161#208—41no—
162#208—41no—
163C2×C3⋊C8—22no—
164C2×C3⋊C8—41no—
165C2×C3⋊C8—41no—
166C2×C3⋊C8—41no—
167C2×C3⋊C8—41no—
168C22×Q8C2×Dic311yescharacteristic
169C22×D4—21no—
170C22×D4C2×Dic311yescharacteristic
171C23×C4C22×D311yescharacteristic
172C2×SD16—21no—
173C2×SD16—21no—
174C2×SD16—42no—
175C2×M16—61no—
176C2×M16—61no—
177C22×C8—42no—
178C22×C8—61no—
179C22×C8C2×Dic311yescharacteristic
180C22×C8C2×Dic311yescharacteristic
181C22×C8#6711yescharacteristic
182C22×C8#6711yescharacteristic
183C22×C8#6711yescharacteristic
184C22×C8#6711yescharacteristic
185C22⋊Q8—21no—
186C22⋊Q8—21no—
187C22⋊Q8—21no—
188C22⋊Q8—21no—
189#121—21no—
190#121—21no—
191#121—21no—
192#121—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4C2×Dic311yescharacteristic
195C2×C4⋊C4C2×Dic311yescharacteristic
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—61no—
199C2×C42—61no—
200#108—122no—
201#106—21no—
202#106—21no—
203#103—21no—
204#103—21no—
205#102—21no—
206#102—21no—
207C22⋊C8—32no—
208C22⋊C8—61no—
209C22⋊C8—61no—
210C22⋊C8—61no—
211C22⋊C8—61no—
212C22⋊C8—61no—
213C4×C8—122no—
214#95—32no—
215C23×C3—22no—
216C23×C3—22no—
217C23×C3—41no—
218C23×C3—41no—
219C23×C3—42no—
220C23×C3—81no—
221C23×C3—81no—
222C23×C3C4×D411yescharacteristic
223C23×C3#12111yescharacteristic
224C23×C3#12411yescharacteristic
225C3×Q8—41no—
226C3×Q8—41no—
227C3×Q8—81no—
228C3×D4—41no—
229C3×D4—41no—
230C3×D4—81no—
231C3×D4—82no—
232C3×D4—82no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—42no—
246C2×C3×C4—81no—
247C2×C3×C4—81no—
248C2×C3×C4C2×C22⋊C411yescharacteristic
249C2×C3×C4C4×D411yescharacteristic
250C2×C3×C4#12111yescharacteristic
251C2×C3×C4#12711yescharacteristic
252C2×Dic3—24no—
253C2×Dic3—42no—
254C2×Dic3—42no—
255C3×C8—21no—
256C3×C8—21no—
257C3×C8—21no—
258C3×C8#9912yes—
259C3⋊C8—42no—
260C3⋊C8—81no—
261C3⋊C8—81no—
262C24—21no—
263C24#23711yescharacteristic
264C2×Q8—21no—
265C2×Q8—21no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—82no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—62no—
281C22×C4D3×D411yescharacteristic
282C22×C4D3×D411yescharacteristic
283C22×C4#23711yescharacteristic
284C22×C4C22×Dic311yescharacteristic
285C22×C4C2×#6711yescharacteristic
286C22×C4C2×#6711yescharacteristic
287SD16—42no—
288SD16—42no—
289M16—121no—
290M16—121no—
291M16—121no—
292M16—121no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—62no—
300C2×C8—121no—
301C2×C8—121no—
302C2×C8—121no—
303C2×C8—121no—
304C2×C8#21711yescharacteristic
305C2×C8#21711yescharacteristic
306C2×C8#21711yescharacteristic
307C2×C8#21711yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—64no—
319C42—121no—
320C42—121no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—44no—
329C22×C3—81no—
330C22×C3—81no—
331C22×C3—81no—
332C22×C3—162no—
333C22×C3#39011yescharacteristic
334C22×C3#44312yes—
335C22×C3#47112yes—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—81no—
340C3×C4—81no—
341C3×C4—81no—
342C3×C4C2×#9911yescharacteristic
343C3×C4#41011yescharacteristic
344Dic3—44no—
345C23—22no—
346C23—22no—
347C23—41no—
348C23—41no—
349C23—42no—
350C23—81no—
351C23—81no—
352C23D4×Dic311yescharacteristic
353C23#93611yescharacteristic
354C23#93811yescharacteristic
355Q8—41no—
356Q8—41no—
357Q8—81no—
358D4—41no—
359D4—41no—
360D4—81no—
361D4—82no—
362D4—82no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—64no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—122no—
380C2×C4—122no—
381C2×C4D4×Dic311yescharacteristic
382C2×C4#93811yescharacteristic
383C2×C4#94011yescharacteristic
384C2×C4C2×#21711yescharacteristic
385C8—21no—
386C8—21no—
387C8—21no—
388C8—122no—
389C8—241no—
390C8—241no—
391C8#83412yes—
392C2×C3—22no—
393C2×C3—22no—
394C2×C3—41no—
395C2×C3—161no—
396C2×C3#196411yescharacteristic
397C2×C3#197411yescharacteristic
398C2×C3#197711yescharacteristic
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—44no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—162no—
411C22#517712yes—
412C22#519612yes—
413C22#522111yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—81no—
418C4—81no—
419C4—81no—
420C4—124no—
421C4C2×#83411yescharacteristic
422C4#524011yescharacteristic
423C3#981311yescharacteristic, complemented, Sylow
424C2—22no—
425C2—22no—
426C2—41no—
427C2—161no—
428C2#7168611yescharacteristic
429C2#7205211yescharacteristic
430C2#7208411yescharacteristic
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#9813
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. #71977C2semidirect
3. C8×#217C2semidirect
4. #71691C2semidirect
5. #71688C2semidirect
9. C22×C3×SD16C4semidirect
17. C2×C3×#106C4semidirect
18. C2×C3×#103C4semidirect
19. C2×C3×#102C4semidirect
86. C22×SD16Dic3semidirect
95. C2×#106Dic3semidirect
96. C2×#103Dic3semidirect
97. C2×#102Dic3semidirect
423. C3#9813semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4652
#719772
C8×#2172
#716912
#716882
#716762
#716732
#98133