Finite Group Databasebeta

← #10945091

Subgroups — #10945091

Order 768.

1098 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#459C211yesmaximal, characteristic
2#83184C211yesmaximal, characteristic, complemented
3C4×D4×Dic3C211yesmaximal, characteristic, complemented
4C2×#5007C211yesmaximal, characteristic, complemented
5#72037C214yesmaximal, complemented
6#71761C211yesmaximal, characteristic, complemented
7#70460C211yesmaximal, characteristic, complemented
8#70443C211yesmaximal, characteristic, complemented
9#70173C214yesmaximal, complemented
10#33024—31nomaximal, Sylow
11C22×C3×D8C2212yes—
12C2×C3×#127C2211yescharacteristic
13C2×C3×C4×D4C2211yescharacteristic
14C2×#936—21no—
15C2×#936—21no—
16C2×D4×Dic3C2212yes—
17C22×C4×Dic3—21no—
18C2×#869C2211yescharacteristic
19C3×#459—24no—
20C2×C3×C4⋊C8C2211yescharacteristic
21C2×C3×#102C2212yes—
22#5221—22no—
23#5221C2214yes—
24C2×#832C2214yes—
25C2×#832C2212yes—
26C4×#217—21no—
27#5007—24no—
28Dic3×C4⋊C4C2212yes—
29C2×#807C2212yes—
30Dic3×C22⋊C4—22no—
31C4×C4⋊Dic3C2211yescharacteristic
32C42×Dic3C2211yescharacteristic
33C2×C4⋊(C3⋊C8)C2211yescharacteristic
34#4551C2214yes—
35#4550C2214yes—
36C2×#474—31no—
37C2×#459D311yescharacteristic
38#2858—31no—
39C42×D4—31no—
40#1988—31no—
41#1982—31no—
42#1925—34no—
43#1879—34no—
44#1831—31no—
45C22×C3×D4—21no—
46C22×C3×D4—21no—
47C22×C3×D4C2312yes—
48C22×C3×D4C2311yescharacteristic
49C23×C3×C4—21no—
50C23×Dic3—22no—
51C22×Dic6—21no—
52C2×C3×D8—24no—
53C2×C3×D8—28no—
54C22×C3×C8C2312yes—
55C3×#127—22no—
56C3×#127—22no—
57C3×C4×D4—22no—
58C3×C4×D4D414yes—
59C2×C3×C4⋊C4C2311yescharacteristic
60C2×C3×C22⋊C4—21no—
61C2×C3×C42C2311yescharacteristic
62C2×#217—22no—
63C2×#217—24no—
64C2×#217—24no—
65#936—44no—
66#936—44no—
67D4×Dic3—24no—
68D4×Dic3—24no—
69C2×C4⋊Dic3—21no—
70C2×C4⋊Dic3C2312yes—
71C2×C4⋊Dic3C2312yes—
72C2×C4×Dic3—21no—
73C2×C4×Dic3—22no—
74C2×C4×Dic3—44no—
75C2×C4×Dic3D411yescharacteristic
76C2×C4×Dic3D412yes—
77C2×C4×Dic3D411yescharacteristic
78C2×C4×Dic3C2312yes—
79C22×C3⋊C8C2312yes—
80#869—22no—
81#869—22no—
82C3×C4⋊C8—22no—
83C3×#102—24no—
84#832—24no—
85#832—28no—
86C3⋊#95—21no—
87C3⋊#95—22no—
88#807—24no—
89C4⋊(C3⋊C8)—22no—
90C22×D8C2×D312yes—
91C2×C4⋊Q8—31no—
92C2×#127C2×D311yescharacteristic
93C2×#124—61no—
94C2×#124—61no—
95C2×C4×D4—32no—
96C2×C4×D4C2×D311yescharacteristic
97C22×C42—61no—
98#474—64no—
99#459—24no—
100C2×#107—32no—
101C2×C4⋊C8—31no—
102C2×C4⋊C8C2×D311yescharacteristic
103C2×#102—32no—
104C2×#102—34no—
105C2×#102C2×D312yes—
106#390—34no—
107#390—62no—
108C4×C4⋊C4—32no—
109C4×C4⋊C4—31no—
110C4×C22⋊C4—62no—
111C4×C22⋊C4—61no—
112C43—31no—
113#340—34no—
114#336—34no—
115C24×C3—21no—
116C24×C3—22no—
117C2×C3×D4—22no—
118C2×C3×D4—22no—
119C2×C3×D4—21no—
120C2×C3×D4—24no—
121C2×C3×D4—22no—
122C2×C3×D4—24no—
123C2×C3×D4—44no—
124C2×C3×D4—42no—
125C2×C3×D4—42no—
126C2×C3×D4—42no—
127C2×C3×D4—42no—
128C2×C3×D4C2×D412yes—
129C2×C3×D4#4114yes—
130C22×C3×C4—21no—
131C22×C3×C4—24no—
132C22×C3×C4—42no—
133C22×C3×C4—42no—
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2411yescharacteristic
137C22×Dic3—28no—
138C22×Dic3—22no—
139C22×Dic3—22no—
140C22×Dic3—44no—
141C22×Dic3—44no—
142C22×Dic3C2×D412yes—
143C22×Dic3C2×D412yes—
144C2×Dic6—42no—
145C2×Dic6—42no—
146C2×Dic6—42no—
147C3×D8—48no—
148C2×C3×C8—22no—
149C2×C3×C8—24no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4C2×D412yes—
152C3×C22⋊C4—24no—
153C3×C42C2×D412yes—
154C3×C42#4112yes—
155#217—28no—
156#217—48no—
157#217—48no—
158C4⋊Dic3—22no—
159C4⋊Dic3—22no—
160C4⋊Dic3—22no—
161C4⋊Dic3—42no—
162C4⋊Dic3#4114yes—
163C4×Dic3—22no—
164C4×Dic3—22no—
165C4×Dic3—22no—
166C4×Dic3—42no—
167C4×Dic3—42no—
168C4×Dic3—44no—
169C4×Dic3D814yescomplemented
170C2×C3⋊C8—22no—
171C2×C3⋊C8—24no—
172C22×Q8—61no—
173C22×D4—21no—
174C22×D4—21no—
175C22×D4C22×D312yes—
176C22×D4C22×D311yescharacteristic
177C23×C4—21no—
178C23×C4—62no—
179C2×D8—28no—
180C2×D8—24no—
181C22×C8—32no—
182C22×C8C22×D312yes—
183C4⋊Q8—62no—
184C4⋊Q8—62no—
185#127—22no—
186#127—22no—
187#124—124no—
188#124—124no—
189C4×D4—22no—
190C4×D4—64no—
191C4×D4—64no—
192C4×D4#6714yes—
193C2×C4⋊C4—32no—
194C2×C4⋊C4—32no—
195C2×C4⋊C4—61no—
196C2×C4⋊C4C22×D311yescharacteristic
197C2×C22⋊C4—21no—
198C2×C22⋊C4—62no—
199C2×C22⋊C4—64no—
200C2×C22⋊C4—64no—
201C2×C42—31no—
202C2×C42—31no—
203C2×C42—32no—
204C2×C42—32no—
205C2×C42—62no—
206C2×C42—61no—
207C2×C42—124no—
208C2×C42C22×D311yescharacteristic
209#107—64no—
210C4⋊C8—22no—
211C4⋊C8—62no—
212#102—24no—
213#102—68no—
214#102—64no—
215#95—62no—
216#95—61no—
217C23×C3—22no—
218C23×C3—24no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3—42no—
222C23×C3—42no—
223C23×C3—44no—
224C23×C3—42no—
225C23×C3—42no—
226C23×C3—44no—
227C23×C3C22×D411yescharacteristic
228C3×D4—24no—
229C3×D4—22no—
230C3×D4—42no—
231C3×D4—44no—
232C3×D4—42no—
233C3×D4—84no—
234C3×D4—84no—
235C3×D4#12414yescomplemented
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—48no—
241C2×C3×C4#12112yes—
242C2×C3×C4#12112yes—
243C2×C3×C4C2×D812yes—
244C2×C3×C4AGL(1,ℤ/8)12yes—
245C2×C3×C4C22×D411yescharacteristic
246C2×C3×C4C2×#4112yes—
247C2×C3×C4C2×#4111yescharacteristic
248C2×C3×C4C2×#4112yes—
249C2×Dic3—22no—
250C2×Dic3—22no—
251C2×Dic3—22no—
252C2×Dic3—24no—
253C2×Dic3—22no—
254C2×Dic3—22no—
255C2×Dic3—416no—
256C2×Dic3—42no—
257C2×Dic3—42no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260C2×Dic3—42no—
261C2×Dic3C2×D814yes—
262Dic6—84no—
263C3×C8—42no—
264C3⋊C8—42no—
265C24—21no—
266C24—22no—
267C2×Q8—122no—
268C2×Q8—122no—
269C2×Q8—122no—
270C2×D4—21no—
271C2×D4—22no—
272C2×D4—24no—
273C2×D4—22no—
274C2×D4—24no—
275C2×D4—22no—
276C2×D4—44no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4#23714yes—
282C2×D4C2×#6712yes—
283C22×C4—21no—
284C22×C4—24no—
285C22×C4—32no—
286C22×C4—32no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—68no—
290C22×C4—62no—
291C22×C4—62no—
292C22×C4—124no—
293C22×C4—124no—
294C22×C4D3×D411yescharacteristic
295C22×C4D3×D411yescharacteristic
296C22×C4C23×D311yescharacteristic
297D8—48no—
298C2×C8—24no—
299C2×C8—22no—
300C2×C8—64no—
301C2×C8—62no—
302C4⋊C4—21no—
303C4⋊C4—34no—
304C4⋊C4—62no—
305C4⋊C4—62no—
306C4⋊C4—62no—
307C4⋊C4—122no—
308C4⋊C4C2×#6712yes—
309C22⋊C4—24no—
310C22⋊C4—68no—
311C22⋊C4—128no—
312C22⋊C4—128no—
313C42—34no—
314C42—62no—
315C42—62no—
316C42—62no—
317C42—122no—
318C42—124no—
319C42—122no—
320C42#23712yes—
321C42C2×#6712yes—
322C22×C3—24no—
323C22×C3—44no—
324C22×C3—44no—
325C22×C3—42no—
326C22×C3—42no—
327C22×C3—84no—
328C22×C3—82no—
329C22×C3—84no—
330C22×C3—82no—
331C22×C3—82no—
332C22×C3C2×#12111yescharacteristic
333C22×C3#52512yes—
334C22×C3C22×D811yescharacteristic
335C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
336C22×C3#57512yes—
337C3×C4—22no—
338C3×C4—42no—
339C3×C4#45914yes—
340C3×C4C2×#12411yescharacteristic
341C3×C4#52912yes—
342C3×C4#53211yescharacteristic
343Dic3—24no—
344Dic3—44no—
345Dic3—42no—
346Dic3—82no—
347C23—22no—
348C23—24no—
349C23—41no—
350C23—41no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—44no—
355C23—42no—
356C23—44no—
357C23C2×D3×D411yescharacteristic
358Q8—244no—
359D4—22no—
360D4—24no—
361D4—42no—
362D4—42no—
363D4—44no—
364D4—84no—
365D4—84no—
366D4#93614yescomplemented
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—34no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—48no—
373C2×C4—62no—
374C2×C4—62no—
375C2×C4—62no—
376C2×C4—62no—
377C2×C4—62no—
378C2×C4—64no—
379C2×C4—1216no—
380C2×C4—122no—
381C2×C4—122no—
382C2×C4—122no—
383C2×C4—122no—
384C2×C4—122no—
385C2×C4D3×D812yes—
386C2×C4#91112yes—
387C2×C4#93912yes—
388C2×C4#93912yes—
389C2×C4C2×#23712yes—
390C2×C4C2×#23711yescharacteristic
391C2×C4D3×#4112yes—
392C2×C4C22×#6711yescharacteristic
393C8—42no—
394C8—122no—
395C2×C3—44no—
396C2×C3—162no—
397C2×C3#285011yescharacteristic
398C2×C3C2×#45911yescharacteristic
399C2×C3#304011yescharacteristic
400C2×C3#310012yes—
401C2×C3#311512yes—
402C22—24no—
403C22—42no—
404C22—44no—
405C22—44no—
406C22—42no—
407C22—82no—
408C22—82no—
409C22—84no—
410C22—84no—
411C22—82no—
412C22#560612yes—
413C22C2×D3×D811yescharacteristic
414C22C2×#91111yescharacteristic
415C22#577312yes—
416C22C2×#93911yescharacteristic
417C4—22no—
418C4—42no—
419C4—64no—
420C4—124no—
421C4—122no—
422C4—242no—
423C4#515014yes—
424C4#560012yes—
425C4C2×#93611yescharacteristic
426C4#583211yescharacteristic
427C3#3302411yescharacteristic, complemented, Sylow
428C2—44no—
429C2—162no—
430C2#8031212yes—
431C2#8031512yes—
432C2#8269811yescharacteristic
433C2C2×#515011yescharacteristic
434C2#8393311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#459C2
socleC23×C3C22×D4
hypercenterC2×#459D3
hypocenterC3#33024
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
2. #83184C2semidirect
3. C4×D4×Dic3C2semidirect
4. C2×#5007C2semidirect
5. #72037C2semidirect
6. #71761C2semidirect
7. #70460C2semidirect
8. #70443C2semidirect
9. #70173C2semidirect
169. C4×Dic3D8semidirect
235. C3×D4#124semidirect
366. D4#936semidirect
427. C3#33024semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4592
#831842
C4×D4×Dic32
C2×#50072
#720372
#717612
#704602
#704432
#701732
#330243