Finite Group Databasebeta

← #10689372

Subgroups — #10689372

Order 768.

618 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#74360C211yesmaximal, characteristic
2C3×#1844C211yesmaximal, characteristic, complemented
3C2×#4459C211yesmaximal, characteristic
4#69584C211yesmaximal, characteristic
5#69568C211yesmaximal, characteristic, complemented
6#67655C211yesmaximal, characteristic, complemented
7#67634C211yesmaximal, characteristic, complemented
8#8886—31nomaximal, Sylow
9C2×#893—21no—
10C2×D3×C4⋊C4—21no—
11C2×C4×D12C2211yescharacteristic
12C2×C3×#106—22no—
13C2×C3×C4×C8C2211yescharacteristic
14C3×#384C2211yescharacteristic
15C3×#384C2211yescharacteristic
16#4989—22no—
17#4987—21no—
18#4972—21no—
19C2×#809—21no—
20C2×#808—21no—
21C4×C4⋊Dic3C2211yescharacteristic
22C2×C4⋊(C3⋊C8)C2211yescharacteristic
23C2×C4⋊(C3⋊C8)C2211yescharacteristic
24#4529—21no—
25#4529—21no—
26#4459C414yescomplemented
27D4×C4⋊C4—31no—
28#1948—31no—
29#1844D311yescharacteristic, complemented
30#1802—31no—
31C2×D4⋊C8—31no—
32#1296—31no—
33#1279—31no—
34C22×D12D411yescharacteristic
35C22×C4×D3—21no—
36C22×C4×D3—22no—
37C22×C3×C8—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4D412yes—
44C2×C3×C42C2311yescharacteristic
45C2×(C2×D3)⋊C4—21no—
46C2×(C2×D3)⋊C4—22no—
47C2×C4⋊Dic3D411yescharacteristic
48C2×C4⋊Dic3D411yescharacteristic
49C2×C4⋊Dic3D411yescharacteristic
50C2×Dic3⋊C4—22no—
51C2×C4×Dic3—21no—
52C22×C3⋊C8—21no—
53C22×C3⋊C8—21no—
54#893—42no—
55#893—42no—
56D3×C4⋊C4—42no—
57D3×C4⋊C4—42no—
58C4×D12—21no—
59C4×D12—21no—
60C4×D12C2×C412yes—
61C4×D12Q812yes—
62C3×#106—44no—
63C3×C4×C8—21no—
64C3×C4×C8C2×C412yescomplemented
65C3⋊#95—22no—
66C3⋊#95—21no—
67#809—42no—
68#808—42no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)C2×C412yescomplemented
73C2×C4×D4—31no—
74C2×C4×D4—61no—
75C22×C4⋊C4—61no—
76C2×#106—22no—
77C2×#106—61no—
78C2×C4⋊C8—31no—
79C2×C4⋊C8—31no—
80C2×#102—61no—
81C2×C4×C8C2×D311yescharacteristic
82#389—61no—
83#385—62no—
84#384C2×D311yescharacteristic
85#384C2×D311yescharacteristic
86#380—61no—
87C4×C4⋊C4—31no—
88#340—61no—
89#340—61no—
90D4⋊C8—34no—
91C23×D3—21no—
92C22×C3×C4—22no—
93C22×C3×C4—41no—
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×Dic3—21no—
98C22×Dic3—21no—
99C22×Dic3—21no—
100C2×D12—21no—
101C2×D12—21no—
102C2×D12—21no—
103C2×D12C4⋊C412yes—
104C2×D12SD1612yes—
105C2×D12SD1612yes—
106C2×C4×D3—22no—
107C2×C4×D3—22no—
108C2×C4×D3—41no—
109C2×C4×D3—41no—
110C2×C4×D3—41no—
111C2×C4×D3—41no—
112C2×C4×D3—44no—
113C2×C4×D3—44no—
114C2×C4×D3—82no—
115C2×C3×C8—22no—
116C2×C3×C8—22no—
117C2×C3×C8—22no—
118C3×C4⋊C4—42no—
119C3×C4⋊C4—42no—
120C3×C4⋊C4—41no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—41no—
127C3×C4⋊C4—41no—
128C3×C42C22×C411yescharacteristic
129C3×C42C2×Q811yescharacteristic
130C3×C42#4111yescharacteristic
131C3×C42#4111yescharacteristic
132(C2×D3)⋊C4—22no—
133(C2×D3)⋊C4—22no—
134(C2×D3)⋊C4—44no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3C4⋊C412yes—
140C4⋊Dic3SD1612yes—
141Dic3⋊C4—44no—
142C4×Dic3—41no—
143C4×Dic3—41no—
144C2×C3⋊C8—22no—
145C2×C3⋊C8—41no—
146C2×C3⋊C8—41no—
147C2×C3⋊C8—41no—
148C2×C3⋊C8—41no—
149C2×C3⋊C8—41no—
150C22×D4—31no—
151C23×C4—62no—
152C23×C4—61no—
153C22×C8—21no—
154C22×C8—61no—
155C22×C8—61no—
156C4×D4—32no—
157C4×D4—32no—
158C4×D4—61no—
159C4×D4—61no—
160C4×D4—122no—
161C4×D4—122no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—31no—
168C2×C4⋊C4—31no—
169C2×C4⋊C4—31no—
170C2×C4⋊C4—62no—
171C2×C4⋊C4—122no—
172C2×C4⋊C4—122no—
173C2×C4⋊C4#6712yes—
174C2×C22⋊C4—61no—
175C2×C22⋊C4—62no—
176C2×C42—61no—
177C2×C42C22×D311yescharacteristic
178#106—44no—
179#106—122no—
180C4⋊C8—32no—
181C4⋊C8—61no—
182C4⋊C8—61no—
183C4⋊C8—61no—
184#102—122no—
185C4×C8—21no—
186C4×C8C4×D312yescomplemented
187#95—62no—
188#95—61no—
189C23×C3#12011yescharacteristic
190C22×D3—22no—
191C22×D3—22no—
192C22×D3—22no—
193C22×D3—41no—
194C22×D3—41no—
195C22×D3—41no—
196C22×D3—41no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4—42no—
202C2×C3×C4—81no—
203C2×C3×C4—81no—
204C2×C3×C4—81no—
205C2×C3×C4C2×C4⋊C411yescharacteristic
206C2×C3×C4C4×D411yescharacteristic
207C2×C3×C4C4×D411yescharacteristic
208C2×C3×C4C22⋊Q811yescharacteristic
209C2×C3×C4C22⋊Q811yescharacteristic
210C2×C3×C4#12311yescharacteristic
211C2×C3×C4C2×SD1611yescharacteristic
212C2×C3×C4C2×SD1611yescharacteristic
213C2×C3×C4C2×SD1611yescharacteristic
214C2×C3×C4#13511yescharacteristic
215C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
216C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
217C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
218C2×C3×C4#13711yescharacteristic
219C2×Dic3—22no—
220C2×Dic3—22no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228D12—22no—
229D12—22no—
230D12—22no—
231D12#10614yescomplemented
232C4×D3—42no—
233C4×D3—42no—
234C4×D3—42no—
235C4×D3—42no—
236C4×D3—88no—
237C3×C8—24no—
238C3⋊C8—42no—
239C3⋊C8—81no—
240C24—61no—
241C2×D4—32no—
242C2×D4—32no—
243C2×D4—32no—
244C2×D4—61no—
245C2×D4—61no—
246C2×D4—61no—
247C22×C4—22no—
248C22×C4—41no—
249C22×C4—61no—
250C22×C4—61no—
251C22×C4—62no—
252C22×C4—62no—
253C22×C4—61no—
254C22×C4—124no—
255C22×C4—124no—
256C22×C4—121no—
257C22×C4—121no—
258C22×C4—121no—
259C22×C4—121no—
260C22×C4—242no—
261C22×C4D3×D411yescharacteristic
262C22×C4D3×D411yescharacteristic
263C22×C4C2×#6711yescharacteristic
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—62no—
268C2×C8—121no—
269C2×C8—121no—
270C2×C8—121no—
271C2×C8—121no—
272C2×C8—121no—
273C4⋊C4—32no—
274C4⋊C4—32no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—61no—
286C4⋊C4—61no—
287C4⋊C4—61no—
288C4⋊C4—61no—
289C4⋊C4—124no—
290C22⋊C4—62no—
291C22⋊C4—62no—
292C22⋊C4—124no—
293C42—121no—
294C42—121no—
295C42C2×C4×D311yescharacteristic
296C42#23511yescharacteristic
297C42D3×Q811yescharacteristic
298C42#23911yescharacteristic
299C22×C3#38511yescharacteristic
300C22×C3#44811yescharacteristic
301C22×C3#44911yescharacteristic
302C22×C3#45011yescharacteristic
303C22×C3#45011yescharacteristic
304C22×C3#45311yescharacteristic
305C22×C3#45611yescharacteristic
306C2×D3—24no—
307C2×D3—42no—
308C2×D3—42no—
309C2×D3—42no—
310C2×D3—42no—
311C2×D3—42no—
312C2×D3—42no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—82no—
316C3×C4—161no—
317C3×C4C2×#10611yescharacteristic
318C3×C4#42811yescharacteristic
319C3×C4C4×SD1611yescharacteristic
320C3×C4#44211yescharacteristic
321C3×C4#47611yescharacteristic
322C3×C4#47811yescharacteristic
323C3×C4#48111yescharacteristic
324C3×C4#48311yescharacteristic
325Dic3—42no—
326Dic3—42no—
327Dic3—81no—
328C23—62no—
329C23—62no—
330C23—62no—
331C23—121no—
332C23—121no—
333C23—121no—
334C23—121no—
335C23#93711yescharacteristic
336D4—34no—
337D4—62no—
338D4—62no—
339D4—62no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—62no—
346C2×C4—62no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4—121no—
353C2×C4—121no—
354C2×C4—122no—
355C2×C4—122no—
356C2×C4—122no—
357C2×C4—122no—
358C2×C4—121no—
359C2×C4—121no—
360C2×C4—121no—
361C2×C4—248no—
362C2×C4D3×C4⋊C411yescharacteristic
363C2×C4#89311yescharacteristic
364C2×C4#89411yescharacteristic
365C2×C4#89611yescharacteristic
366C2×C4D3×SD1611yescharacteristic
367C2×C4D3×SD1611yescharacteristic
368C2×C4D3×SD1611yescharacteristic
369C2×C4#91411yescharacteristic
370C2×C4#91411yescharacteristic
371C2×C4#91411yescharacteristic
372C2×C4#91511yescharacteristic
373C2×C4#91611yescharacteristic
374C2×C4C4×#6711yescharacteristic
375C2×C4#94611yescharacteristic
376C8—24no—
377C8—122no—
378C8—241no—
379C2×C3#162211yescharacteristic
380C2×C3#162911yescharacteristic
381C2×C3#165111yescharacteristic
382C2×C3#165411yescharacteristic
383C2×C3#185311yescharacteristic
384C2×C3#185911yescharacteristic
385C2×C3#186411yescharacteristic
386D3—44no—
387C22—64no—
388C22—122no—
389C22—122no—
390C22—122no—
391C22—122no—
392C22—122no—
393C22—122no—
394C22#498911yescharacteristic
395C22#507711yescharacteristic
396C22#509511yescharacteristic
397C22#516911yescharacteristic
398C22#516911yescharacteristic
399C22#517011yescharacteristic
400C22#517211yescharacteristic
401C4—21no—
402C4—21no—
403C4—82no—
404C4—122no—
405C4—122no—
406C4—161no—
407C4—241no—
408C4#485211yescharacteristic
409C4D3×#10611yescharacteristic
410C4#486111yescharacteristic
411C4#486211yescharacteristic
412C4#486311yescharacteristic
413C4#486911yescharacteristic
414C4#487011yescharacteristic
415C4#487111yescharacteristic
416C3#888611yescharacteristic, complemented, Sylow
417C2—124no—
418C2#6959211yescharacteristic
419C2#7057911yescharacteristic
420C2#7058411yescharacteristic
421C2#7126411yescharacteristic
422C2#7126611yescharacteristic
423C2#7134211yescharacteristic
424C2#7134311yescharacteristic
subgroupG/H
centerC23#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1844C2
socleC23×C3#120
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenter#1844D3
hypocenterC3#8886
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1844

Complements

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

NH ≅ G/Ntype
2. C3×#1844C2semidirect
5. #69568C2semidirect
6. #67655C2semidirect
7. #67634C2semidirect
26. #4459C4semidirect
29. #1844D3semidirect
64. C3×C4×C8C2×C4semidirect
72. C4⋊(C3⋊C8)C2×C4semidirect
186. C4×C8C4×D3semidirect
231. D12#106semidirect
416. C3#8886semidirect

Maximal subgroups

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

subgroupindex
#743602
C3×#18442
C2×#44592
#695842
#695682
#676552
#676342
#88863