Finite Group Databasebeta

← #10690467

Subgroups — #10690467

Order 768.

618 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83229C211yesmaximal, characteristic, complemented
2C2×#5088C211yesmaximal, characteristic
3C2×C3×#332C211yesmaximal, characteristic, complemented
4#70457C211yesmaximal, characteristic
5#70457C211yesmaximal, characteristic
6#67704C211yesmaximal, characteristic, complemented
7#67704C211yesmaximal, characteristic, complemented
8#9062—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×Q8×Dic3—21no—
11C2×#874C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14#5229—21no—
15C2×#821—21no—
16C2×#821—21no—
17C2×#821—21no—
18C2×#821—21no—
19#5088—22no—
20#5088C414yescomplemented
21#4984—21no—
22#4980—22no—
23Dic3×C4⋊C4—21no—
24C2×#809—22no—
25C4×C4⋊Dic3C2211yescharacteristic
26C4×C4⋊Dic3C2211yescharacteristic
27C2×C4×C3⋊C8C2211yescharacteristic
28C3×#332—22no—
29C2×#486—31no—
30#2365—31no—
31#1982—31no—
32#1982—31no—
33C2×#332D311yescharacteristic, complemented
34#1279—31no—
35#1279—31no—
36C22×C3×Q8—21no—
37C22×D12—21no—
38C22×D12—21no—
39C22×D12—21no—
40C22×C3×C8—21no—
41C22×C3×C8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8C2×C412yes—
45C3×C4⋊Q8D412yes—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4D412yes—
48C2×C3×C42C2311yescharacteristic
49Q8×Dic3—42no—
50Q8×Dic3—42no—
51C2×C4⋊Dic3—22no—
52C2×C4⋊Dic3D411yescharacteristic
53C2×C4⋊Dic3D411yescharacteristic
54C2×C4⋊Dic3D411yescharacteristic
55C2×C4⋊Dic3D411yescharacteristic
56C2×C4×Dic3—22no—
57C2×C4×Dic3—21no—
58C2×C4×Dic3—21no—
59C22×C3⋊C8—21no—
60#874—21no—
61#874—21no—
62#874—21no—
63#874C2×C412yes—
64C3×C4⋊C8—21no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3⋊#95—22no—
69C3⋊#95—22no—
70C3⋊#95—21no—
71C3⋊#95—21no—
72#821—42no—
73#821—42no—
74#821—42no—
75#821—42no—
76#809—28no—
77C4×C3⋊C8—21no—
78C4×C3⋊C8C2×C412yes—
79C2×C4⋊Q8C2×D311yescharacteristic
80C2×#127—31no—
81C2×C4×Q8—61no—
82#486—34no—
83#486—62no—
84C2×C4⋊C8C2×D311yescharacteristic
85C2×C4⋊C8C2×D311yescharacteristic
86C2×#102—62no—
87C2×#102—61no—
88C2×#102—61no—
89C2×#102—61no—
90C2×#102—61no—
91C2×C4×C8—31no—
92#391—61no—
93#389—61no—
94#387—62no—
95C4×C4⋊C4—31no—
96C4×C4⋊C4—31no—
97C4×C4⋊C4—61no—
98#332—22no—
99C23×D3—41no—
100C2×C3×Q8—22no—
101C2×C3×Q8—22no—
102C2×C3×Q8—41no—
103C2×C3×Q8—41no—
104C2×C3×Q8—41no—
105C2×C3×Q8—41no—
106C22×C3×C4—22no—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×Dic3—21no—
111C22×Dic3—21no—
112C22×Dic3—21no—
113C22×Dic3—21no—
114C2×D12—22no—
115C2×D12—22no—
116C2×D12—22no—
117C2×D12—42no—
118C2×D12—42no—
119C2×D12—41no—
120C2×D12—41no—
121C2×D12—41no—
122C2×D12—41no—
123C2×D12—41no—
124C2×D12—41no—
125C2×D12—41no—
126C2×D12—41no—
127C2×D12—41no—
128C2×D12—41no—
129C2×D12—41no—
130C2×C3×C8—41no—
131C2×C3×C8—41no—
132C2×C3×C8—41no—
133C2×C3×C8—41no—
134C2×C3×C8—41no—
135C2×C3×C8—41no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4—22no—
139C3×C4⋊C4C22⋊C414yes—
140C3×C42C22×C411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42#4111yescharacteristic
143C3×C42#4111yescharacteristic
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—44no—
151C4⋊Dic3D812yes—
152C4⋊Dic3SD1612yes—
153C4×Dic3—44no—
154C4×Dic3—41no—
155C4×Dic3—41no—
156C4×Dic3—41no—
157C4×Dic3—41no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—41no—
160C2×C3⋊C8—41no—
161C22×Q8—21no—
162C22×D4—61no—
163C22×D4—61no—
164C22×D4—61no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—61no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8C4×D312yes—
171C4⋊Q8#6712yes—
172#127—32no—
173#127—61no—
174#127—61no—
175#127—61no—
176C4×Q8—122no—
177C4×Q8—122no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—31no—
180C2×C4⋊C4—31no—
181C2×C4⋊C4—31no—
182C2×C4⋊C4—31no—
183C2×C4⋊C4—62no—
184C2×C4⋊C4D1212yes—
185C2×C42—62no—
186C2×C42—61no—
187C2×C42—61no—
188C2×C42C22×D311yescharacteristic
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192C4⋊C8—21no—
193#102—68no—
194#102—122no—
195#102—122no—
196#102—122no—
197#102—122no—
198C4×C8—32no—
199C4×C8—61no—
200#95—62no—
201#95—62no—
202#95—61no—
203#95—61no—
204C23×C3#12011yescharacteristic
205C22×D3—42no—
206C22×D3—81no—
207C22×D3—81no—
208C22×D3—81no—
209C22×D3—81no—
210C22×D3—81no—
211C22×D3—81no—
212C3×Q8—42no—
213C3×Q8—42no—
214C3×Q8—42no—
215C3×Q8—42no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—24no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4C2×C22⋊C411yescharacteristic
222C2×C3×C4C4×D411yescharacteristic
223C2×C3×C4C4×D411yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12311yescharacteristic
227C2×C3×C4C2×D811yescharacteristic
228C2×C3×C4C2×SD1611yescharacteristic
229C2×C3×C4#13511yescharacteristic
230C2×C3×C4#13511yescharacteristic
231C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
232C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
233C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
234C2×C3×C4#13711yescharacteristic
235C2×Dic3—22no—
236C2×Dic3—22no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247D12—44no—
248D12—42no—
249D12—42no—
250D12—42no—
251D12—42no—
252D12—82no—
253D12—81no—
254D12—81no—
255D12—81no—
256D12—81no—
257C3×C8—81no—
258C3×C8—81no—
259C3⋊C8—42no—
260C24—121no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×D4—62no—
268C2×D4—62no—
269C2×D4—62no—
270C2×D4—122no—
271C2×D4—122no—
272C2×D4—121no—
273C2×D4—121no—
274C2×D4—121no—
275C2×D4—121no—
276C2×D4—121no—
277C2×D4—121no—
278C2×D4—121no—
279C2×D4—121no—
280C2×D4—121no—
281C2×D4—121no—
282C2×D4—121no—
283C22×C4—22no—
284C22×C4—61no—
285C22×C4—61no—
286C22×C4—61no—
287C22×C4—61no—
288C22×C4C2×D1211yescharacteristic
289C22×C4D3×D411yescharacteristic
290C22×C4D3×D411yescharacteristic
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—62no—
298C2×C8—121no—
299C2×C8—121no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—32no—
304C4⋊C4—32no—
305C4⋊C4—61no—
306C4⋊C4—61no—
307C4⋊C4—61no—
308C4⋊C4—61no—
309C4⋊C4—61no—
310C4⋊C4—61no—
311C4⋊C4—124no—
312C4⋊C4(C2×D3)⋊C414yes—
313C42—124no—
314C42—121no—
315C42—121no—
316C42—121no—
317C42—121no—
318C42C2×C4×D311yescharacteristic
319C42#23711yescharacteristic
320C42#23711yescharacteristic
321C42C2×#6711yescharacteristic
322C22×C3#38611yescharacteristic
323C22×C3#44711yescharacteristic
324C22×C3#44811yescharacteristic
325C22×C3#44911yescharacteristic
326C22×C3#44911yescharacteristic
327C22×C3#45411yescharacteristic
328C22×C3#45411yescharacteristic
329C2×D3—82no—
330C2×D3—82no—
331C2×D3—82no—
332C2×D3—161no—
333C2×D3—161no—
334C2×D3—161no—
335C2×D3—161no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—44no—
339C3×C4C4×D811yescharacteristic
340C3×C4C4×SD1611yescharacteristic
341C3×C4#44011yescharacteristic
342C3×C4#44211yescharacteristic
343C3×C4#45911yescharacteristic
344C3×C4#46011yescharacteristic
345C3×C4#46311yescharacteristic
346C3×C4#46411yescharacteristic
347Dic3—42no—
348Dic3—42no—
349Dic3—81no—
350Dic3—81no—
351C23—122no—
352C23—241no—
353C23—241no—
354C23—241no—
355C23—241no—
356C23—241no—
357C23—241no—
358C23#88211yescharacteristic
359Q8—42no—
360Q8—42no—
361Q8—42no—
362Q8—42no—
363D4—124no—
364D4—122no—
365D4—122no—
366D4—122no—
367D4—122no—
368D4—242no—
369D4—241no—
370D4—241no—
371D4—241no—
372D4—241no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—24no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—62no—
379C2×C4—62no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4#67⋊C411yescharacteristic
391C2×C4#67⋊C411yescharacteristic
392C2×C4#88611yescharacteristic
393C2×C4D3×D811yescharacteristic
394C2×C4#91111yescharacteristic
395C2×C4D3×SD1611yescharacteristic
396C2×C4#91411yescharacteristic
397C2×C4#91411yescharacteristic
398C2×C4#91611yescharacteristic
399C2×C4#91811yescharacteristic
400C2×C4#91911yescharacteristic
401C2×C4C2×(C2×D3)⋊C411yescharacteristic
402C2×C4#93911yescharacteristic
403C2×C4#93911yescharacteristic
404C8—81no—
405C8—81no—
406C8—122no—
407C2×C3#162511yescharacteristic
408C2×C3#162611yescharacteristic
409C2×C3#162711yescharacteristic
410C2×C3#162811yescharacteristic
411C2×C3#187911yescharacteristic
412C2×C3#188011yescharacteristic
413C2×C3#188811yescharacteristic
414D3—162no—
415C22—242no—
416C22—242no—
417C22—242no—
418C22—481no—
419C22—481no—
420C22—481no—
421C22—481no—
422C22#475311yescharacteristic
423C22#475311yescharacteristic
424C22#477311yescharacteristic
425C22#478111yescharacteristic
426C22#480611yescharacteristic
427C22#481011yescharacteristic
428C22#495811yescharacteristic
429C4—21no—
430C4—21no—
431C4—44no—
432C4—122no—
433C4—122no—
434C4—241no—
435C4—241no—
436C4#475611yescharacteristic
437C4#478511yescharacteristic
438C4#478811yescharacteristic
439C4#481711yescharacteristic
440C4#515011yescharacteristic
441C4#516311yescharacteristic
442C4#516511yescharacteristic
443C4#518511yescharacteristic
444C3#906211yescharacteristic, complemented, Sylow
445C2—482no—
446C2#6843211yescharacteristic
447C2#6843311yescharacteristic
448C2#6845611yescharacteristic
449C2#6845711yescharacteristic
450C2#7010411yescharacteristic
451C2#7017211yescharacteristic
452C2#7024011yescharacteristic
subgroupG/H
centerC23#882
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC2×C3×#332C2
socleC23×C3#120
prime radicalC2×#874C22
cosocleC2×C3×C42C23
hypercenterC2×#332D3
hypocenterC3#9062
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
1. #83229C2semidirect
3. C2×C3×#332C2semidirect
6. #67704C2semidirect
7. #67704C2semidirect
20. #5088C4semidirect
33. C2×#332D3semidirect
444. C3#9062semidirect

Maximal subgroups

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

subgroupindex
#832292
C2×#50882
C2×C3×#3322
#704572
#677042
#90623