Finite Group Databasebeta

← #10690197

Subgroups — #10690197

Order 768.

1296 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#392C211yesmaximal, characteristic
2#82094C211yesmaximal, characteristic, complemented
3#72128C212yesmaximal, complemented
4C2×#4525C211yesmaximal, characteristic, complemented
5#67612C212yesmaximal, complemented
6#9017—31nomaximal, Sylow
7C23×C3×D4—21no—
8C23×C3×D4C2211yescharacteristic
9C22×C3×C22⋊C4—21no—
10C22×C3×C22⋊C4C2211yescharacteristic
11C22×#217—21no—
12C3×#392—22no—
13C3×#392C414yescomplemented
14C2×C3×#95C2211yescharacteristic
15C2×#832—24no—
16#5212—21no—
17#5212C2212yes—
18C2×#830C2212yes—
19#4950—24no—
20#4525—22no—
21C2×#392D311yescharacteristic
22#2315—31no—
23#1786—32no—
24C2×#323—31no—
25#1294—32no—
26C25×C3—21no—
27C25×C3D411yescharacteristic
28C22×C3×D4—24no—
29C22×C3×D4—22no—
30C22×C3×D4—21no—
31C22×C3×D4—22no—
32C22×C3×D4—44no—
33C22×C3×D4—44no—
34C22×C3×D4—44no—
35C22×C3×D4—42no—
36C22×C3×D4—42no—
37C22×C3×D4C2×C412yescomplemented
38C23×C3×C4—21no—
39C23×C3×C4D411yescharacteristic
40C23×C3×C4C2311yescharacteristic
41C23×Dic3—22no—
42C2×C3×C22⋊C4—22no—
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4—24no—
45C2×C3×C22⋊C4—44no—
46C2×C3×C22⋊C4C2×C412yes—
47C2×C3×C22⋊C4D412yes—
48C2×#217—22no—
49C2×#217—22no—
50C2×#217—42no—
51C2×#217—42no—
52C2×C4⋊Dic3—22no—
53C2×C4⋊Dic3D414yescomplemented
54C22×C3⋊C8—22no—
55C3×#95—21no—
56C3×#95C2×C412yescomplemented
57#832—48no—
58C3⋊#95—22no—
59C3⋊#95—22no—
60#830—22no—
61#830—22no—
62C23×D4—21no—
63C23×D4C2×D311yescharacteristic
64C22×C22⋊C4—21no—
65C22×C22⋊C4—61no—
66C22×C22⋊C4C2×D311yescharacteristic
67C2×#102—64no—
68C2×C22⋊C8—32no—
69#392—22no—
70#392Dic314yescomplemented
71#385—64no—
72#380—32no—
73#380—61no—
74C2×#95C2×D311yescharacteristic
75#323—62no—
76C24×C3—21no—
77C24×C3—21no—
78C24×C3—21no—
79C24×C3—21no—
80C24×C3—24no—
81C24×C3—22no—
82C24×C3—42no—
83C24×C3—42no—
84C24×C3—42no—
85C24×C3—42no—
86C24×C3—42no—
87C24×C3—41no—
88C24×C3—41no—
89C24×C3—41no—
90C24×C3—41no—
91C24×C3—41no—
92C24×C3—41no—
93C24×C3C22⋊C412yes—
94C24×C3D812yes—
95C24×C3SD1612yes—
96C24×C3C2×D411yescharacteristic
97C2×C3×D4—28no—
98C2×C3×D4—24no—
99C2×C3×D4—44no—
100C2×C3×D4—48no—
101C2×C3×D4—44no—
102C2×C3×D4—48no—
103C2×C3×D4—44no—
104C2×C3×D4—48no—
105C2×C3×D4—416no—
106C2×C3×D4—416no—
107C2×C3×D4—48no—
108C2×C3×D4—42no—
109C22×C3×C4—24no—
110C22×C3×C4—22no—
111C22×C3×C4—21no—
112C22×C3×C4—21no—
113C22×C3×C4—42no—
114C22×C3×C4—42no—
115C22×C3×C4—41no—
116C22×C3×C4—41no—
117C22×C3×C4—42no—
118C22×C3×C4C22⋊C412yes—
119C22×C3×C4C22×C411yescharacteristic
120C22×C3×C4C2×D412yes—
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4#4112yes—
123C22×Dic3—24no—
124C22×Dic3—44no—
125C22×Dic3—82no—
126C22×Dic3—82no—
127C3×C22⋊C4—24no—
128C3×C22⋊C4—24no—
129C3×C22⋊C4—48no—
130#217—44no—
131#217—42no—
132#217—42no—
133C4⋊Dic3—44no—
134C4⋊Dic3—82no—
135C2×C3⋊C8—44no—
136C2×C3⋊C8—42no—
137C25—21no—
138C25#6711yescharacteristic
139C22×D4—24no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—21no—
143C22×D4—44no—
144C22×D4—44no—
145C22×D4—44no—
146C22×D4—42no—
147C22×D4—42no—
148C22×D4C2×Dic312yescomplemented
149C23×C4—21no—
150C23×C4—62no—
151C23×C4#6711yescharacteristic
152C23×C4C22×D311yescharacteristic
153C22×C8—62no—
154C2×C4⋊C4—34no—
155C2×C4⋊C4—62no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—44no—
160C2×C22⋊C4—62no—
161C2×C22⋊C4—62no—
162C2×C22⋊C4—122no—
163C2×C22⋊C4—122no—
164C2×C22⋊C4C2×Dic312yes—
165C2×C22⋊C4#6712yes—
166#102—128no—
167C22⋊C8—62no—
168C22⋊C8—62no—
169#95—21no—
170#95—62no—
171#95—62no—
172#95C2×Dic312yescomplemented
173C23×C3—22no—
174C23×C3—22no—
175C23×C3—22no—
176C23×C3—22no—
177C23×C3—22no—
178C23×C3—22no—
179C23×C3—22no—
180C23×C3—22no—
181C23×C3—24no—
182C23×C3—42no—
183C23×C3—42no—
184C23×C3—44no—
185C23×C3—44no—
186C23×C3—42no—
187C23×C3—44no—
188C23×C3—44no—
189C23×C3—42no—
190C23×C3—42no—
191C23×C3—44no—
192C23×C3—44no—
193C23×C3—44no—
194C23×C3—42no—
195C23×C3—42no—
196C23×C3—42no—
197C23×C3—42no—
198C23×C3—42no—
199C23×C3—44no—
200C23×C3—82no—
201C23×C3—82no—
202C23×C3—81no—
203C23×C3—81no—
204C23×C3—82no—
205C23×C3—82no—
206C23×C3—82no—
207C23×C3—82no—
208C23×C3—82no—
209C23×C3—81no—
210C23×C3—81no—
211C23×C3—82no—
212C23×C3—81no—
213C23×C3—81no—
214C23×C3#10214yescomplemented
215C23×C3C2×C22⋊C411yescharacteristic
216C23×C3#12011yescharacteristic
217C23×C3#12311yescharacteristic
218C23×C3C2×D811yescharacteristic
219C23×C3C2×SD1611yescharacteristic
220C23×C3AGL(1,ℤ/8)11yescharacteristic
221C23×C3#13711yescharacteristic
222C3×D4—28no—
223C3×D4—48no—
224C3×D4—44no—
225C3×D4—432no—
226C2×C3×C4—24no—
227C2×C3×C4—24no—
228C2×C3×C4—22no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—42no—
232C2×C3×C4—44no—
233C2×C3×C4—44no—
234C2×C3×C4—42no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4—81no—
239C2×C3×C4C4×D412yes—
240C2×C3×C4#12112yes—
241C2×Dic3—44no—
242C2×Dic3—44no—
243C2×Dic3—44no—
244C2×Dic3—88no—
245C3⋊C8—82no—
246C24—21no—
247C24—21no—
248C24—21no—
249C24—21no—
250C24—24no—
251C24—22no—
252C24—42no—
253C24—42no—
254C24—42no—
255C24—42no—
256C24—42no—
257C24—41no—
258C24—41no—
259C24—41no—
260C24—41no—
261C24—41no—
262C24—41no—
263C24#21312yes—
264C24#21412yes—
265C24#21712yes—
266C24C2×#6711yescharacteristic
267C2×D4—28no—
268C2×D4—24no—
269C2×D4—44no—
270C2×D4—48no—
271C2×D4—44no—
272C2×D4—48no—
273C2×D4—44no—
274C2×D4—48no—
275C2×D4—416no—
276C2×D4—416no—
277C2×D4—48no—
278C2×D4—42no—
279C22×C4—24no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—64no—
289C22×C4—124no—
290C22×C4—242no—
291C22×C4—242no—
292C22×C4#21712yes—
293C22×C4D3×D412yes—
294C22×C4#23712yes—
295C22×C4C22×Dic311yescharacteristic
296C22×C4C2×#6711yescharacteristic
297C2×C8—124no—
298C2×C8—122no—
299C4⋊C4—124no—
300C4⋊C4—242no—
301C22⋊C4—24no—
302C22⋊C4—24no—
303C22⋊C4—48no—
304C22⋊C4—124no—
305C22⋊C4—122no—
306C22⋊C4—122no—
307C22×C3—22no—
308C22×C3—22no—
309C22×C3—22no—
310C22×C3—22no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—21no—
315C22×C3—24no—
316C22×C3—44no—
317C22×C3—44no—
318C22×C3—44no—
319C22×C3—44no—
320C22×C3—44no—
321C22×C3—42no—
322C22×C3—42no—
323C22×C3—44no—
324C22×C3—82no—
325C22×C3—82no—
326C22×C3—82no—
327C22×C3—82no—
328C22×C3—82no—
329C22×C3—84no—
330C22×C3—84no—
331C22×C3—84no—
332C22×C3—84no—
333C22×C3—82no—
334C22×C3—82no—
335C22×C3#38611yescharacteristic
336C22×C3C2×#10211yescharacteristic
337C22×C3#41711yescharacteristic
338C22×C3#44711yescharacteristic
339C22×C3#44811yescharacteristic
340C22×C3#45312yes—
341C22×C3B(4,𝔽2)12yes—
342C22×C3#48011yescharacteristic
343C22×C3#48111yescharacteristic
344C3×C4—22no—
345C3×C4—22no—
346C3×C4—44no—
347C3×C4—42no—
348Dic3—84no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—24no—
358C23—44no—
359C23—44no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—44no—
364C23—44no—
365C23—42no—
366C23—42no—
367C23—44no—
368C23—44no—
369C23—44no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—44no—
376C23—82no—
377C23—82no—
378C23—81no—
379C23—81no—
380C23—82no—
381C23—82no—
382C23—82no—
383C23—82no—
384C23—82no—
385C23—81no—
386C23—81no—
387C23—81no—
388C23—81no—
389C23—82no—
390C23#83214yescomplemented
391C23C2×#21311yescharacteristic
392C23C2×#21411yescharacteristic
393C23#93511yescharacteristic
394C23#93711yescharacteristic
395C23#94911yescharacteristic
396C23#95111yescharacteristic
397C23C2×#21711yescharacteristic
398D4—28no—
399D4—48no—
400D4—44no—
401D4—432no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—81no—
415C2×C4—124no—
416C2×C4—124no—
417C2×C4—124no—
418C2×C4—248no—
419C2×C4D4×Dic312yes—
420C2×C4#93912yes—
421C8—242no—
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—44no—
427C2×C3—84no—
428C2×C3#159512yes—
429C2×C3#159612yes—
430C2×C3#187411yescharacteristic
431C2×C3#188711yescharacteristic
432C2×C3#188911yescharacteristic
433C22—22no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—24no—
442C22—44no—
443C22—44no—
444C22—44no—
445C22—44no—
446C22—44no—
447C22—42no—
448C22—42no—
449C22—44no—
450C22—82no—
451C22—82no—
452C22—82no—
453C22—82no—
454C22—82no—
455C22—84no—
456C22—84no—
457C22—84no—
458C22—84no—
459C22—82no—
460C22—82no—
461C22#503212yes—
462C22#503311yescharacteristic
463C22#503411yescharacteristic
464C22#503611yescharacteristic
465C22#504011yescharacteristic
466C22#507712yes—
467C22C2×#83211yescharacteristic
468C22#522011yescharacteristic
469C22#523211yescharacteristic
470C4—22no—
471C4—22no—
472C4—44no—
473C4—42no—
474C4—244no—
475C3#901711yescharacteristic, complemented, Sylow
476C2—21no—
477C2—21no—
478C2—21no—
479C2—21no—
480C2—44no—
481C2—84no—
482C2#6969612yes—
483C2#6969812yes—
484C2#7166911yescharacteristic
485C2#7167711yescharacteristic
486C2#7176311yescharacteristic
subgroupG/H
centerC23#937
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
Fitting, prime radicalC2×C3×#392C2
socleC23×C3#120
cosocleC23×C3×C4C23
hypercenterC2×#392D3
hypocenterC3#9017
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#392

Complements

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

NH ≅ G/Ntype
2. #82094C2semidirect
3. #72128C2semidirect
4. C2×#4525C2semidirect
5. #67612C2semidirect
13. C3×#392C4semidirect
37. C22×C3×D4C2×C4semidirect
53. C2×C4⋊Dic3D4semidirect
56. C3×#95C2×C4semidirect
70. #392Dic3semidirect
148. C22×D4C2×Dic3semidirect
172. #95C2×Dic3semidirect
214. C23×C3#102semidirect
390. C23#832semidirect
475. C3#9017semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3922
#820942
#721282
C2×#45252
#676122
#90173