Finite Group Databasebeta

← #10690194

Subgroups — #10690194

Order 768.

1296 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 1296 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4960C211yesmaximal, characteristic
2#82094C211yesmaximal, characteristic, complemented
3C2×C3×#323C211yesmaximal, characteristic, complemented
4#69959C212yesmaximal, complemented
5#67675C212yesmaximal, complemented
6#9017—31nomaximal, Sylow
7C23×#67—21no—
8C23×D12C2211yescharacteristic
9C22×C3×C22⋊C4C2211yescharacteristic
10C22×#217—21no—
11C22×(C2×D3)⋊C4—21no—
12C2×C3×C22⋊C8C2212yes—
13#5212—21no—
14#5212C2212yes—
15C2×C3⋊#95C2211yescharacteristic
16C2×#821—24no—
17#4960—22no—
18#4960C414yescomplemented
19#4950—24no—
20C3×#323—22no—
21C2×#392—31no—
22#2315—31no—
23#1786—32no—
24C2×#323D311yescharacteristic, complemented
25#1294—32no—
26C25×C3D411yescharacteristic
27C24×D3—21no—
28C23×C3×C4D411yescharacteristic
29C23×C3×C4C2311yescharacteristic
30C22×#67—24no—
31C22×#67—44no—
32C22×#67—44no—
33C22×#67—42no—
34C22×#67—42no—
35C23×Dic3—21no—
36C23×Dic3—22no—
37C22×D12—22no—
38C22×D12—22no—
39C22×D12—21no—
40C22×D12—44no—
41C22×D12C2×C412yescomplemented
42C22×C3×C8—22no—
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4—22no—
45C2×C3×C22⋊C4C2×C412yes—
46C2×C3×C22⋊C4D412yes—
47C2×#217—22no—
48C2×#217—22no—
49C2×#217—42no—
50C2×#217—42no—
51C2×(C2×D3)⋊C4—24no—
52C2×(C2×D3)⋊C4—44no—
53C2×C4⋊Dic3—22no—
54C2×C4⋊Dic3D414yescomplemented
55C3×C22⋊C8—22no—
56C3×C22⋊C8—22no—
57C3⋊#95—22no—
58C3⋊#95—22no—
59C3⋊#95—21no—
60C3⋊#95C2×C412yescomplemented
61#821—48no—
62C23×D4—31no—
63C23×D4—61no—
64C22×C22⋊C4—61no—
65C22×C22⋊C4—61no—
66C22×C22⋊C4C2×D311yescharacteristic
67C2×#102—64no—
68C2×C22⋊C8C2×D312yes—
69#392—34no—
70#392—62no—
71#385—64no—
72#380—32no—
73#380—61no—
74C2×#95—31no—
75#323—22no—
76C24×C3—21no—
77C24×C3—21no—
78C24×C3—21no—
79C24×C3—21no—
80C24×C3—42no—
81C24×C3—42no—
82C24×C3C22⋊C412yes—
83C24×C3D812yes—
84C24×C3SD1612yes—
85C24×C3C2×D411yescharacteristic
86C23×D3—24no—
87C23×D3—22no—
88C23×D3—42no—
89C23×D3—42no—
90C23×D3—42no—
91C23×D3—41no—
92C23×D3—41no—
93C23×D3—41no—
94C23×D3—41no—
95C23×D3—41no—
96C23×D3—41no—
97C22×C3×C4—21no—
98C22×C3×C4—21no—
99C22×C3×C4—42no—
100C22×C3×C4—42no—
101C22×C3×C4C22⋊C412yes—
102C22×C3×C4C22×C411yescharacteristic
103C22×C3×C4C2×D412yes—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4#4112yes—
106C2×#67—48no—
107C2×#67—48no—
108C2×#67—416no—
109C2×#67—416no—
110C2×#67—48no—
111C22×Dic3—24no—
112C22×Dic3—22no—
113C22×Dic3—24no—
114C22×Dic3—42no—
115C22×Dic3—41no—
116C22×Dic3—41no—
117C22×Dic3—44no—
118C22×Dic3—82no—
119C22×Dic3—82no—
120C2×D12—28no—
121C2×D12—24no—
122C2×D12—44no—
123C2×D12—48no—
124C2×D12—44no—
125C2×D12—44no—
126C2×D12—42no—
127C2×C3×C8—44no—
128C2×C3×C8—42no—
129C3×C22⋊C4—24no—
130C3×C22⋊C4—24no—
131#217—44no—
132#217—42no—
133#217—42no—
134(C2×D3)⋊C4—48no—
135C4⋊Dic3—44no—
136C4⋊Dic3—82no—
137C25—61no—
138C25D1211yescharacteristic
139C22×D4—32no—
140C22×D4—64no—
141C22×D4—62no—
142C22×D4—62no—
143C22×D4—61no—
144C22×D4—124no—
145C22×D4—124no—
146C22×D4—124no—
147C22×D4—122no—
148C22×D4—122no—
149C23×C4—61no—
150C23×C4—62no—
151C23×C4D1211yescharacteristic
152C23×C4C22×D311yescharacteristic
153C22×C8—22no—
154C2×C4⋊C4—34no—
155C2×C4⋊C4—62no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—62no—
159C2×C22⋊C4—62no—
160C2×C22⋊C4—64no—
161C2×C22⋊C4—124no—
162C2×C22⋊C4—122no—
163C2×C22⋊C4—122no—
164C2×C22⋊C4C4×D312yes—
165C2×C22⋊C4#6712yes—
166#102—128no—
167C22⋊C8—22no—
168C22⋊C8—22no—
169#95—32no—
170#95—62no—
171#95—62no—
172#95—61no—
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—42no—
182C23×C3—42no—
183C23×C3—44no—
184C23×C3—44no—
185C23×C3—42no—
186C23×C3—42no—
187C23×C3—82no—
188C23×C3—82no—
189C23×C3—81no—
190C23×C3—81no—
191C23×C3#10214yescomplemented
192C23×C3C2×C22⋊C411yescharacteristic
193C23×C3#12011yescharacteristic
194C23×C3#12311yescharacteristic
195C23×C3C2×D811yescharacteristic
196C23×C3C2×SD1611yescharacteristic
197C23×C3AGL(1,ℤ/8)11yescharacteristic
198C23×C3#13711yescharacteristic
199C22×D3—24no—
200C22×D3—42no—
201C22×D3—42no—
202C22×D3—44no—
203C22×D3—44no—
204C22×D3—44no—
205C22×D3—42no—
206C22×D3—44no—
207C22×D3—44no—
208C22×D3—42no—
209C22×D3—42no—
210C22×D3—42no—
211C22×D3—44no—
212C22×D3—82no—
213C22×D3—82no—
214C22×D3—82no—
215C22×D3—82no—
216C22×D3—82no—
217C22×D3—81no—
218C22×D3—81no—
219C22×D3—82no—
220C22×D3—81no—
221C22×D3—81no—
222C2×C3×C4—24no—
223C2×C3×C4—22no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4—81no—
230C2×C3×C4C4×D412yes—
231C2×C3×C4#12112yes—
232#67—432no—
233C2×Dic3—24no—
234C2×Dic3—44no—
235C2×Dic3—44no—
236C2×Dic3—42no—
237C2×Dic3—42no—
238C2×Dic3—44no—
239C2×Dic3—44no—
240C2×Dic3—44no—
241C2×Dic3—88no—
242D12—28no—
243D12—48no—
244D12—44no—
245C3×C8—82no—
246C24—21no—
247C24—21no—
248C24—21no—
249C24—21no—
250C24—42no—
251C24—42no—
252C24—64no—
253C24—62no—
254C24—122no—
255C24—122no—
256C24—122no—
257C24—121no—
258C24—121no—
259C24—121no—
260C24—121no—
261C24—121no—
262C24—121no—
263C24#20412yes—
264C24D2412yes—
265C24(C2×D3)⋊C412yes—
266C24C2×D1211yescharacteristic
267C2×D4—68no—
268C2×D4—64no—
269C2×D4—124no—
270C2×D4—124no—
271C2×D4—128no—
272C2×D4—124no—
273C2×D4—128no—
274C2×D4—128no—
275C2×D4—1216no—
276C2×D4—1216no—
277C2×D4—128no—
278C2×D4—122no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—64no—
284C22×C4—62no—
285C22×C4—64no—
286C22×C4—122no—
287C22×C4—121no—
288C22×C4—121no—
289C22×C4—124no—
290C22×C4—242no—
291C22×C4—242no—
292C22×C4(C2×D3)⋊C412yes—
293C22×C4C2×C4×D311yescharacteristic
294C22×C4D3×D412yes—
295C22×C4#23712yes—
296C22×C4C2×#6711yescharacteristic
297C2×C8—44no—
298C2×C8—42no—
299C4⋊C4—124no—
300C4⋊C4—242no—
301C22⋊C4—24no—
302C22⋊C4—24no—
303C22⋊C4—128no—
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—42no—
320C22×C3—42no—
321C22×C3—82no—
322C22×C3—82no—
323C22×C3—82no—
324C22×C3#38611yescharacteristic
325C22×C3C2×#10211yescharacteristic
326C22×C3#41711yescharacteristic
327C22×C3#44711yescharacteristic
328C22×C3#44811yescharacteristic
329C22×C3#45312yes—
330C22×C3B(4,𝔽2)12yes—
331C22×C3#48011yescharacteristic
332C22×C3#48111yescharacteristic
333C2×D3—44no—
334C2×D3—44no—
335C2×D3—44no—
336C2×D3—82no—
337C2×D3—82no—
338C2×D3—84no—
339C2×D3—84no—
340C2×D3—84no—
341C2×D3—84no—
342C2×D3—82no—
343C2×D3—82no—
344C3×C4—22no—
345C3×C4—22no—
346C3×C4—42no—
347Dic3—44no—
348Dic3—84no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—44no—
358C23—44no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—64no—
364C23—82no—
365C23—82no—
366C23—81no—
367C23—81no—
368C23—122no—
369C23—124no—
370C23—124no—
371C23—122no—
372C23—124no—
373C23—122no—
374C23—124no—
375C23—124no—
376C23—122no—
377C23—122no—
378C23—124no—
379C23—122no—
380C23—242no—
381C23—242no—
382C23—242no—
383C23—242no—
384C23—242no—
385C23—241no—
386C23—241no—
387C23—241no—
388C23—241no—
389C23—242no—
390C23#82114yescomplemented
391C23#88211yescharacteristic
392C23#88611yescharacteristic
393C23C2×#20411yescharacteristic
394C23C2×D2411yescharacteristic
395C23#90811yescharacteristic
396C23#90911yescharacteristic
397C23C2×(C2×D3)⋊C411yescharacteristic
398D4—68no—
399D4—128no—
400D4—124no—
401D4—1232no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—64no—
410C2×C4—81no—
411C2×C4—124no—
412C2×C4—124no—
413C2×C4—122no—
414C2×C4—122no—
415C2×C4—124no—
416C2×C4—124no—
417C2×C4—124no—
418C2×C4—248no—
419C2×C4#67⋊C412yes—
420C2×C4#93912yes—
421C8—82no—
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—44no—
427C2×C3#159512yes—
428C2×C3#159612yes—
429C2×C3#187411yescharacteristic
430C2×C3#188711yescharacteristic
431C2×C3#188911yescharacteristic
432D3—84no—
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—42no—
446C22—42no—
447C22—82no—
448C22—82no—
449C22—82no—
450C22—124no—
451C22—124no—
452C22—124no—
453C22—242no—
454C22—242no—
455C22—244no—
456C22—244no—
457C22—244no—
458C22—244no—
459C22—242no—
460C22—242no—
461C22#473211yescharacteristic
462C22#473511yescharacteristic
463C22#473611yescharacteristic
464C22#473811yescharacteristic
465C22#474112yes—
466C22#474712yes—
467C22#495811yescharacteristic
468C22C2×#82111yescharacteristic
469C22#513311yescharacteristic
470C4—22no—
471C4—22no—
472C4—42no—
473C4—124no—
474C4—244no—
475C3#901711yescharacteristic, complemented, Sylow
476C2—21no—
477C2—21no—
478C2—21no—
479C2—21no—
480C2—44no—
481C2—244no—
482C2#6810612yes—
483C2#6811412yes—
484C2#6993211yescharacteristic
485C2#7002211yescharacteristic
486C2#7007011yescharacteristic
subgroupG/H
centerC23#882
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#323C2
socleC23×C3#120
prime radicalC2×#4960C2
cosocleC23×C3×C4C23
hypercenterC2×#323D3
hypocenterC3#9017
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
2. #82094C2semidirect
3. C2×C3×#323C2semidirect
4. #69959C2semidirect
5. #67675C2semidirect
18. #4960C4semidirect
24. C2×#323D3semidirect
41. C22×D12C2×C4semidirect
54. C2×C4⋊Dic3D4semidirect
60. C3⋊#95C2×C4semidirect
191. C23×C3#102semidirect
390. C23#821semidirect
475. C3#9017semidirect

Maximal subgroups

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

subgroupindex
C2×#49602
#820942
C2×C3×#3232
#699592
#676752
#90173