Finite Group Databasebeta

← #10690196

Subgroups — #10690196

Order 768.

1296 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1296 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C3×#2315C211yesmaximal, characteristic, complemented
2C2×#4960C211yesmaximal, characteristic
3C2×#4525C211yesmaximal, characteristic, complemented
4#69521C212yesmaximal, complemented
5#67624C212yesmaximal, complemented
6#9017—31nomaximal, Sylow
7C23×#67—21no—
8C23×D12C2211yescharacteristic
9C22×C3×C22⋊C4—21no—
10C22×C3×C22⋊C4C2211yescharacteristic
11C22×(C2×D3)⋊C4—21no—
12C3×#385—24no—
13C3×#380—21no—
14C3×#380C2212yes—
15C2×C3⋊#95C2211yescharacteristic
16C2×#830C2212yes—
17C2×#809—24no—
18#4960—22no—
19#4960C414yescomplemented
20#4525—22no—
21C2×#392—31no—
22#2315D311yescharacteristic, complemented
23#1786—32no—
24C2×#323—31no—
25#1294—32no—
26C25×C3D411yescharacteristic
27C24×D3—21no—
28C23×C3×C4—22no—
29C23×C3×C4D411yescharacteristic
30C23×C3×C4C2311yescharacteristic
31C22×#67—24no—
32C22×#67—44no—
33C22×#67—44no—
34C22×#67—42no—
35C22×#67—42no—
36C23×Dic3—21no—
37C22×D12—22no—
38C22×D12—22no—
39C22×D12—21no—
40C22×D12—44no—
41C22×D12C2×C412yescomplemented
42C2×C3×C4⋊C4—22no—
43C2×C3×C4⋊C4D414yescomplemented
44C2×C3×C22⋊C4—22no—
45C2×C3×C22⋊C4—22no—
46C2×C3×C22⋊C4—22no—
47C2×C3×C22⋊C4—22no—
48C2×C3×C22⋊C4—42no—
49C2×C3×C22⋊C4—42no—
50C2×C3×C22⋊C4C2×C412yes—
51C2×C3×C22⋊C4D412yes—
52C2×(C2×D3)⋊C4—24no—
53C2×(C2×D3)⋊C4—44no—
54C22×C3⋊C8—22no—
55C3×#95—22no—
56C3×#95—22no—
57C3⋊#95—21no—
58C3⋊#95C2×C412yescomplemented
59#830—22no—
60#830—22no—
61#809—48no—
62C23×D4—31no—
63C23×D4—61no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—61no—
66C22×C22⋊C4C2×D311yescharacteristic
67C2×#102—64no—
68C2×C22⋊C8—32no—
69#392—34no—
70#392—62no—
71#385—24no—
72#380—21no—
73#380C2×D312yes—
74C2×#95—31no—
75#323—62no—
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—24no—
100C22×C3×C4—42no—
101C22×C3×C4—42no—
102C22×C3×C4—44no—
103C22×C3×C4—82no—
104C22×C3×C4—82no—
105C22×C3×C4C22⋊C412yes—
106C22×C3×C4C22×C411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D412yes—
109C22×C3×C4#4112yes—
110C2×#67—48no—
111C2×#67—48no—
112C2×#67—416no—
113C2×#67—416no—
114C2×#67—48no—
115C22×Dic3—24no—
116C22×Dic3—22no—
117C22×Dic3—42no—
118C22×Dic3—41no—
119C22×Dic3—41no—
120C2×D12—28no—
121C2×D12—24no—
122C2×D12—44no—
123C2×D12—48no—
124C2×D12—44no—
125C2×D12—44no—
126C2×D12—42no—
127C3×C4⋊C4—44no—
128C3×C4⋊C4—82no—
129C3×C22⋊C4—24no—
130C3×C22⋊C4—24no—
131C3×C22⋊C4—44no—
132C3×C22⋊C4—42no—
133C3×C22⋊C4—42no—
134(C2×D3)⋊C4—48no—
135C2×C3⋊C8—44no—
136C2×C3⋊C8—42no—
137C25—61no—
138C25#6711yescharacteristic
139C22×D4—32no—
140C22×D4—64no—
141C22×D4—61no—
142C22×D4—62no—
143C22×D4—62no—
144C22×D4—124no—
145C22×D4—124no—
146C22×D4—124no—
147C22×D4—122no—
148C22×D4—122no—
149C23×C4—22no—
150C23×C4—61no—
151C23×C4#6711yescharacteristic
152C23×C4C22×D311yescharacteristic
153C22×C8—62no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4#6714yescomplemented
156C2×C22⋊C4—22no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—42no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—64no—
163C2×C22⋊C4—124no—
164C2×C22⋊C4C4×D312yes—
165C2×C22⋊C4D1212yes—
166#102—128no—
167C22⋊C8—62no—
168C22⋊C8—62no—
169#95—22no—
170#95—22no—
171#95—32no—
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—44no—
230C2×C3×C4—44no—
231C2×C3×C4—44no—
232C2×C3×C4—81no—
233C2×C3×C4—88no—
234C2×C3×C4C4×D412yes—
235C2×C3×C4#12112yes—
236#67—432no—
237C2×Dic3—24no—
238C2×Dic3—44no—
239C2×Dic3—44no—
240C2×Dic3—42no—
241C2×Dic3—42no—
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(C2×D3)⋊C412yes—
264C24#21312yes—
265C24#21512yes—
266C24C2×#6711yescharacteristic
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—24no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—64no—
286C22×C4—62no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4—122no—
290C22×C4—121no—
291C22×C4—121no—
292C22×C4(C2×D3)⋊C412yes—
293C22×C4C2×C4×D311yescharacteristic
294C22×C4C2×D1211yescharacteristic
295C22×C4#23512yes—
296C22×C4C2×#6712yes—
297C2×C8—124no—
298C2×C8—122no—
299C4⋊C4—44no—
300C4⋊C4—82no—
301C22⋊C4—24no—
302C22⋊C4—24no—
303C22⋊C4—44no—
304C22⋊C4—42no—
305C22⋊C4—42no—
306C22⋊C4—128no—
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—
347C3×C4—84no—
348Dic3—44no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—42no—
358C23—42no—
359C23—44no—
360C23—44no—
361C23—42no—
362C23—42no—
363C23—64no—
364C23—82no—
365C23—82no—
366C23—81no—
367C23—81no—
368C23—122no—
369C23—124no—
370C23—124no—
371C23—122no—
372C23—122no—
373C23—124no—
374C23—124no—
375C23—124no—
376C23—122no—
377C23—122no—
378C23—122no—
379C23—124no—
380C23—242no—
381C23—242no—
382C23—242no—
383C23—242no—
384C23—242no—
385C23—241no—
386C23—241no—
387C23—241no—
388C23—241no—
389C23—242no—
390C23#80914yescomplemented
391C23C2×(C2×D3)⋊C411yescharacteristic
392C23#92911yescharacteristic
393C23C2×#21311yescharacteristic
394C23#93211yescharacteristic
395C23C2×#21511yescharacteristic
396C23#94211yescharacteristic
397C23#95311yescharacteristic
398D4—68no—
399D4—128no—
400D4—124no—
401D4—1232no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—64no—
413C2×C4—81no—
414C2×C4—88no—
415C2×C4—124no—
416C2×C4—124no—
417C2×C4—122no—
418C2×C4—122no—
419C2×C4C4×#6712yes—
420C2×C4#93012yes—
421C8—242no—
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—242no—
458C22—244no—
459C22—244no—
460C22—242no—
461C22C2×#80911yescharacteristic
462C22#496611yescharacteristic
463C22#496711yescharacteristic
464C22#496811yescharacteristic
465C22#521311yescharacteristic
466C22#521711yescharacteristic
467C22#522711yescharacteristic
468C22#524112yes—
469C22#524412yes—
470C4—22no—
471C4—22no—
472C4—42no—
473C4—84no—
474C4—124no—
475C3#901711yescharacteristic, complemented, Sylow
476C2—21no—
477C2—21no—
478C2—21no—
479C2—21no—
480C2—44no—
481C2—244no—
482C2#7156011yescharacteristic
483C2#7165011yescharacteristic
484C2#7165411yescharacteristic
485C2#7184512yes—
486C2#7189212yes—
subgroupG/H
centerC23#953
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2315C2
socleC23×C3#120
prime radicalC2×#4960C2
cosocleC23×C3×C4C23
hypercenter#2315D3
hypocenterC3#9017
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2315

Complements

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

NH ≅ G/Ntype
1. C3×#2315C2semidirect
3. C2×#4525C2semidirect
4. #69521C2semidirect
5. #67624C2semidirect
19. #4960C4semidirect
22. #2315D3semidirect
41. C22×D12C2×C4semidirect
43. C2×C3×C4⋊C4D4semidirect
58. C3⋊#95C2×C4semidirect
155. C2×C4⋊C4#67semidirect
191. C23×C3#102semidirect
390. C23#809semidirect
475. C3#9017semidirect

Maximal subgroups

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

subgroupindex
C3×#23152
C2×#49602
C2×#45252
#695212
#676242
#90173