Finite Group Databasebeta

← #10692399

Subgroups — #10692399

Order 768.

774 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 774 conjugacy classes of subgroups — too many to draw. Only its 115 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×#3006C211yesmaximal, characteristic
2#84089C211yesmaximal, characteristic, complemented
3#72138C211yesmaximal, characteristic, complemented
4#71686C211yesmaximal, characteristic, complemented
5#71673C211yesmaximal, characteristic, complemented
6#67684C211yesmaximal, characteristic, complemented
7#67683C211yesmaximal, characteristic, complemented
8#9368—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×C3×AGL(1,ℤ/8)C412yescomplemented
11C2×C3×#121C2211yescharacteristic
12C2×Q8×Dic3—21no—
13C3×#449—22no—
14C3×#448—21no—
15C3×#447—21no—
16C3×#417C412yescomplemented
17C3×#407C2211yescharacteristic
18C2×#837—22no—
19#5221—21no—
20#5221—22no—
21C2×#834—22no—
22#5212—21no—
23#5212C2211yescharacteristic
24C4×#217—21no—
25C4×#217C2211yescharacteristic
26C4×#217C2211yescharacteristic
27#5207C2211yescharacteristic
28#4537—21no—
29#4537—21no—
30#3006D311yescharacteristic
31#2341—31no—
32#1964—31no—
33#1789—31no—
34#1507—31no—
35#1305—31no—
36#1297—31no—
37C2×C3×#41—21no—
38C2×C3×#41—21no—
39C2×C3×#41—44no—
40C2×C3×#41C2×C412yes—
41C2×C3×#41D412yes—
42C22×C3×Q8D411yescharacteristic
43C22×C3×D4—21no—
44C22×C3×D4—21no—
45C22×C3×D4C2×C411yescharacteristic
46C22×C3×D4D411yescharacteristic
47C23×C3×C4—21no—
48C23×C3×C4C2311yescharacteristic
49C3×AGL(1,ℤ/8)—44no—
50C2×C3×SD16—22no—
51C2×C3×D8—22no—
52C2×C3×M16C2×C412yes—
53C3×#121—21no—
54C3×#121—21no—
55C3×#121—21no—
56C3×#121—21no—
57C2×C3×C4⋊C4C2×C411yescharacteristic
58C2×C3×C22⋊C4—21no—
59C2×#217—21no—
60C2×#217—22no—
61C2×#217—21no—
62C2×#217—22no—
63C2×#217D412yes—
64Q8×Dic3—44no—
65C2×C4⋊Dic3—22no—
66C2×C4⋊Dic3—21no—
67C2×C4×Dic3—22no—
68C2×C4×Dic3—21no—
69C2×C4×Dic3—21no—
70C2×#208—21no—
71C3×#103—22no—
72C3×#102—22no—
73C3×C22⋊C8—21no—
74C3×C22⋊C8—21no—
75#837—44no—
76#834—44no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95D411yescharacteristic
80C3⋊#95D411yescharacteristic
81#830—21no—
82#830—21no—
83C22×#41C2×D311yescharacteristic
84C2×AGL(1,ℤ/8)Dic312yescomplemented
85C2×#121C2×D311yescharacteristic
86C2×C4×Q8—61no—
87#449—22no—
88#448—21no—
89#447—21no—
90C2×#104—62no—
91#417Dic312yescomplemented
92C2×#99—62no—
93#407—31no—
94#407C2×D311yescharacteristic
95#390—62no—
96#390—61no—
97#380—31no—
98#380—61no—
99C4×C22⋊C4—31no—
100C4×C22⋊C4—31no—
101C4×C22⋊C4—61no—
102#327—61no—
103#327—61no—
104C24×C3—21no—
105C24×C3—21no—
106C24×C3C2×D411yescharacteristic
107C3×#41—24no—
108C3×#41—24no—
109C3×#41—42no—
110C3×#41—42no—
111C3×#41—44no—
112C3×#41—44no—
113C2×C3×Q8—24no—
114C2×C3×Q8—21no—
115C2×C3×Q8C22⋊C412yes—
116C2×C3×D4—22no—
117C2×C3×D4—22no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—42no—
122C2×C3×D4—44no—
123C2×C3×D4—42no—
124C2×C3×D4—42no—
125C2×C3×D4—42no—
126C2×C3×D4—41no—
127C2×C3×D4—41no—
128C2×C3×D4C22⋊C412yes—
129C22×C3×C4—21no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—21no—
133C22×C3×C4—42no—
134C22×C3×C4—44no—
135C22×C3×C4C22×C411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4#4111yescharacteristic
140C22×C3×C4#4111yescharacteristic
141C22×Dic3—21no—
142C22×Dic3—22no—
143C22×Dic3—21no—
144C22×Dic3—41no—
145C3×SD16—44no—
146C3×D8—44no—
147C3×M16—42no—
148C2×C3×C8—22no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C22⋊C4—41no—
152C3×C22⋊C4—41no—
153#217—22no—
154#217—42no—
155#217—44no—
156#217—41no—
157#217—42no—
158#217—82no—
159C4⋊Dic3—44no—
160C4⋊Dic3—42no—
161C4×Dic3—44no—
162C4×Dic3—42no—
163C4×Dic3—42no—
164#208—42no—
165C2×C3⋊C8—41no—
166C2×#41—21no—
167C2×#41—21no—
168C2×#41—44no—
169C2×#41C2×Dic312yes—
170C2×#41#6712yes—
171C22×Q8#6711yescharacteristic
172C22×D4—21no—
173C22×D4—21no—
174C22×D4C2×Dic311yescharacteristic
175C22×D4#6711yescharacteristic
176C23×C4—21no—
177C23×C4C22×D311yescharacteristic
178AGL(1,ℤ/8)—44no—
179C2×SD16—22no—
180C2×D8—22no—
181C2×M16—61no—
182C2×M16C2×Dic312yes—
183#121—21no—
184#121—21no—
185#121—21no—
186#121—21no—
187C4×Q8—124no—
188C2×C4⋊C4—62no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4C2×Dic311yescharacteristic
191C2×C22⋊C4—21no—
192C2×C22⋊C4—32no—
193C2×C22⋊C4—61no—
194C2×C22⋊C4—62no—
195C2×C22⋊C4—61no—
196C2×C22⋊C4—62no—
197C2×C42—62no—
198C2×C42—61no—
199C2×C42—61no—
200#104—124no—
201#103—22no—
202#102—22no—
203#99—124no—
204C22⋊C8—21no—
205C22⋊C8—21no—
206C22⋊C8—61no—
207C22⋊C8—61no—
208#95—31no—
209#95—31no—
210#95—61no—
211#95—61no—
212C23×C3—21no—
213C23×C3—21no—
214C23×C3—22no—
215C23×C3—41no—
216C23×C3—41no—
217C23×C3—42no—
218C23×C3—42no—
219C23×C3—42no—
220C23×C3—41no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—81no—
224C23×C3—81no—
225C23×C3C4×D411yescharacteristic
226C23×C3#12011yescharacteristic
227C23×C3#12111yescharacteristic
228C3×Q8—24no—
229C3×Q8—22no—
230C3×Q8#9914yescomplemented
231C3×D4—22no—
232C3×D4—22no—
233C3×D4—44no—
234C3×D4—44no—
235C3×D4—42no—
236C3×D4—42no—
237C3×D4—81no—
238C3×D4—82no—
239C3×D4—81no—
240C3×D4—81no—
241C2×C3×C4—24no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—44no—
250C2×C3×C4—44no—
251C2×C3×C4—42no—
252C2×C3×C4—41no—
253C2×C3×C4C2×C22⋊C411yescharacteristic
254C2×C3×C4C4×D411yescharacteristic
255C2×C3×C4#12111yescharacteristic
256C2×C3×C4#12311yescharacteristic
257C2×Dic3—22no—
258C2×Dic3—41no—
259C2×Dic3—41no—
260C2×Dic3—42no—
261C2×Dic3—41no—
262C2×Dic3—42no—
263C2×Dic3—42no—
264C2×Dic3—41no—
265C2×Dic3—42no—
266C2×Dic3—41no—
267C2×Dic3—81no—
268C2×Dic3—81no—
269C3×C8—42no—
270C3⋊C8—81no—
271C24—21no—
272C24—21no—
273C24D3×D411yescharacteristic
274#41—24no—
275#41—24no—
276#41—42no—
277#41—42no—
278#41—44no—
279#41—44no—
280C2×Q8—24no—
281C2×Q8—21no—
282C2×Q8#21712yes—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—42no—
289C2×D4—44no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4#21712yes—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—42no—
301C22×C4—44no—
302C22×C4—61no—
303C22×C4—61no—
304C22×C4—62no—
305C22×C4—121no—
306C22×C4D3×D411yescharacteristic
307C22×C4#23711yescharacteristic
308C22×C4#23711yescharacteristic
309C22×C4C22×Dic311yescharacteristic
310C22×C4C2×#6711yescharacteristic
311C22×C4C2×#6711yescharacteristic
312SD16—44no—
313D8—44no—
314M16—42no—
315M16—122no—
316C2×C8—22no—
317C2×C8—121no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—124no—
321C4⋊C4—122no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—62no—
325C22⋊C4—122no—
326C22⋊C4—124no—
327C22⋊C4—121no—
328C22⋊C4—122no—
329C22⋊C4—242no—
330C42—124no—
331C42—122no—
332C42—122no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—42no—
342C22×C3—42no—
343C22×C3—42no—
344C22×C3—81no—
345C22×C3—81no—
346C22×C3—82no—
347C22×C3—82no—
348C22×C3—81no—
349C22×C3—81no—
350C22×C3—81no—
351C22×C3—162no—
352C22×C3#38611yescharacteristic
353C22×C3#45311yescharacteristic
354C22×C3#45411yescharacteristic
355C3×C4—21no—
356C3×C4—24no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—81no—
360C3×C4C2×#9912yes—
361Dic3—42no—
362Dic3—82no—
363Dic3—81no—
364Dic3—82no—
365C23—21no—
366C23—21no—
367C23—22no—
368C23—41no—
369C23—41no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—81no—
377C23—81no—
378C23D4×Dic311yescharacteristic
379C23#93711yescharacteristic
380C23#93911yescharacteristic
381Q8—24no—
382Q8—22no—
383Q8#83414yescomplemented
384D4—22no—
385D4—22no—
386D4—44no—
387D4—44no—
388D4—42no—
389D4—42no—
390D4—81no—
391D4—82no—
392D4—81no—
393D4—81no—
394C2×C4—24no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—62no—
407C2×C4—121no—
408C2×C4—121no—
409C2×C4—121no—
410C2×C4—122no—
411C2×C4—122no—
412C2×C4—121no—
413C2×C4—122no—
414C2×C4—122no—
415C2×C4—121no—
416C2×C4—241no—
417C2×C4—241no—
418C2×C4D4×Dic311yescharacteristic
419C2×C4#93511yescharacteristic
420C2×C4#93911yescharacteristic
421C2×C4C2×#21711yescharacteristic
422C8—42no—
423C8—241no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—41no—
427C2×C3—41no—
428C2×C3—82no—
429C2×C3—161no—
430C2×C3#188311yescharacteristic
431C2×C3#189811yescharacteristic
432C2×C3#190611yescharacteristic
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—81no—
445C22—81no—
446C22—82no—
447C22—82no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22—162no—
452C22#519811yescharacteristic
453C22#520111yescharacteristic
454C22#522011yescharacteristic
455C4—21no—
456C4—24no—
457C4—21no—
458C4—21no—
459C4—81no—
460C4—122no—
461C4—242no—
462C4—241no—
463C4—242no—
464C4C2×#83412yes—
465C3#936811yescharacteristic, complemented, Sylow
466C2—21no—
467C2—21no—
468C2—41no—
469C2—41no—
470C2—82no—
471C2—161no—
472C2#7167011yescharacteristic
473C2#7208811yescharacteristic
474C2#7211411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#3006C2
socleC22×C3#386
cosocleC23×C3×C4C23
hypercenter#3006D3
hypocenterC3#9368
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3006

Complements

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

NH ≅ G/Ntype
2. #84089C2semidirect
3. #72138C2semidirect
4. #71686C2semidirect
5. #71673C2semidirect
6. #67684C2semidirect
7. #67683C2semidirect
10. C2×C3×AGL(1,ℤ/8)C4semidirect
16. C3×#417C4semidirect
84. C2×AGL(1,ℤ/8)Dic3semidirect
91. #417Dic3semidirect
230. C3×Q8#99semidirect
383. Q8#834semidirect
465. C3#9368semidirect

Maximal subgroups

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

subgroupindex
C3×#30062
#840892
#721382
#716862
#716732
#676842
#676832
#93683