Finite Group Databasebeta

← #10692379

Subgroups — #10692379

Order 768.

802 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 97 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
1#82624C211yesmaximal, characteristic
2#74008C211yesmaximal, characteristic, complemented
3C3×#1964C211yesmaximal, characteristic, complemented
4#71673C211yesmaximal, characteristic, complemented
5#69454C211yesmaximal, characteristic, complemented
6#67626C211yesmaximal, characteristic, complemented
7#67618C211yesmaximal, characteristic, complemented
8#9365—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C2×C3×#121C2211yescharacteristic
11C2×#932C412yescomplemented
12C2×#893—21no—
13C2×C3×#99—22no—
14C3×#380C2211yescharacteristic
15C3×C4×C22⋊C4C2211yescharacteristic
16C4×#217C2211yescharacteristic
17#5207C2211yescharacteristic
18#5207C2211yescharacteristic
19#5040—21no—
20#5037—22no—
21#5036—21no—
22#4973—21no—
23#4966C412yescomplemented
24#4956—22no—
25C2×#805—22no—
26#4664—21no—
27D3×#95—21no—
28#4537—21no—
29#4537—21no—
30#3006—31no—
31#2339—31no—
32#1964D311yescharacteristic, complemented
33#1789—31no—
34#1507—31no—
35#1305—31no—
36#1297—31no—
37C22×C3×D4—21no—
38C22×C3×D4C2×C411yescharacteristic, complemented
39C23×C3×C4C2311yescharacteristic
40C22×#67—21no—
41C2×#235—21no—
42C2×#235—21no—
43C2×#235—44no—
44C2×#235C2×C412yes—
45C2×#235D412yes—
46C22×D12D411yescharacteristic
47C22×C4×D3—21no—
48C22×C4×D3—22no—
49C22×Dic6D411yescharacteristic
50C3×#121—21no—
51C3×#121—21no—
52C3×#121—21no—
53C3×#121—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4C2×C411yescharacteristic, complemented
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4D412yes—
59C2×C3×C42—21no—
60C2×#217—21no—
61C2×#214—22no—
62#932—44no—
63C2×#213—22no—
64C2×(C2×D3)⋊C4—22no—
65C2×C4×Dic3—21no—
66C2×#208—21no—
67C2×#208C2×C412yescomplemented
68#893—44no—
69C3×#99—44no—
70C3×#95—21no—
71C3×#95—21no—
72C3⋊#95—22no—
73C3⋊#95—22no—
74C3⋊#95D411yescharacteristic
75C3⋊#95D411yescharacteristic
76#830—21no—
77#830—21no—
78#830—21no—
79#830—21no—
80#810—22no—
81#809—22no—
82#805—44no—
83C22×#41—31no—
84C2×AGL(1,ℤ/8)—32no—
85C2×#121C2×D311yescharacteristic
86C2×C4×D4—61no—
87#449—62no—
88#448—61no—
89#447—61no—
90C2×#104—62no—
91#417—32no—
92C2×#99—22no—
93#407—31no—
94#407—31no—
95#391—61no—
96#386—62no—
97#381—61no—
98#380C2×D311yescharacteristic
99C4×C22⋊C4—31no—
100C4×C22⋊C4C2×D311yescharacteristic
101C2×#95—61no—
102#327—61no—
103#327—61no—
104C24×C3—21no—
105C24×C3C2×D411yescharacteristic
106C23×D3—21no—
107C2×C3×D4—21no—
108C2×C3×D4—21no—
109C2×C3×D4—42no—
110C2×C3×D4—42no—
111C2×C3×D4—42no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C22×C3×C4—21no—
115C22×C3×C4—22no—
116C22×C3×C4—42no—
117C22×C3×C4—41no—
118C22×C3×C4C22×C411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4#4111yescharacteristic
123C22×C3×C4#4111yescharacteristic
124C2×#67—22no—
125C2×#67—22no—
126C2×#67—44no—
127C22×Dic3—21no—
128C22×Dic3—21no—
129C22×Dic3—21no—
130#235—24no—
131#235—24no—
132#235—42no—
133#235—42no—
134#235—44no—
135#235—44no—
136C2×D12—21no—
137C2×D12—24no—
138C2×D12C22⋊C412yes—
139C2×C4×D3—22no—
140C2×C4×D3—22no—
141C2×C4×D3—44no—
142C2×C4×D3—44no—
143C2×C4×D3—44no—
144C2×C4×D3—82no—
145C2×Dic6—21no—
146C2×Dic6—42no—
147C2×Dic6C22⋊C412yes—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—42no—
151C3×C22⋊C4—22no—
152C3×C22⋊C4—41no—
153C3×C22⋊C4—41no—
154C3×C22⋊C4—41no—
155C3×C22⋊C4—42no—
156C3×C42—42no—
157#217—42no—
158#214—44no—
159#213—44no—
160(C2×D3)⋊C4—44no—
161C4×Dic3—42no—
162#208—42no—
163#208—42no—
164C2×C3⋊C8—22no—
165C2×C3⋊C8—41no—
166C2×#41—32no—
167C2×#41—32no—
168C2×#41—61no—
169C2×#41—61no—
170C2×#41—124no—
171C22×Q8—31no—
172C22×D4—21no—
173C22×D4—31no—
174C22×D4—61no—
175C22×D4C4×D311yescharacteristic, complemented
176C23×C4—62no—
177C23×C4—61no—
178C23×C4C22×D311yescharacteristic
179AGL(1,ℤ/8)—124no—
180C2×SD16—62no—
181C2×D8—62no—
182C2×M16—32no—
183C2×M16—61no—
184#121—21no—
185#121—21no—
186#121—21no—
187#121—21no—
188C4×D4—124no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4C4×D311yescharacteristic, complemented
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—62no—
194C2×C22⋊C4—61no—
195C2×C22⋊C4#6712yes—
196C2×C42—21no—
197C2×C42—61no—
198#104—124no—
199#103—62no—
200#102—62no—
201#99—44no—
202C22⋊C8—61no—
203C22⋊C8—61no—
204C22⋊C8—61no—
205C22⋊C8—61no—
206#95—21no—
207#95—21no—
208#95—31no—
209#95—31no—
210#95—62no—
211#95—62no—
212C23×C3—21no—
213C23×C3—21no—
214C23×C3—41no—
215C23×C3—41no—
216C23×C3—42no—
217C23×C3—41no—
218C23×C3—81no—
219C23×C3—81no—
220C23×C3C4×D411yescharacteristic
221C23×C3#12011yescharacteristic
222C23×C3#12111yescharacteristic
223C22×D3—24no—
224C22×D3—22no—
225C22×D3—42no—
226C22×D3—41no—
227C22×D3—41no—
228C3×D4—42no—
229C3×D4—81no—
230C3×D4—82no—
231C3×D4—81no—
232C3×D4—81no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—41no—
244C2×C3×C4—42no—
245C2×C3×C4—81no—
246C2×C3×C4—81no—
247C2×C3×C4C2×C22⋊C411yescharacteristic
248C2×C3×C4C4×D411yescharacteristic
249C2×C3×C4#12111yescharacteristic
250C2×C3×C4#12311yescharacteristic
251#67—44no—
252#67—44no—
253C2×Dic3—22no—
254C2×Dic3—22no—
255C2×Dic3—41no—
256C2×Dic3—42no—
257C2×Dic3—41no—
258C2×Dic3—41no—
259C2×Dic3—41no—
260C2×Dic3—41no—
261D12—24no—
262D12—22no—
263D12#9914yescomplemented
264C4×D3—44no—
265C4×D3—44no—
266C4×D3—88no—
267Dic6—22no—
268Dic6—22no—
269Dic6—42no—
270C3⋊C8—42no—
271C3⋊C8—81no—
272C24—21no—
273C24—61no—
274C24C2×#6711yescharacteristic
275#41—64no—
276#41—64no—
277#41—122no—
278#41—122no—
279#41—124no—
280#41—124no—
281C2×Q8—32no—
282C2×Q8—61no—
283C2×Q8—122no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—32no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—64no—
293C2×D4—62no—
294C2×D4—62no—
295C2×D4—61no—
296C2×D4—124no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—61no—
302C22×C4—61no—
303C22×C4—62no—
304C22×C4—62no—
305C22×C4—61no—
306C22×C4—124no—
307C22×C4—124no—
308C22×C4—124no—
309C22×C4—242no—
310C22×C4C2×C4×D311yescharacteristic
311C22×C4#23511yescharacteristic
312C22×C4D3×D411yescharacteristic
313C22×C4D3×D411yescharacteristic
314C22×C4D3×D411yescharacteristic
315C22×C4#23711yescharacteristic
316SD16—124no—
317D8—124no—
318M16—122no—
319M16—122no—
320C2×C8—62no—
321C2×C8—121no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—42no—
325C22⋊C4—22no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—42no—
330C22⋊C4—124no—
331C22⋊C4—122no—
332C42—42no—
333C42—122no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—81no—
343C22×C3—81no—
344C22×C3—81no—
345C22×C3—81no—
346C22×C3—81no—
347C22×C3—162no—
348C22×C3#38611yescharacteristic
349C22×C3#45311yescharacteristic
350C22×C3#45411yescharacteristic
351C2×D3—24no—
352C2×D3—44no—
353C2×D3—44no—
354C2×D3—42no—
355C2×D3—42no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—82no—
360C3×C4—82no—
361C3×C4—81no—
362C3×C4C2×#9912yes—
363Dic3—42no—
364Dic3—42no—
365Dic3—81no—
366C23—21no—
367C23—21no—
368C23—41no—
369C23—41no—
370C23—42no—
371C23—41no—
372C23—64no—
373C23—62no—
374C23—81no—
375C23—81no—
376C23—122no—
377C23—121no—
378C23—121no—
379C23C4×#6711yescharacteristic
380C23#93711yescharacteristic
381C23#93811yescharacteristic
382Q8—62no—
383Q8—62no—
384Q8—122no—
385D4—34no—
386D4—42no—
387D4—64no—
388D4—62no—
389D4—81no—
390D4—82no—
391D4—81no—
392D4—81no—
393D4—124no—
394D4—124no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—62no—
408C2×C4—62no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—124no—
412C2×C4—124no—
413C2×C4—121no—
414C2×C4—122no—
415C2×C4—121no—
416C2×C4—121no—
417C2×C4—121no—
418C2×C4—121no—
419C2×C4—248no—
420C2×C4D3×C22⋊C411yescharacteristic
421C2×C4#67⋊C411yescharacteristic
422C2×C4#88311yescharacteristic
423C2×C4#88411yescharacteristic
424C8—122no—
425C8—241no—
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—41no—
429C2×C3—41no—
430C2×C3—161no—
431C2×C3#188311yescharacteristic
432C2×C3#189811yescharacteristic
433C2×C3#190611yescharacteristic
434D3—44no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—64no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22—124no—
450C22—124no—
451C22—122no—
452C22—122no—
453C22—162no—
454C22#495611yescharacteristic
455C22#506211yescharacteristic
456C22#507711yescharacteristic
457C4—21no—
458C4—21no—
459C4—21no—
460C4—82no—
461C4—82no—
462C4—81no—
463C4—122no—
464C4—122no—
465C4—241no—
466C4D3×#9912yes—
467C3#936511yescharacteristic, complemented, Sylow
468C2—21no—
469C2—21no—
470C2—41no—
471C2—41no—
472C2—124no—
473C2—161no—
474C2#6966311yescharacteristic
475C2#7003611yescharacteristic
476C2#7005511yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1964C2
socleC22×C3#386
prime radical#82624C2
cosocleC23×C3×C4C23
hypercenter#1964D3
hypocenterC3#9365
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1964

Complements

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

NH ≅ G/Ntype
2. #74008C2semidirect
3. C3×#1964C2semidirect
4. #71673C2semidirect
5. #69454C2semidirect
6. #67626C2semidirect
7. #67618C2semidirect
11. C2×#932C4semidirect
23. #4966C4semidirect
32. #1964D3semidirect
38. C22×C3×D4C2×C4semidirect
55. C2×C3×C4⋊C4C2×C4semidirect
67. C2×#208C2×C4semidirect
175. C22×D4C4×D3semidirect
190. C2×C4⋊C4C4×D3semidirect
263. D12#99semidirect
467. C3#9365semidirect

Maximal subgroups

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

subgroupindex
#826242
#740082
C3×#19642
#716732
#694542
#676262
#676182
#93653