Finite Group Databasebeta

← #10692394

Subgroups — #10692394

Order 768.

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

normal subgroups only

This group has 774 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#73454C211yesmaximal, characteristic, complemented
3C3×#1507C211yesmaximal, characteristic, complemented
4#71686C211yesmaximal, characteristic, complemented
5#69991C211yesmaximal, characteristic, complemented
6#67685C211yesmaximal, characteristic, complemented
7#67618C211yesmaximal, characteristic, complemented
8#9368—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C2×C3×#121C2211yescharacteristic
11C2×#932C412yescomplemented
12C2×C4×Dic6—21no—
13C3×#407C2211yescharacteristic
14C3×C4×C22⋊C4C2211yescharacteristic
15C2×#834—22no—
16#5212C2211yescharacteristic
17C4×#217C2211yescharacteristic
18#5207C2211yescharacteristic
19C2×#825—22no—
20#5040—21no—
21#5037—22no—
22#5036—21no—
23#4966C412yescomplemented
24#4957—22no—
25#4939—21no—
26#4667—21no—
27#4666—21no—
28C3×#327—21no—
29C3×#327—21no—
30#3006—31no—
31#2341—31no—
32#1964—31no—
33#1789—31no—
34#1507D311yescharacteristic, complemented
35#1305—31no—
36#1297—31no—
37C22×C3×D4—21no—
38C22×C3×D4C2×C411yescharacteristic
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×Dic6D411yescharacteristic
49C2×C3×M16—21no—
50C3×#121—21no—
51C3×#121—21no—
52C3×#121—21no—
53C3×#121—21no—
54C2×C3×C4⋊C4C2×C411yescharacteristic
55C2×C3×C22⋊C4—21no—
56C2×C3×C22⋊C4—21no—
57C2×C3×C42—21no—
58C2×#217—21no—
59C2×#217D412yes—
60C2×#214—22no—
61#932—44no—
62C2×#213—22no—
63C2×(C2×D3)⋊C4—22no—
64C2×(C2×D3)⋊C4—22no—
65C2×C4⋊Dic3—21no—
66C2×Dic3⋊C4—22no—
67C2×C4×Dic3—22no—
68C2×C4×Dic3—21no—
69C2×#208C2×C412yes—
70C4×Dic6—44no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×#95D411yescharacteristic
74C3×#95D411yescharacteristic
75#834—44no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78#830—21no—
79#830—21no—
80#825—44no—
81#810—22no—
82#809—22no—
83C22×#41—31no—
84C2×AGL(1,ℤ/8)—32no—
85C2×#121C2×D311yescharacteristic
86C2×C4×Q8—61no—
87#449—62no—
88#448—61no—
89#447—61no—
90C2×#104—62no—
91#417—32no—
92C2×#99—62no—
93#407—31no—
94#407C2×D311yescharacteristic
95#390—61no—
96#390—62no—
97#380—31no—
98#380—61no—
99C4×C22⋊C4—31no—
100C4×C22⋊C4—61no—
101C4×C22⋊C4C2×D311yescharacteristic
102#327—21no—
103#327—21no—
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—21no—
116C22×C3×C4—21no—
117C22×C3×C4—42no—
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—22no—
129C22×Dic3—41no—
130#235—24no—
131#235—24no—
132#235—42no—
133#235—42no—
134#235—44no—
135#235—44no—
136C2×D12—21no—
137C2×D12—42no—
138C2×D12C22⋊C412yes—
139C2×C4×D3—22no—
140C2×C4×D3—22no—
141C2×C4×D3—44no—
142C2×Dic6—21no—
143C2×Dic6—24no—
144C2×Dic6C22⋊C412yes—
145C3×M16—42no—
146C2×C3×C8—41no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—21no—
149C3×C22⋊C4—42no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C42—42no—
153#217—22no—
154#217—41no—
155#217—42no—
156#214—44no—
157#213—44no—
158(C2×D3)⋊C4—44no—
159(C2×D3)⋊C4—82no—
160C4⋊Dic3—42no—
161Dic3⋊C4—44no—
162C4×Dic3—44no—
163C4×Dic3—42no—
164#208—42no—
165C2×C3⋊C8—22no—
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
176C23×C4—61no—
177C23×C4C22×D311yescharacteristic
178AGL(1,ℤ/8)—124no—
179C2×SD16—62no—
180C2×D8—62no—
181C2×M16—21no—
182C2×M16—32no—
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⋊C4C4×D311yescharacteristic
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—32no—
194C2×C22⋊C4—61no—
195C2×C22⋊C4—62no—
196C2×C22⋊C4—62no—
197C2×C42—21no—
198C2×C42—62no—
199C2×C42—61no—
200#104—124no—
201#103—62no—
202#102—62no—
203#99—124no—
204C22⋊C8—21no—
205C22⋊C8—21no—
206C22⋊C8—61no—
207C22⋊C8—61no—
208#95—61no—
209#95—61no—
210#95#6711yescharacteristic
211#95#6711yescharacteristic
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—22no—
224C22×D3—42no—
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—22no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—42no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
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—24no—
254C2×Dic3—22no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3—42no—
258C2×Dic3—42no—
259C2×Dic3—81no—
260C2×Dic3—81no—
261D12—22no—
262D12—22no—
263D12—42no—
264C4×D3—44no—
265C4×D3—44no—
266Dic6—24no—
267Dic6—22no—
268Dic6#9914yescomplemented
269C3×C8—81no—
270C3⋊C8—42no—
271C24—21no—
272C24—61no—
273C24D3×D411yescharacteristic
274#41—64no—
275#41—64no—
276#41—122no—
277#41—122no—
278#41—124no—
279#41—124no—
280C2×Q8—32no—
281C2×Q8—64no—
282C2×Q8—61no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—32no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—62no—
292C2×D4—62no—
293C2×D4—61no—
294C2×D4—122no—
295C2×D4—124no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—42no—
300C22×C4—61no—
301C22×C4—62no—
302C22×C4—62no—
303C22×C4—62no—
304C22×C4—124no—
305C22×C4—121no—
306C22×C4C2×C4×D311yescharacteristic
307C22×C4#23511yescharacteristic
308C22×C4D3×D411yescharacteristic
309C22×C4D3×D411yescharacteristic
310C22×C4#23711yescharacteristic
311C22×C4C2×#6711yescharacteristic
312SD16—124no—
313D8—124no—
314M16—42no—
315M16—122no—
316C2×C8—41no—
317C2×C8—62no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—122no—
321C4⋊C4—124no—
322C22⋊C4—42no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—62no—
326C22⋊C4—124no—
327C22⋊C4—121no—
328C22⋊C4—122no—
329C22⋊C4—242no—
330C42—42no—
331C42—122no—
332C42—124no—
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—81no—
342C22×C3—81no—
343C22×C3—81no—
344C22×C3—81no—
345C22×C3—81no—
346C22×C3—162no—
347C22×C3#38611yescharacteristic
348C22×C3#45311yescharacteristic
349C22×C3#45411yescharacteristic
350C2×D3—42no—
351C2×D3—42no—
352C2×D3—42no—
353C2×D3—82no—
354C2×D3—82no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—42no—
359C3×C4—81no—
360C3×C4—81no—
361C3×C4C2×#9912yes—
362Dic3—24no—
363Dic3—82no—
364Dic3—82no—
365C23—21no—
366C23—21no—
367C23—41no—
368C23—41no—
369C23—42no—
370C23—41no—
371C23—62no—
372C23—81no—
373C23—81no—
374C23—122no—
375C23—122no—
376C23—121no—
377C23—121no—
378C23#67⋊C411yescharacteristic
379C23#88411yescharacteristic
380C23#93711yescharacteristic
381Q8—34no—
382Q8—64no—
383Q8—62no—
384D4—42no—
385D4—62no—
386D4—62no—
387D4—81no—
388D4—82no—
389D4—81no—
390D4—81no—
391D4—122no—
392D4—124no—
393D4—124no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—64no—
409C2×C4—62no—
410C2×C4—124no—
411C2×C4—124no—
412C2×C4—122no—
413C2×C4—122no—
414C2×C4—122no—
415C2×C4—122no—
416C2×C4—241no—
417C2×C4—241no—
418C2×C4D3×C22⋊C411yescharacteristic
419C2×C4#88311yescharacteristic
420C2×C4C4×#6711yescharacteristic
421C2×C4#93811yescharacteristic
422C8—81no—
423C8—122no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—41no—
427C2×C3—41no—
428C2×C3—161no—
429C2×C3#188311yescharacteristic
430C2×C3#189811yescharacteristic
431C2×C3#190611yescharacteristic
432D3—82no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22—122no—
447C22—122no—
448C22—122no—
449C22—162no—
450C22—242no—
451C22—242no—
452C22#495611yescharacteristic
453C22#519811yescharacteristic
454C22#520111yescharacteristic
455C4—21no—
456C4—21no—
457C4—21no—
458C4—42no—
459C4—64no—
460C4—81no—
461C4—81no—
462C4—242no—
463C4—242no—
464C4D3×#9912yes—
465C3#936811yescharacteristic, complemented, Sylow
466C2—21no—
467C2—21no—
468C2—41no—
469C2—41no—
470C2—161no—
471C2—242no—
472C2#7003611yescharacteristic
473C2#7032811yescharacteristic
474C2#7039111yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1507C2
socleC22×C3#386
prime radical#82624C2
cosocleC23×C3×C4C23
hypercenter#1507D3
hypocenterC3#9368
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1507

Complements

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

NH ≅ G/Ntype
2. #73454C2semidirect
3. C3×#1507C2semidirect
4. #71686C2semidirect
5. #69991C2semidirect
6. #67685C2semidirect
7. #67618C2semidirect
11. C2×#932C4semidirect
23. #4966C4semidirect
34. #1507D3semidirect
268. Dic6#99semidirect
465. C3#9368semidirect

Maximal subgroups

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

subgroupindex
#826242
#734542
C3×#15072
#716862
#699912
#676852
#676182
#93683