Finite Group Databasebeta

← #10692398

Subgroups — #10692398

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#78639C211yesmaximal, characteristic
2#73454C211yesmaximal, characteristic, complemented
3#69456C211yesmaximal, characteristic, complemented
4#69454C211yesmaximal, characteristic, complemented
5#69444C211yesmaximal, characteristic, complemented
6C3×#1297C211yesmaximal, characteristic, complemented
7#67684C211yesmaximal, characteristic, complemented
8#9368—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C2×#908C412yescomplemented
12C2×C4×Dic6—21no—
13C3×#407C2211yescharacteristic
14C3×C4×C22⋊C4C2211yescharacteristic
15C3×C4×C22⋊C4C2211yescharacteristic
16#5212C2211yescharacteristic
17#5207C2211yescharacteristic
18#5133C412yescomplemented
19#4957—22no—
20C2×#806—22no—
21#4939—21no—
22C2×#805—22no—
23#4738—21no—
24#4734—22no—
25#4732—21no—
26#4667—21no—
27#4666—21no—
28#4451—21no—
29#4451—21no—
30#3006—31no—
31#2341—31no—
32#1964—31no—
33#1789—31no—
34#1507—31no—
35#1305—31no—
36#1297D311yescharacteristic, complemented
37C23×C3×C4C2311yescharacteristic
38C22×#67—21no—
39C22×#67—21no—
40C2×#235—21no—
41C2×#235—21no—
42C2×#235—44no—
43C2×#235C2×C412yes—
44C2×#235D412yes—
45C22×D12C2×C411yescharacteristic
46C22×D12D411yescharacteristic
47C22×C4×D3—21no—
48C22×Dic6D411yescharacteristic
49C2×C3×M16C2×C412yes—
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C42—21no—
53C2×C3×C42—21no—
54C2×#217D412yes—
55#930—21no—
56#930—21no—
57#930—21no—
58#930—21no—
59C2×(C2×D3)⋊C4—22no—
60C2×(C2×D3)⋊C4—21no—
61C2×(C2×D3)⋊C4—22no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3C2×C411yescharacteristic
64C2×Dic3⋊C4—22no—
65C2×C4×Dic3—22no—
66C2×#208—21no—
67#908—44no—
68C2×D24—22no—
69C2×#204—22no—
70C4×Dic6—44no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×#95—21no—
74C3×#95D411yescharacteristic
75C3×#95D411yescharacteristic
76C3⋊#95—21no—
77#830—21no—
78#830—21no—
79#821—22no—
80#816—22no—
81#806—44no—
82#805—44no—
83C22×#41—31no—
84C2×AGL(1,ℤ/8)—32no—
85C2×#121—31no—
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—61no—
100C4×C22⋊C4C2×D311yescharacteristic
101C4×C22⋊C4C2×D311yescharacteristic
102#327—61no—
103#327—61no—
104C24×C3C2×D411yescharacteristic
105C23×D3—21no—
106C23×D3—21no—
107C22×C3×C4—21no—
108C22×C3×C4—21no—
109C22×C3×C4—42no—
110C22×C3×C4—41no—
111C22×C3×C4C22×C411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4#4111yescharacteristic
116C22×C3×C4#4111yescharacteristic
117C2×#67—22no—
118C2×#67—22no—
119C2×#67—44no—
120C2×#67—42no—
121C2×#67—42no—
122C2×#67—41no—
123C2×#67—41no—
124C22×Dic3—21no—
125C22×Dic3—21no—
126C22×Dic3—22no—
127#235—24no—
128#235—24no—
129#235—42no—
130#235—42no—
131#235—44no—
132#235—44no—
133C2×D12—21no—
134C2×D12—21no—
135C2×D12—21no—
136C2×D12—42no—
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—22no—
147C3×C22⋊C4—22no—
148C3×C22⋊C4—42no—
149C3×C22⋊C4—41no—
150C3×C42—42no—
151C3×C42—42no—
152#217—42no—
153(C2×D3)⋊C4—44no—
154(C2×D3)⋊C4—41no—
155(C2×D3)⋊C4—41no—
156(C2×D3)⋊C4—82no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—42no—
160Dic3⋊C4—44no—
161C4×Dic3—44no—
162#208—42no—
163C2×C3⋊C8—41no—
164D24—44no—
165#204—44no—
166C2×#41—32no—
167C2×#41—32no—
168C2×#41—61no—
169C2×#41—61no—
170C2×#41—124no—
171C22×Q8—31no—
172C22×D4—31no—
173C22×D4—31no—
174C22×D4—61no—
175C22×D4—61no—
176C23×C4—61no—
177C23×C4C22×D311yescharacteristic
178AGL(1,ℤ/8)—124no—
179C2×SD16—62no—
180C2×D8—62no—
181C2×M16—61no—
182C2×M16C4×D312yes—
183#121—61no—
184#121—61no—
185#121—61no—
186#121—61no—
187C4×Q8—124no—
188C2×C4⋊C4—31no—
189C2×C4⋊C4—62no—
190C2×C4⋊C4—61no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—32no—
194C2×C22⋊C4—62no—
195C2×C22⋊C4—61no—
196C2×C22⋊C4—62no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—62no—
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—21no—
209#95—61no—
210#95D1211yescharacteristic
211#95D1211yescharacteristic
212C23×C3—21no—
213C23×C3—21no—
214C23×C3—41no—
215C23×C3—41no—
216C23×C3C4×D411yescharacteristic
217C23×C3#12011yescharacteristic
218C23×C3#12111yescharacteristic
219C22×D3—22no—
220C22×D3—42no—
221C22×D3—42no—
222C22×D3—41no—
223C22×D3—41no—
224C22×D3—41no—
225C22×D3—42no—
226C22×D3—81no—
227C22×D3—81no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—42no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—81no—
242C2×C3×C4—81no—
243C2×C3×C4C2×C22⋊C411yescharacteristic
244C2×C3×C4C4×D411yescharacteristic
245C2×C3×C4#12111yescharacteristic
246C2×C3×C4#12311yescharacteristic
247#67—44no—
248#67—44no—
249#67—82no—
250#67—81no—
251#67—81no—
252C2×Dic3—24no—
253C2×Dic3—22no—
254C2×Dic3—42no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3—41no—
258C2×Dic3—42no—
259D12—22no—
260D12—22no—
261D12—42no—
262D12—42no—
263D12—81no—
264C4×D3—44no—
265C4×D3—44no—
266Dic6—24no—
267Dic6—22no—
268Dic6#9914yescomplemented
269C3×C8—42no—
270C3⋊C8—81no—
271C24—61no—
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—32no—
284C2×D4—62no—
285C2×D4—62no—
286C2×D4—61no—
287C2×D4—61no—
288C2×D4—61no—
289C2×D4—122no—
290C2×D4—122no—
291C2×D4—124no—
292C2×D4—122no—
293C2×D4—122no—
294C2×D4—121no—
295C2×D4—121no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—61no—
301C22×C4—62no—
302C22×C4—62no—
303C22×C4—62no—
304C22×C4—61no—
305C22×C4—124no—
306C22×C4C2×C4×D311yescharacteristic
307C22×C4C2×D1211yescharacteristic
308C22×C4#23511yescharacteristic
309C22×C4D3×D411yescharacteristic
310C22×C4D3×D411yescharacteristic
311C22×C4#23911yescharacteristic
312SD16—124no—
313D8—124no—
314M16—42no—
315M16—122no—
316C2×C8—22no—
317C2×C8—121no—
318C4⋊C4—61no—
319C4⋊C4—61no—
320C4⋊C4—122no—
321C4⋊C4—124no—
322C22⋊C4—22no—
323C22⋊C4—42no—
324C22⋊C4—41no—
325C22⋊C4—124no—
326C22⋊C4—121no—
327C22⋊C4—121no—
328C22⋊C4—122no—
329C22⋊C4—242no—
330C42—42no—
331C42—42no—
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#38611yescharacteristic
344C22×C3#45311yescharacteristic
345C22×C3#45411yescharacteristic
346C2×D3—42no—
347C2×D3—42no—
348C2×D3—42no—
349C2×D3—82no—
350C2×D3—82no—
351C2×D3—81no—
352C2×D3—81no—
353C2×D3—81no—
354C2×D3—162no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—42no—
359C3×C4—81no—
360C3×C4—82no—
361C3×C4C2×#9912yes—
362Dic3—24no—
363Dic3—82no—
364Dic3—81no—
365C23—21no—
366C23—21no—
367C23—41no—
368C23—41no—
369C23—62no—
370C23—122no—
371C23—122no—
372C23—122no—
373C23—121no—
374C23—121no—
375C23—121no—
376C23—241no—
377C23—241no—
378C23#88211yescharacteristic
379C23#88411yescharacteristic
380C23#89311yescharacteristic
381Q8—34no—
382Q8—64no—
383Q8—62no—
384D4—62no—
385D4—62no—
386D4—122no—
387D4—124no—
388D4—124no—
389D4—122no—
390D4—241no—
391D4—242no—
392D4—241no—
393D4—241no—
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—42no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—64no—
408C2×C4—62no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—124no—
412C2×C4—124no—
413C2×C4—122no—
414C2×C4—122no—
415C2×C4—122no—
416C2×C4—121no—
417C2×C4—122no—
418C2×C4C4×D1211yescharacteristic
419C2×C4D3×C22⋊C411yescharacteristic
420C2×C4#89411yescharacteristic
421C2×C4#89511yescharacteristic
422C8—42no—
423C8—241no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—41no—
427C2×C3—41no—
428C2×C3#188311yescharacteristic
429C2×C3#189811yescharacteristic
430C2×C3#190611yescharacteristic
431D3—82no—
432D3—161no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—81no—
442C22—81no—
443C22—122no—
444C22—122no—
445C22—122no—
446C22—242no—
447C22—242no—
448C22—241no—
449C22—241no—
450C22—241no—
451C22—482no—
452C22#466511yescharacteristic
453C22#482211yescharacteristic
454C22#482711yescharacteristic
455C4—21no—
456C4—21no—
457C4—21no—
458C4—42no—
459C4—64no—
460C4—81no—
461C4—82no—
462C4—242no—
463C4—241no—
464C4D3×#9912yes—
465C3#936811yescharacteristic, complemented, Sylow
466C2—21no—
467C2—21no—
468C2—41no—
469C2—41no—
470C2—242no—
471C2—481no—
472C2#6877611yescharacteristic
473C2#6877811yescharacteristic
474C2#6897011yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1297C2
socleC22×C3#386
prime radical#78639C2
cosocleC23×C3×C4C23
hypercenter#1297D3
hypocenterC3#9368
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1297

Complements

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

NH ≅ G/Ntype
2. #73454C2semidirect
3. #69456C2semidirect
4. #69454C2semidirect
5. #69444C2semidirect
6. C3×#1297C2semidirect
7. #67684C2semidirect
11. C2×#908C4semidirect
18. #5133C4semidirect
36. #1297D3semidirect
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
#786392
#734542
#694562
#694542
#694442
C3×#12972
#676842
#93683