Finite Group Databasebeta

← #10690469

Subgroups — #10690469

Order 768.

618 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2365C211yesmaximal, characteristic, complemented
2C2×#5088C211yesmaximal, characteristic
3C2×#4547C211yesmaximal, characteristic, complemented
4#69487C211yesmaximal, characteristic
5#69487C211yesmaximal, characteristic
6#67619C211yesmaximal, characteristic, complemented
7#67619C211yesmaximal, characteristic, complemented
8#9062—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4×Q8—21no—
11C2×#874C2211yescharacteristic
12C3×#391—21no—
13C3×#389—21no—
14C3×#387—22no—
15C3×C4×C4⋊C4—21no—
16C3×C4×C4⋊C4C2211yescharacteristic
17C3×C4×C4⋊C4C2211yescharacteristic
18#5088—22no—
19#5088C414yescomplemented
20C2×#809—22no—
21C2×#809—21no—
22C2×#809—21no—
23C2×#809—21no—
24C2×#809—21no—
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26C2×C4⋊(C3⋊C8)C2211yescharacteristic
27C2×C4×C3⋊C8C2211yescharacteristic
28#4547—22no—
29C2×#486—31no—
30#2365D311yescharacteristic, complemented
31#1982—31no—
32#1982—31no—
33C2×#332—31no—
34#1279—31no—
35#1279—31no—
36C22×C3×Q8—21no—
37C22×D12—21no—
38C22×D12—21no—
39C22×D12—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8C2×C412yes—
43C3×C4⋊Q8D412yes—
44C3×C4×Q8—42no—
45C3×C4×Q8—42no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—22no—
48C2×C3×C4⋊C4D411yescharacteristic
49C2×C3×C4⋊C4D411yescharacteristic
50C2×C3×C4⋊C4D412yes—
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C4⋊C4D411yescharacteristic
53C2×C3×C42—22no—
54C2×C3×C42—21no—
55C2×C3×C42—21no—
56C2×C3×C42C2311yescharacteristic
57C22×C3⋊C8—21no—
58C22×C3⋊C8—21no—
59C22×C3⋊C8—21no—
60#874—21no—
61#874—21no—
62#874—21no—
63#874C2×C412yes—
64C3×#95—22no—
65C3×#95—22no—
66C3×#95—21no—
67C3×#95—21no—
68#809—28no—
69#809—42no—
70#809—42no—
71#809—42no—
72#809—42no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77C4×C3⋊C8—21no—
78C4×C3⋊C8C2×C412yes—
79C2×C4⋊Q8C2×D311yescharacteristic
80C2×#127—31no—
81C2×C4×Q8—21no—
82#486—34no—
83#486—62no—
84C2×C4⋊C8—31no—
85C2×C4⋊C8—31no—
86C2×#102—62no—
87C2×#102—61no—
88C2×#102—61no—
89C2×#102—61no—
90C2×#102—61no—
91C2×C4×C8—31no—
92#391—21no—
93#389—21no—
94#387—22no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4C2×D311yescharacteristic
97C4×C4⋊C4C2×D311yescharacteristic
98#332—62no—
99C23×D3—41no—
100C2×C3×Q8—22no—
101C2×C3×Q8—22no—
102C2×C3×Q8—41no—
103C2×C3×Q8—41no—
104C2×C3×Q8—41no—
105C2×C3×Q8—41no—
106C22×C3×C4—21no—
107C22×C3×C4—22no—
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C2×D12—22no—
115C2×D12—22no—
116C2×D12—22no—
117C2×D12—42no—
118C2×D12—42no—
119C2×D12—41no—
120C2×D12—41no—
121C2×D12—41no—
122C2×D12—41no—
123C2×D12—41no—
124C2×D12—41no—
125C2×D12—41no—
126C2×D12—41no—
127C2×D12—41no—
128C2×D12—41no—
129C2×D12—41no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—22no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—44no—
140C3×C4⋊C4C22⋊C414yes—
141C3×C4⋊C4D812yes—
142C3×C4⋊C4SD1612yes—
143C3×C42—44no—
144C3×C42—41no—
145C3×C42—41no—
146C3×C42—41no—
147C3×C42—41no—
148C3×C42C22×C411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42#4111yescharacteristic
151C3×C42#4111yescharacteristic
152C2×C3⋊C8—22no—
153C2×C3⋊C8—41no—
154C2×C3⋊C8—41no—
155C2×C3⋊C8—41no—
156C2×C3⋊C8—41no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159C2×C3⋊C8—41no—
160C2×C3⋊C8—41no—
161C22×Q8—21no—
162C22×D4—61no—
163C22×D4—61no—
164C22×D4—61no—
165C22×C8—61no—
166C22×C8—61no—
167C22×C8—61no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8C4×D312yes—
171C4⋊Q8D1212yes—
172#127—32no—
173#127—61no—
174#127—61no—
175#127—61no—
176C4×Q8—42no—
177C4×Q8—42no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4#6711yescharacteristic
181C2×C4⋊C4#6711yescharacteristic
182C2×C4⋊C4#6712yes—
183C2×C4⋊C4#6711yescharacteristic
184C2×C4⋊C4#6711yescharacteristic
185C2×C42—22no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42C22×D311yescharacteristic
189C4⋊C8—61no—
190C4⋊C8—61no—
191C4⋊C8—61no—
192C4⋊C8—61no—
193#102—68no—
194#102—122no—
195#102—122no—
196#102—122no—
197#102—122no—
198C4×C8—32no—
199C4×C8—61no—
200#95—22no—
201#95—22no—
202#95—21no—
203#95—21no—
204C23×C3#12011yescharacteristic
205C22×D3—42no—
206C22×D3—81no—
207C22×D3—81no—
208C22×D3—81no—
209C22×D3—81no—
210C22×D3—81no—
211C22×D3—81no—
212C3×Q8—42no—
213C3×Q8—42no—
214C3×Q8—42no—
215C3×Q8—42no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—24no—
219C2×C3×C4—22no—
220C2×C3×C4—22no—
221C2×C3×C4—41no—
222C2×C3×C4—42no—
223C2×C3×C4—41no—
224C2×C3×C4—42no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4C2×C22⋊C411yescharacteristic
234C2×C3×C4C4×D411yescharacteristic
235C2×C3×C4C4×D411yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12311yescharacteristic
239C2×C3×C4C2×D811yescharacteristic
240C2×C3×C4C2×SD1611yescharacteristic
241C2×C3×C4#13511yescharacteristic
242C2×C3×C4#13511yescharacteristic
243C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
244C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
245C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
246C2×C3×C4#13711yescharacteristic
247D12—44no—
248D12—42no—
249D12—42no—
250D12—42no—
251D12—42no—
252D12—82no—
253D12—81no—
254D12—81no—
255D12—81no—
256D12—81no—
257C3⋊C8—42no—
258C3⋊C8—81no—
259C3⋊C8—81no—
260C24—121no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×D4—62no—
268C2×D4—62no—
269C2×D4—62no—
270C2×D4—122no—
271C2×D4—122no—
272C2×D4—121no—
273C2×D4—121no—
274C2×D4—121no—
275C2×D4—121no—
276C2×D4—121no—
277C2×D4—121no—
278C2×D4—121no—
279C2×D4—121no—
280C2×D4—121no—
281C2×D4—121no—
282C2×D4—121no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4C2×#6711yescharacteristic
289C22×C4C2×#6711yescharacteristic
290C22×C4C2×#6711yescharacteristic
291C2×C8—62no—
292C2×C8—121no—
293C2×C8—121no—
294C2×C8—121no—
295C2×C8—121no—
296C2×C8—121no—
297C2×C8—121no—
298C2×C8—121no—
299C2×C8—121no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—44no—
310C4⋊C4(C2×D3)⋊C414yes—
311C4⋊C4#21312yes—
312C4⋊C4#21512yes—
313C42—44no—
314C42—41no—
315C42—41no—
316C42—41no—
317C42—41no—
318C42C2×C4×D311yescharacteristic
319C42C2×D1211yescharacteristic
320C42#23511yescharacteristic
321C42#23511yescharacteristic
322C22×C3#38611yescharacteristic
323C22×C3#44711yescharacteristic
324C22×C3#44811yescharacteristic
325C22×C3#44911yescharacteristic
326C22×C3#44911yescharacteristic
327C22×C3#45411yescharacteristic
328C22×C3#45411yescharacteristic
329C2×D3—82no—
330C2×D3—82no—
331C2×D3—82no—
332C2×D3—161no—
333C2×D3—161no—
334C2×D3—161no—
335C2×D3—161no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—44no—
339C3×C4—42no—
340C3×C4—42no—
341C3×C4—81no—
342C3×C4—81no—
343C3×C4C4×D811yescharacteristic
344C3×C4C4×SD1611yescharacteristic
345C3×C4#44011yescharacteristic
346C3×C4#44211yescharacteristic
347C3×C4#45911yescharacteristic
348C3×C4#46011yescharacteristic
349C3×C4#46311yescharacteristic
350C3×C4#46411yescharacteristic
351C23—122no—
352C23—241no—
353C23—241no—
354C23—241no—
355C23—241no—
356C23—241no—
357C23—241no—
358C23#95311yescharacteristic
359Q8—42no—
360Q8—42no—
361Q8—42no—
362Q8—42no—
363D4—124no—
364D4—122no—
365D4—122no—
366D4—122no—
367D4—122no—
368D4—242no—
369D4—241no—
370D4—241no—
371D4—241no—
372D4—241no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—24no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4C2×(C2×D3)⋊C411yescharacteristic
391C2×C4C4×#6711yescharacteristic
392C2×C4C4×#6711yescharacteristic
393C2×C4#92911yescharacteristic
394C2×C4#93011yescharacteristic
395C2×C4#93011yescharacteristic
396C2×C4C2×#21311yescharacteristic
397C2×C4#93211yescharacteristic
398C2×C4C2×#21511yescharacteristic
399C2×C4#94211yescharacteristic
400C2×C4#94911yescharacteristic
401C2×C4#94911yescharacteristic
402C2×C4#95011yescharacteristic
403C2×C4#95011yescharacteristic
404C8—122no—
405C8—241no—
406C8—241no—
407C2×C3#162511yescharacteristic
408C2×C3#162611yescharacteristic
409C2×C3#162711yescharacteristic
410C2×C3#162811yescharacteristic
411C2×C3#187911yescharacteristic
412C2×C3#188011yescharacteristic
413C2×C3#188811yescharacteristic
414D3—162no—
415C22—242no—
416C22—242no—
417C22—242no—
418C22—481no—
419C22—481no—
420C22—481no—
421C22—481no—
422C22#521311yescharacteristic
423C22#521711yescharacteristic
424C22#522711yescharacteristic
425C22#523811yescharacteristic
426C22#523811yescharacteristic
427C22#524511yescharacteristic
428C22#524511yescharacteristic
429C4—21no—
430C4—21no—
431C4—44no—
432C4—42no—
433C4—42no—
434C4—81no—
435C4—81no—
436C4C4×#21311yescharacteristic
437C4#501411yescharacteristic
438C4#501511yescharacteristic
439C4#501611yescharacteristic
440C4C4×#21511yescharacteristic
441C4#502611yescharacteristic
442C4#502711yescharacteristic
443C4#502811yescharacteristic
444C3#906211yescharacteristic, complemented, Sylow
445C2—482no—
446C2#7161311yescharacteristic
447C2#7163311yescharacteristic
448C2#7166111yescharacteristic
449C2#7191011yescharacteristic
450C2#7191411yescharacteristic
451C2#7193111yescharacteristic
452C2#7193511yescharacteristic
subgroupG/H
centerC23#953
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2365C2
socleC23×C3#120
prime radicalC2×#874C22
cosocleC2×C3×C42C23
hypercenter#2365D3
hypocenterC3#9062
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2365

Complements

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

NH ≅ G/Ntype
1. C3×#2365C2semidirect
3. C2×#4547C2semidirect
6. #67619C2semidirect
7. #67619C2semidirect
19. #5088C4semidirect
30. #2365D3semidirect
444. C3#9062semidirect

Maximal subgroups

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

subgroupindex
C3×#23652
C2×#50882
C2×#45472
#694872
#676192
#90623