Finite Group Databasebeta

← #10690000

Subgroups — #10690000

Order 768.

592 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5087C211yesmaximal, characteristic
2#73638C211yesmaximal, characteristic
3C3×#1981C211yesmaximal, characteristic, complemented
4#71787C211yesmaximal, characteristic, complemented
5C2×#4483C211yesmaximal, characteristic
6#67702C211yesmaximal, characteristic, complemented
7#67619C211yesmaximal, characteristic, complemented
8#8984—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#892—21no—
11C2×C4×D12C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×#103—21no—
14C2×C3×#103—21no—
15C3×C4×C4⋊C4C2211yescharacteristic
16#5087—22no—
17#5087C414yescomplemented
18#4987—21no—
19C2×#809—21no—
20C2×#807—21no—
21#4939—21no—
22#4935C2211yescharacteristic
23C4×C4⋊Dic3—21no—
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25C2×C4×C3⋊C8C2211yescharacteristic
26#4670—22no—
27#4529—22no—
28#4483—22no—
29C2×#474—31no—
30#2354—31no—
31#1990—31no—
32#1981D311yescharacteristic, complemented
33C2×D4⋊C8—31no—
34#1296—31no—
35#1279—31no—
36C22×C3×Q8—21no—
37C22×D12D411yescharacteristic
38C22×C4×D3—21no—
39C22×C3×C8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8C2×C412yes—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4D411yescharacteristic
47C2×C3×C4⋊C4D411yescharacteristic
48C2×C3×C42—21no—
49C2×C3×C42C2311yescharacteristic
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—22no—
52C2×(C2×D3)⋊C4—22no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3D411yescharacteristic
56C2×C4⋊Dic3D412yes—
57C2×C4×Dic3—22no—
58C22×C3⋊C8—21no—
59C22×C3⋊C8—21no—
60#892—42no—
61#892—42no—
62C4×D12—21no—
63C4×D12—21no—
64C4×D12C2×C412yes—
65C4×D12D412yes—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×#103—42no—
69C3×#103—42no—
70C3×#95—21no—
71C3⋊#95—22no—
72#809—24no—
73#807—24no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)C2×C412yes—
76C4×C3⋊C8—21no—
77C4×C3⋊C8—21no—
78C2×C4⋊Q8C2×D311yescharacteristic
79C2×C4×D4—31no—
80C2×#117—61no—
81#474—34no—
82#474—62no—
83C2×#107—61no—
84C2×C4⋊C8—31no—
85C2×C4⋊C8C2×D311yescharacteristic
86C2×#103—21no—
87C2×#103—21no—
88C2×#102—61no—
89C2×C4×C8—31no—
90#390—61no—
91#388—62no—
92#384—31no—
93#380—61no—
94C4×C4⋊C4—61no—
95C4×C4⋊C4C2×D311yescharacteristic
96#340—62no—
97D4⋊C8—62no—
98C23×D3—21no—
99C2×C3×Q8—22no—
100C2×C3×Q8—41no—
101C2×C3×Q8—41no—
102C2×C3×Q8—41no—
103C2×C3×Q8—41no—
104C2×C3×Q8—41no—
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4—41no—
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×Dic3—21no—
112C22×Dic3—22no—
113C2×D12—21no—
114C2×D12—21no—
115C2×D12—21no—
116C2×D12—21no—
117C2×D12—21no—
118C2×D12C22⋊C412yes—
119C2×C4×D3—22no—
120C2×C4×D3—22no—
121C2×C4×D3—41no—
122C2×C4×D3—41no—
123C2×C4×D3—41no—
124C2×C4×D3—41no—
125C2×C3×C8—41no—
126C2×C3×C8—41no—
127C2×C3×C8—41no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4D812yes—
136C3×C42—41no—
137C3×C42—41no—
138C3×C42C22×C411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42#4111yescharacteristic
141C3×C42#4111yescharacteristic
142(C2×D3)⋊C4—22no—
143(C2×D3)⋊C4—22no—
144(C2×D3)⋊C4—44no—
145(C2×D3)⋊C4—82no—
146C4⋊Dic3—21no—
147C4⋊Dic3—41no—
148C4⋊Dic3—41no—
149C4⋊Dic3—41no—
150C4⋊Dic3—41no—
151C4⋊Dic3—42no—
152C4⋊Dic3C22⋊C412yes—
153C4×Dic3—44no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—41no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159C2×C3⋊C8—41no—
160C22×Q8—21no—
161C22×D4—31no—
162C23×C4—61no—
163C22×C8—21no—
164C22×C8—61no—
165C22×C8—61no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8C4×D312yes—
170C4×D4—32no—
171C4×D4—32no—
172C4×D4—61no—
173C4×D4—61no—
174#117—122no—
175#117—122no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—32no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4#6711yescharacteristic
183C2×C4⋊C4#6711yescharacteristic
184C2×C22⋊C4—61no—
185C2×C22⋊C4—62no—
186C2×C22⋊C4—62no—
187C2×C42—21no—
188C2×C42—62no—
189C2×C42C22×D311yescharacteristic
190#107—64no—
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—32no—
194C4⋊C8—61no—
195#103—42no—
196#103—42no—
197#102—64no—
198C4×C8—61no—
199C4×C8—61no—
200#95—21no—
201#95—62no—
202C23×C3#12011yescharacteristic
203C22×D3—22no—
204C22×D3—41no—
205C22×D3—41no—
206C22×D3—41no—
207C22×D3—41no—
208C22×D3—41no—
209C22×D3—41no—
210C3×Q8—42no—
211C3×Q8—42no—
212C3×Q8—81no—
213C3×Q8—81no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—22no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—42no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—81no—
224C2×C3×C4—81no—
225C2×C3×C4C2×C22⋊C411yescharacteristic
226C2×C3×C4C4×D411yescharacteristic
227C2×C3×C4C4×D411yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12311yescharacteristic
231C2×C3×C4C2×D811yescharacteristic
232C2×C3×C4#13511yescharacteristic
233C2×C3×C4#13511yescharacteristic
234C2×C3×C4#13511yescharacteristic
235C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
236C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
237C2×C3×C4#13711yescharacteristic
238C2×C3×C4#13711yescharacteristic
239C2×Dic3—22no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—42no—
243C2×Dic3—42no—
244C2×Dic3—42no—
245D12—22no—
246D12—22no—
247D12—41no—
248D12—41no—
249C4×D3—42no—
250C4×D3—42no—
251C4×D3—42no—
252C4×D3—42no—
253C3×C8—81no—
254C3⋊C8—42no—
255C3⋊C8—42no—
256C24—61no—
257C2×Q8—22no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×D4—32no—
264C2×D4—61no—
265C2×D4—61no—
266C2×D4—61no—
267C2×D4—61no—
268C2×D4—61no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—41no—
272C22×C4—61no—
273C22×C4—62no—
274C22×C4—62no—
275C22×C4—62no—
276C22×C4—121no—
277C22×C4—121no—
278C22×C4—121no—
279C22×C4—121no—
280C22×C4D3×D411yescharacteristic
281C22×C4D3×D411yescharacteristic
282C22×C4C2×#6711yescharacteristic
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—62no—
287C2×C8—62no—
288C2×C8—121no—
289C2×C8—121no—
290C2×C8—121no—
291C2×C8—121no—
292C4⋊C4—21no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—32no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—61no—
301C4⋊C4—122no—
302C4⋊C4—121no—
303C4⋊C4—121no—
304C4⋊C4—121no—
305C4⋊C4—121no—
306C4⋊C4#21312yes—
307C22⋊C4—62no—
308C22⋊C4—62no—
309C22⋊C4—124no—
310C22⋊C4—242no—
311C42—41no—
312C42—41no—
313C42—124no—
314C42C2×C4×D311yescharacteristic
315C42#23511yescharacteristic
316C42D3×D411yescharacteristic
317C42#23711yescharacteristic
318C22×C3#38611yescharacteristic
319C22×C3#44711yescharacteristic
320C22×C3#44911yescharacteristic
321C22×C3#45211yescharacteristic
322C22×C3#45211yescharacteristic
323C22×C3#45511yescharacteristic
324C22×C3#45611yescharacteristic
325C2×D3—42no—
326C2×D3—42no—
327C2×D3—42no—
328C2×D3—81no—
329C2×D3—81no—
330C2×D3—81no—
331C2×D3—81no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—42no—
335C3×C4—82no—
336C3×C4—81no—
337C3×C4#41611yescharacteristic
338C3×C4#41911yescharacteristic
339C3×C4C4×D811yescharacteristic
340C3×C4#44211yescharacteristic
341C3×C4#45911yescharacteristic
342C3×C4#46311yescharacteristic
343C3×C4#48511yescharacteristic
344C3×C4#48511yescharacteristic
345Dic3—42no—
346Dic3—82no—
347C23—62no—
348C23—121no—
349C23—121no—
350C23—121no—
351C23—121no—
352C23—121no—
353C23—121no—
354C23#93711yescharacteristic
355Q8—42no—
356Q8—42no—
357Q8—81no—
358Q8—81no—
359D4—62no—
360D4—62no—
361D4—121no—
362D4—121no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—62no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—121no—
376C2×C4—121no—
377C2×C4—122no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4—122no—
381C2×C4—122no—
382C2×C4—122no—
383C2×C4—122no—
384C2×C4D3×C22⋊C411yescharacteristic
385C2×C4#67⋊C411yescharacteristic
386C2×C4#88311yescharacteristic
387C2×C4#88411yescharacteristic
388C2×C4#91511yescharacteristic
389C2×C4#91611yescharacteristic
390C2×C4#91811yescharacteristic
391C2×C4#91911yescharacteristic
392C2×C4C4×#6711yescharacteristic
393C2×C4C2×#21311yescharacteristic
394C2×C4#93211yescharacteristic
395C2×C4#93811yescharacteristic
396C2×C4#94911yescharacteristic
397C2×C4#95011yescharacteristic
398C8—81no—
399C8—122no—
400C8—122no—
401C2×C3#164111yescharacteristic
402C2×C3#164611yescharacteristic
403C2×C3#164711yescharacteristic
404C2×C3#165211yescharacteristic
405C2×C3#187311yescharacteristic
406C2×C3#187911yescharacteristic
407C2×C3#188211yescharacteristic
408D3—82no—
409C22—122no—
410C22—122no—
411C22—122no—
412C22—241no—
413C22—241no—
414C22—241no—
415C22—241no—
416C22#495611yescharacteristic
417C22#503611yescharacteristic
418C22#503711yescharacteristic
419C22#517311yescharacteristic
420C22#518711yescharacteristic
421C22#520211yescharacteristic
422C22#520511yescharacteristic
423C4—21no—
424C4—21no—
425C4—42no—
426C4—82no—
427C4—81no—
428C4—122no—
429C4—242no—
430C4#480211yescharacteristic
431C4#480411yescharacteristic
432C4#481411yescharacteristic
433C4#481511yescharacteristic
434C4C4×#21311yescharacteristic
435C4#501411yescharacteristic
436C4#505711yescharacteristic
437C4#507211yescharacteristic
438C3#898411yescharacteristic, complemented, Sylow
439C2—242no—
440C2#6965711yescharacteristic
441C2#7026911yescharacteristic
442C2#7034811yescharacteristic
443C2#7104711yescharacteristic
444C2#7104811yescharacteristic
445C2#7106111yescharacteristic
446C2#7106611yescharacteristic
subgroupG/H
centerC23#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1981C2
socleC23×C3#120
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenter#1981D3
hypocenterC3#8984
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1981

Complements

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

NH ≅ G/Ntype
3. C3×#1981C2semidirect
4. #71787C2semidirect
6. #67702C2semidirect
7. #67619C2semidirect
17. #5087C4semidirect
32. #1981D3semidirect
438. C3#8984semidirect

Maximal subgroups

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

subgroupindex
C2×#50872
#736382
C3×#19812
#717872
C2×#44832
#677022
#676192
#89843