Finite Group Databasebeta

← #10690027

Subgroups — #10690027

Order 768.

608 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#461C211yesmaximal, characteristic
2#82337C211yesmaximal, characteristic
3#71787C211yesmaximal, characteristic, complemented
4#71780C211yesmaximal, characteristic, complemented
5C2×D4⋊(C3⋊C8)C211yesmaximal, characteristic
6#67704C211yesmaximal, characteristic, complemented
7#67702C211yesmaximal, characteristic, complemented
8#8988—31nomaximal, Sylow
9C22×C3×SD16—21no—
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×C4×D4C2211yescharacteristic
12C2×#926—21no—
13C3×#461—22no—
14C3×#461C414yescomplemented
15C2×C3×C4⋊C8C2211yescharacteristic
16C2×C3×#103—21no—
17#5221—21no—
18#5212—21no—
19Dic3×C4⋊C4—21no—
20#4951—22no—
21#4935C2211yescharacteristic
22C4×C4⋊Dic3C2211yescharacteristic
23C2×C4⋊(C3⋊C8)C2211yescharacteristic
24C2×C4×C3⋊C8C2211yescharacteristic
25D4⋊(C3⋊C8)—22no—
26#4529—21no—
27#4529—21no—
28#4529—22no—
29C2×#461D311yescharacteristic
30#2354—31no—
31#1990—31no—
32#1988—31no—
33C2×D4⋊C8—31no—
34#1296—31no—
35#1279—31no—
36C22×C3×Q8—21no—
37C22×C3×D4D411yescharacteristic
38C23×C3×C4—21no—
39C2×C3×SD16—24no—
40C2×C3×SD16—42no—
41C2×C3×SD16—42no—
42C22×C3×C8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8C2×C412yes—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C3×C4×D4C2×C412yes—
50C3×C4×D4D412yes—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4D411yescharacteristic
54C2×C3×C22⋊C4—21no—
55C2×C3×C42C2311yescharacteristic
56C2×#217—22no—
57C2×#217—22no—
58#926—42no—
59#926—42no—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3D411yescharacteristic
63C2×C4⋊Dic3D411yescharacteristic
64C2×C4⋊Dic3D412yes—
65C2×C4×Dic3—22no—
66C2×C4×Dic3—21no—
67C22×C3⋊C8—21no—
68C22×C3⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8C2×C412yes—
71C3×#103—24no—
72C3⋊#95—22no—
73C3⋊#95—21no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4×C3⋊C8—21no—
77C4×C3⋊C8—21no—
78C22×SD16—21no—
79C2×C4⋊Q8C2×D311yescharacteristic
80C2×C4×D4C2×D311yescharacteristic
81C2×#117—61no—
82#461—22no—
83#461Dic314yescomplemented
84C2×C4⋊C8—31no—
85C2×C4⋊C8C2×D311yescharacteristic
86C2×#103—21no—
87C2×C4×C8—31no—
88#390—61no—
89#388—62no—
90#384—31no—
91#380—61no—
92C4×C4⋊C4—31no—
93C4×C4⋊C4—61no—
94#340—61no—
95#340—61no—
96#340—62no—
97D4⋊C8—62no—
98C24×C3—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—
105C2×C3×D4—21no—
106C2×C3×D4—21no—
107C2×C3×D4—21no—
108C2×C3×D4—21no—
109C2×C3×D4—21no—
110C2×C3×D4C22⋊C412yes—
111C22×C3×C4—21no—
112C22×C3×C4—22no—
113C22×C3×C4—22no—
114C22×C3×C4—41no—
115C22×C3×C4—41no—
116C22×C3×C4—41no—
117C22×C3×C4—41no—
118C22×C3×C4—41no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×Dic3—21no—
123C22×Dic3—21no—
124C22×Dic3—22no—
125C3×SD16—48no—
126C2×C3×C8—22no—
127C2×C3×C8—41no—
128C2×C3×C8—41no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4C22⋊C412yes—
135C3×C22⋊C4—22no—
136C3×C22⋊C4—22no—
137C3×C42C22×C411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42#4111yescharacteristic
140C3×C42#4111yescharacteristic
141#217—44no—
142#217—82no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—41no—
147C4⋊Dic3—41no—
148C4⋊Dic3—41no—
149C4⋊Dic3—41no—
150C4⋊Dic3—42no—
151C4⋊Dic3SD1612yescomplemented
152C4×Dic3—44no—
153C4×Dic3—41no—
154C4×Dic3—41no—
155C2×C3⋊C8—22no—
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#6711yescharacteristic
163C23×C4—21no—
164C2×SD16—24no—
165C2×SD16—42no—
166C2×SD16—42no—
167C22×C8—21no—
168C22×C8—61no—
169C22×C8—61no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8C2×Dic312yes—
174C4×D4—21no—
175C4×D4—21no—
176C4×D4C2×Dic312yes—
177C4×D4#6712yes—
178#117—122no—
179#117—122no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—31no—
183C2×C4⋊C4—31no—
184C2×C4⋊C4—32no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4#6711yescharacteristic
188C2×C22⋊C4—21no—
189C2×C22⋊C4—62no—
190C2×C22⋊C4—62no—
191C2×C42—62no—
192C2×C42—61no—
193C2×C42C22×D311yescharacteristic
194C4⋊C8—21no—
195C4⋊C8—61no—
196C4⋊C8—61no—
197C4⋊C8C2×Dic312yes—
198#103—24no—
199C4×C8—61no—
200C4×C8—61no—
201#95—62no—
202#95—61no—
203C23×C3—22no—
204C23×C3—41no—
205C23×C3—41no—
206C23×C3—41no—
207C23×C3—41no—
208C23×C3—41no—
209C23×C3—41no—
210C23×C3#12011yescharacteristic
211C3×Q8—42no—
212C3×Q8—42no—
213C3×Q8—81no—
214C3×Q8—81no—
215C3×D4—22no—
216C3×D4—22no—
217C3×D4—41no—
218C3×D4—41no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—22no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—42no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4—81no—
230C2×C3×C4—81no—
231C2×C3×C4C2×C22⋊C411yescharacteristic
232C2×C3×C4C4×D411yescharacteristic
233C2×C3×C4C4×D411yescharacteristic
234C2×C3×C4#12111yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#12311yescharacteristic
237C2×C3×C4C2×SD1611yescharacteristic
238C2×C3×C4#13511yescharacteristic
239C2×C3×C4#13511yescharacteristic
240C2×C3×C4#13511yescharacteristic
241C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
242C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
243C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
244C2×C3×C4#13711yescharacteristic
245C2×Dic3—22no—
246C2×Dic3—41no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—42no—
250C2×Dic3—42no—
251C2×Dic3—41no—
252C2×Dic3—42no—
253C2×Dic3—41no—
254C3×C8—42no—
255C3⋊C8—42no—
256C3⋊C8—81no—
257C24—21no—
258C2×Q8—22no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4#21712yes—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—61no—
279C22×C4—61no—
280C22×C4—62no—
281C22×C4D3×D411yescharacteristic
282C22×C4D3×D411yescharacteristic
283C22×C4C2×#6711yescharacteristic
284SD16—48no—
285C2×C8—22no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—62no—
289C2×C8—121no—
290C2×C8—121no—
291C2×C8—121no—
292C2×C8—121no—
293C2×C8—121no—
294C4⋊C4—21no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—32no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—61no—
301C4⋊C4—61no—
302C4⋊C4—61no—
303C4⋊C4—122no—
304C4⋊C4—121no—
305C4⋊C4—121no—
306C4⋊C4—121no—
307C4⋊C4—121no—
308C4⋊C4#21712yes—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C22⋊C4—124no—
312C22⋊C4—242no—
313C42—124no—
314C42—121no—
315C42—121no—
316C42#23711yescharacteristic
317C42#23711yescharacteristic
318C42C22×Dic311yescharacteristic
319C42C2×#6711yescharacteristic
320C22×C3—42no—
321C22×C3—42no—
322C22×C3—42no—
323C22×C3—81no—
324C22×C3—81no—
325C22×C3—81no—
326C22×C3—81no—
327C22×C3#38611yescharacteristic
328C22×C3#44911yescharacteristic
329C22×C3#44911yescharacteristic
330C22×C3#45011yescharacteristic
331C22×C3#45211yescharacteristic
332C22×C3#45511yescharacteristic
333C22×C3#45611yescharacteristic
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—42no—
337C3×C4—82no—
338C3×C4#41611yescharacteristic
339C3×C4#41811yescharacteristic
340C3×C4C4×SD1611yescharacteristic
341C3×C4#44011yescharacteristic
342C3×C4#46111yescharacteristic
343C3×C4#46311yescharacteristic
344C3×C4#48211yescharacteristic
345C3×C4#48211yescharacteristic
346Dic3—42no—
347Dic3—82no—
348Dic3—81no—
349C23—22no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23#93711yescharacteristic
357Q8—42no—
358Q8—42no—
359Q8—81no—
360Q8—81no—
361D4—22no—
362D4—22no—
363D4—41no—
364D4—41no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—62no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—121no—
379C2×C4—121no—
380C2×C4—122no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—122no—
384C2×C4—121no—
385C2×C4—122no—
386C2×C4D3×SD1611yescharacteristic
387C2×C4#91411yescharacteristic
388C2×C4#91511yescharacteristic
389C2×C4#91611yescharacteristic
390C2×C4#93211yescharacteristic
391C2×C4#93211yescharacteristic
392C2×C4D4×Dic311yescharacteristic
393C2×C4D4×Dic311yescharacteristic
394C2×C4#93511yescharacteristic
395C2×C4#93911yescharacteristic
396C2×C4#93911yescharacteristic
397C2×C4#95011yescharacteristic
398C2×C4#95011yescharacteristic
399C2×C4C2×#21711yescharacteristic
400C8—42no—
401C8—122no—
402C8—241no—
403C2×C3—82no—
404C2×C3#164211yescharacteristic
405C2×C3#164611yescharacteristic
406C2×C3#165011yescharacteristic
407C2×C3#165411yescharacteristic
408C2×C3#187511yescharacteristic
409C2×C3#187711yescharacteristic
410C2×C3#188211yescharacteristic
411C22—42no—
412C22—42no—
413C22—42no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22#503711yescharacteristic
419C22#503711yescharacteristic
420C22#516911yescharacteristic
421C22#517311yescharacteristic
422C22#520211yescharacteristic
423C22#520511yescharacteristic
424C22#522011yescharacteristic
425C4—21no—
426C4—21no—
427C4—42no—
428C4—82no—
429C4—122no—
430C4—242no—
431C4—241no—
432C4#503511yescharacteristic
433C4#503511yescharacteristic
434C4Dic3×SD1611yescharacteristic
435C4#516211yescharacteristic
436C4#516411yescharacteristic
437C4#516511yescharacteristic
438C4#521511yescharacteristic
439C4#523311yescharacteristic
440C3#898811yescharacteristic, complemented, Sylow
441C2—82no—
442C2#7104511yescharacteristic
443C2#7104711yescharacteristic
444C2#7104911yescharacteristic
445C2#7105111yescharacteristic
446C2#7167811yescharacteristic
447C2#7204711yescharacteristic
448C2#7210011yescharacteristic
subgroupG/H
centerC23#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC2×C3×#461C2
socleC23×C3#120
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenterC2×#461D3
hypocenterC3#8988
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
3. #71787C2semidirect
4. #71780C2semidirect
6. #67704C2semidirect
7. #67702C2semidirect
14. C3×#461C4semidirect
83. #461Dic3semidirect
151. C4⋊Dic3SD16semidirect
440. C3#8988semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4612
#823372
#717872
#717802
C2×D4⋊(C3⋊C8)2
#677042
#677022
#89883