Finite Group Databasebeta

← #10690006

Subgroups — #10690006

Order 768.

680 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#461C211yesmaximal, characteristic
2D4×C4⋊Dic3C211yesmaximal, characteristic
3#71787C211yesmaximal, characteristic, complemented
4#71780C211yesmaximal, characteristic, complemented
5C2×D4⋊(C3⋊C8)C211yesmaximal, characteristic
6#67704C211yesmaximal, characteristic, complemented
7#67702C211yesmaximal, characteristic, complemented
8#8985—31nomaximal, Sylow
9C22×C3×SD16—21no—
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×C4×D4C2211yescharacteristic
12C2×D4×Dic3—21no—
13C22×C4⋊Dic3—21no—
14C3×#461—22no—
15C3×#461C414yescomplemented
16C2×C3×C4⋊C8C2211yescharacteristic
17C2×C3×#103—21no—
18#5212—21no—
19#4984—21no—
20#4950—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
30D4×C4⋊C4—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—
39C23×Dic3—22no—
40C2×C3×SD16—24no—
41C2×C3×SD16—42no—
42C2×C3×SD16—42no—
43C22×C3×C8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8C2×C412yescomplemented
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C3×C4×D4C2×C412yes—
51C3×C4×D4D412yes—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C22⋊C4—21no—
56C2×C3×C42C2311yescharacteristic
57C2×#217—22no—
58D4×Dic3—42no—
59D4×Dic3—42no—
60C2×C4⋊Dic3—22no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3—42no—
64C2×C4⋊Dic3—42no—
65C2×C4⋊Dic3D411yescharacteristic
66C2×C4⋊Dic3D411yescharacteristic
67C2×C4⋊Dic3D412yes—
68C2×C4×Dic3—21no—
69C22×C3⋊C8—21no—
70C22×C3⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8C2×C412yescomplemented
73C3×#103—24no—
74C3⋊#95—22no—
75C3⋊#95—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78C4×C3⋊C8—21no—
79C4×C3⋊C8—21no—
80C22×SD16—21no—
81C2×C4⋊Q8C2×D311yescharacteristic
82C2×C4×D4—61no—
83C2×C4×D4C2×D311yescharacteristic
84C22×C4⋊C4—61no—
85#461—22no—
86#461Dic314yescomplemented
87C2×C4⋊C8—31no—
88C2×C4⋊C8C2×D311yescharacteristic
89C2×#103—21no—
90C2×C4×C8—31no—
91#389—61no—
92#385—62no—
93#384—31no—
94#380—61no—
95C4×C4⋊C4—31no—
96#340—61no—
97#340—61no—
98#340—62no—
99D4⋊C8—62no—
100C24×C3—21no—
101C2×C3×Q8—22no—
102C2×C3×Q8—41no—
103C2×C3×Q8—41no—
104C2×C3×Q8—41no—
105C2×C3×Q8—41no—
106C2×C3×Q8—41no—
107C2×C3×D4—21no—
108C2×C3×D4—21no—
109C2×C3×D4—21no—
110C2×C3×D4C22⋊C412yes—
111C2×C3×D4D812yes—
112C2×C3×D4SD1612yes—
113C22×C3×C4—21no—
114C22×C3×C4—22no—
115C22×C3×C4—22no—
116C22×C3×C4—41no—
117C22×C3×C4—41no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4—41no—
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×Dic3—21no—
125C22×Dic3—21no—
126C22×Dic3—22no—
127C22×Dic3—44no—
128C22×Dic3—44no—
129C22×Dic3—82no—
130C3×SD16—48no—
131C2×C3×C8—22no—
132C2×C3×C8—41no—
133C2×C3×C8—41no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4C22⋊C412yes—
140C3×C22⋊C4—22no—
141C3×C22⋊C4—22no—
142C3×C42C22×C411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42#4111yescharacteristic
145C3×C42#4111yescharacteristic
146#217—44no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—44no—
151C4⋊Dic3—41no—
152C4⋊Dic3—41no—
153C4⋊Dic3—41no—
154C4⋊Dic3—41no—
155C4⋊Dic3—42no—
156C4⋊Dic3SD1612yescomplemented
157C4×Dic3—41no—
158C4×Dic3—41no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—41no—
161C2×C3⋊C8—41no—
162C2×C3⋊C8—41no—
163C2×C3⋊C8—41no—
164C2×C3⋊C8—41no—
165C22×Q8—21no—
166C22×D4#6711yescharacteristic
167C23×C4—21no—
168C23×C4—62no—
169C2×SD16—24no—
170C2×SD16—42no—
171C2×SD16—42no—
172C22×C8—21no—
173C22×C8—61no—
174C22×C8—61no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8C2×Dic312yescomplemented
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—122no—
182C4×D4—122no—
183C4×D4C2×Dic312yes—
184C4×D4#6712yes—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—31no—
188C2×C4⋊C4—31no—
189C2×C4⋊C4—32no—
190C2×C4⋊C4—62no—
191C2×C4⋊C4—61no—
192C2×C4⋊C4—61no—
193C2×C4⋊C4—122no—
194C2×C4⋊C4—122no—
195C2×C4⋊C4#6711yescharacteristic
196C2×C22⋊C4—21no—
197C2×C22⋊C4—62no—
198C2×C42—61no—
199C2×C42C22×D311yescharacteristic
200C4⋊C8—21no—
201C4⋊C8—61no—
202C4⋊C8—61no—
203C4⋊C8C2×Dic312yescomplemented
204#103—24no—
205C4×C8—61no—
206C4×C8—61no—
207#95—62no—
208#95—61no—
209C23×C3—22no—
210C23×C3—22no—
211C23×C3—22no—
212C23×C3—41no—
213C23×C3—41no—
214C23×C3—41no—
215C23×C3—41no—
216C23×C3#12011yescharacteristic
217C3×Q8—42no—
218C3×Q8—42no—
219C3×Q8—81no—
220C3×Q8—81no—
221C3×D4—22no—
222C3×D4—22no—
223C3×D4—22no—
224C3×D4#10214yes—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—22no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—42no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4—81no—
236C2×C3×C4—81no—
237C2×C3×C4C2×C22⋊C411yescharacteristic
238C2×C3×C4C4×D411yescharacteristic
239C2×C3×C4C4×D411yescharacteristic
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12311yescharacteristic
243C2×C3×C4C2×D811yescharacteristic
244C2×C3×C4C2×SD1611yescharacteristic
245C2×C3×C4C2×SD1611yescharacteristic
246C2×C3×C4#13511yescharacteristic
247C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
248C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
249C2×C3×C4#13711yescharacteristic
250C2×C3×C4#13711yescharacteristic
251C2×Dic3—22no—
252C2×Dic3—41no—
253C2×Dic3—41no—
254C2×Dic3—41no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3—41no—
258C2×Dic3—42no—
259C2×Dic3—41no—
260C2×Dic3—88no—
261C3×C8—42no—
262C3⋊C8—42no—
263C3⋊C8—81no—
264C24—21no—
265C2×Q8—22no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4#21312yes—
275C2×D4#21412yes—
276C2×D4#21712yes—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—61no—
286C22×C4—61no—
287C22×C4—62no—
288C22×C4—124no—
289C22×C4—124no—
290C22×C4—242no—
291C22×C4D3×D411yescharacteristic
292C22×C4D3×D411yescharacteristic
293C22×C4C2×#6711yescharacteristic
294SD16—48no—
295C2×C8—22no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—62no—
299C2×C8—121no—
300C2×C8—121no—
301C2×C8—121no—
302C2×C8—121no—
303C2×C8—121no—
304C4⋊C4—21no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—32no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—61no—
311C4⋊C4—61no—
312C4⋊C4—61no—
313C4⋊C4—122no—
314C4⋊C4—124no—
315C4⋊C4—121no—
316C4⋊C4—121no—
317C4⋊C4—121no—
318C4⋊C4—121no—
319C4⋊C4#21712yes—
320C22⋊C4—22no—
321C22⋊C4—22no—
322C22⋊C4—124no—
323C42—121no—
324C42—121no—
325C42#23711yescharacteristic
326C42#23711yescharacteristic
327C42C22×Dic311yescharacteristic
328C42C2×#6711yescharacteristic
329C22×C3—24no—
330C22×C3—42no—
331C22×C3—42no—
332C22×C3—42no—
333C22×C3—42no—
334C22×C3—42no—
335C22×C3—42no—
336C22×C3#38611yescharacteristic
337C22×C3#44711yescharacteristic
338C22×C3#44811yescharacteristic
339C22×C3#45011yescharacteristic
340C22×C3#45211yescharacteristic
341C22×C3#45511yescharacteristic
342C22×C3#45611yescharacteristic
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—42no—
346C3×C4—82no—
347C3×C4C2×#10211yescharacteristic
348C3×C4#41711yescharacteristic
349C3×C4C4×SD1611yescharacteristic
350C3×C4#44011yescharacteristic
351C3×C4#46111yescharacteristic
352C3×C4#46311yescharacteristic
353C3×C4#48011yescharacteristic
354C3×C4#48111yescharacteristic
355Dic3—42no—
356Dic3—82no—
357Dic3—81no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23#93711yescharacteristic
366Q8—42no—
367Q8—42no—
368Q8—81no—
369Q8—81no—
370D4—22no—
371D4—22no—
372D4—22no—
373D4#83214yes—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—62no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4—122no—
390C2×C4—121no—
391C2×C4—122no—
392C2×C4—121no—
393C2×C4—121no—
394C2×C4—122no—
395C2×C4—248no—
396C2×C4D3×SD1611yescharacteristic
397C2×C4#91411yescharacteristic
398C2×C4#91511yescharacteristic
399C2×C4#91611yescharacteristic
400C2×C4C2×#21311yescharacteristic
401C2×C4C2×#21411yescharacteristic
402C2×C4D4×Dic311yescharacteristic
403C2×C4D4×Dic311yescharacteristic
404C2×C4#93511yescharacteristic
405C2×C4#93911yescharacteristic
406C2×C4#93911yescharacteristic
407C2×C4#94911yescharacteristic
408C2×C4#95111yescharacteristic
409C2×C4C2×#21711yescharacteristic
410C8—42no—
411C8—122no—
412C8—241no—
413C2×C3—44no—
414C2×C3#163911yescharacteristic
415C2×C3#164311yescharacteristic
416C2×C3#164711yescharacteristic
417C2×C3#165111yescharacteristic
418C2×C3#187411yescharacteristic
419C2×C3#187711yescharacteristic
420C2×C3#188211yescharacteristic
421C22—24no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22#503611yescharacteristic
429C22#504011yescharacteristic
430C22#516911yescharacteristic
431C22#517311yescharacteristic
432C22#520211yescharacteristic
433C22#520511yescharacteristic
434C22#522011yescharacteristic
435C4—21no—
436C4—21no—
437C4—42no—
438C4—82no—
439C4—122no—
440C4—242no—
441C4—241no—
442C4#503311yescharacteristic
443C4#503411yescharacteristic
444C4Dic3×SD1611yescharacteristic
445C4#516211yescharacteristic
446C4#516411yescharacteristic
447C4#516511yescharacteristic
448C4C2×#83211yescharacteristic
449C4#523211yescharacteristic
450C3#898511yescharacteristic, complemented, Sylow
451C2—44no—
452C2#7104411yescharacteristic
453C2#7104611yescharacteristic
454C2#7104811yescharacteristic
455C2#7105011yescharacteristic
456C2#7167711yescharacteristic
457C2#7204711yescharacteristic
458C2#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#8985
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
15. C3×#461C4semidirect
47. C3×C4⋊Q8C2×C4semidirect
72. C3×C4⋊C8C2×C4semidirect
86. #461Dic3semidirect
156. C4⋊Dic3SD16semidirect
178. C4⋊Q8C2×Dic3semidirect
203. C4⋊C8C2×Dic3semidirect
450. C3#8985semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4612
D4×C4⋊Dic32
#717872
#717802
C2×D4⋊(C3⋊C8)2
#677042
#677022
#89853