Finite Group Databasebeta

← #10691468

Subgroups — #10691468

Order 768.

550 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3101C211yesmaximal, characteristic
2#82319C211yesmaximal, characteristic
3#71839C211yesmaximal, characteristic, complemented
4#71835C211yesmaximal, characteristic, complemented
5#71598C211yesmaximal, characteristic
6#67746C211yesmaximal, characteristic, complemented
7#67736C211yesmaximal, characteristic, complemented
8#9235—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C2×#926—21no—
12C3×#483—21no—
13C3×#476C412yescomplemented
14C3×#461C412yescomplemented
15C3×#450—21no—
16C3×C4×SD16—22no—
17C3×#431C2211yescharacteristic
18#5221—21no—
19C4×#217—21no—
20#4984—21no—
21#4951—22no—
22C4×C4⋊Dic3C2211yescharacteristic
23C42×Dic3C2211yescharacteristic
24#4925C2211yescharacteristic
25#4924C2211yescharacteristic
26#4538—22no—
27D4⋊(C3⋊C8)—22no—
28#4532—21no—
29#4532—21no—
30#3101D311yescharacteristic
31#2293—31no—
32#1994—31no—
33#1503—31no—
34#1486—31no—
35#1303—31no—
36#1276—31no—
37C22×C3×D4D411yescharacteristic
38C23×C3×C4—21no—
39C2×C3×SD16—22no—
40C3×C4⋊Q8C2×C411yescharacteristic
41C3×C4⋊Q8C2×C411yescharacteristic
42C3×#125—21no—
43C3×C22⋊Q8—21no—
44C3×C22⋊Q8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C3×C4×D4C2×C412yes—
50C3×C4×D4D412yes—
51C3×#117—21no—
52C2×C3×C4⋊C4D411yescharacteristic
53C2×C3×C22⋊C4—21no—
54C2×C3×C42C2311yescharacteristic
55C2×#217—22no—
56C2×#217—22no—
57#926—42no—
58#926—42no—
59C2×C4⋊Dic3—22no—
60C2×C4⋊Dic3—21no—
61C2×C4×Dic3—21no—
62C2×C4×Dic3—21no—
63C2×C4×Dic3—21no—
64C2×C4×Dic3D411yescharacteristic
65C2×C4×Dic3D411yescharacteristic
66C2×#208—21no—
67C3×#106—22no—
68C3×C4⋊C8C2×C412yes—
69C3×#103—22no—
70C3×#102—22no—
71C3×C22⋊C8—21no—
72C3×C4×C8—21no—
73C3⋊#95—22no—
74C3⋊#95D412yes—
75#830—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4×C3⋊C8—21no—
80#533C2×D311yescharacteristic
81C2×C4×D4C2×D311yescharacteristic
82C2×#117—61no—
83#483—21no—
84#476Dic312yescomplemented
85#461Dic312yescomplemented
86#450—21no—
87C4×SD16—22no—
88#431—31no—
89#431C2×D311yescharacteristic
90#423—31no—
91#390—61no—
92#389—61no—
93#388—62no—
94C4×C4⋊C4—31no—
95C4×C22⋊C4—61no—
96C43—31no—
97#339—61no—
98#339—61no—
99#328—62no—
100D4⋊C8—62no—
101C24×C3—21no—
102C2×C3×Q8—21no—
103C2×C3×Q8—21no—
104C2×C3×D4—21no—
105C2×C3×D4—42no—
106C2×C3×D4C22⋊C412yes—
107C22×C3×C4—21no—
108C22×C3×C4—22no—
109C22×C3×C4—22no—
110C22×C3×C4—41no—
111C22×C3×C4—41no—
112C22×C3×C4—41no—
113C22×C3×C4—41no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×Dic3—21no—
118C22×Dic3—21no—
119C22×Dic3—22no—
120C3×SD16—44no—
121C2×C3×C8—22no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4C22⋊C412yes—
130C3×C22⋊C4—22no—
131C3×C22⋊C4—22no—
132C3×C22⋊C4—41no—
133C3×C42—41no—
134C3×C42C22×C411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42#4111yescharacteristic
137C3×C42#4111yescharacteristic
138#217—44no—
139#217—44no—
140C4⋊Dic3—44no—
141C4⋊Dic3—42no—
142C4×Dic3—21no—
143C4×Dic3—21no—
144C4×Dic3—21no—
145C4×Dic3—42no—
146C4×Dic3—42no—
147C4×Dic3—41no—
148C4×Dic3—41no—
149C4×Dic3SD1612yescomplemented
150#208—41no—
151#208—41no—
152C2×C3⋊C8—41no—
153C2×C3⋊C8—41no—
154C22×D4#6711yescharacteristic
155C23×C4—21no—
156C2×SD16—22no—
157C2×M16—61no—
158C4⋊Q8C2×Dic311yescharacteristic
159C4⋊Q8C2×Dic311yescharacteristic
160#125—21no—
161C22⋊Q8—21no—
162C22⋊Q8—21no—
163C4×Q8—21no—
164C4×Q8—21no—
165C4×D4—21no—
166C4×D4—21no—
167C4×D4C2×Dic312yes—
168C4×D4#6712yes—
169#117—21no—
170#117—122no—
171#117—122no—
172C2×C4⋊C4—62no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4#6711yescharacteristic
175C2×C22⋊C4—21no—
176C2×C22⋊C4—62no—
177C2×C22⋊C4—62no—
178C2×C42—31no—
179C2×C42—31no—
180C2×C42—61no—
181C2×C42—61no—
182C2×C42—61no—
183C2×C42C22×D311yescharacteristic
184#106—22no—
185C4⋊C8—61no—
186C4⋊C8—61no—
187C4⋊C8—61no—
188C4⋊C8C2×Dic312yes—
189#103—22no—
190#102—22no—
191C22⋊C8—21no—
192C22⋊C8—61no—
193C4×C8—21no—
194C4×C8—61no—
195#95—32no—
196#95—62no—
197C23×C3—22no—
198C23×C3—42no—
199C23×C3—42no—
200C23×C3—41no—
201C23×C3—41no—
202C23×C3#12011yescharacteristic
203C3×Q8—42no—
204C3×Q8—81no—
205C3×D4—22no—
206C3×D4—22no—
207C3×D4—42no—
208C2×C3×C4—22no—
209C2×C3×C4—22no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—42no—
214C2×C3×C4—42no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4C2×C22⋊C411yescharacteristic
220C2×C3×C4C4×D411yescharacteristic
221C2×C3×C4C4×D411yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12311yescharacteristic
225C2×C3×C4C2×SD1611yescharacteristic
226C2×C3×C4#13511yescharacteristic
227C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
228C2×C3×C4#13711yescharacteristic
229C2×Dic3—22no—
230C2×Dic3—22no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—42no—
236C2×Dic3—42no—
237C2×Dic3—42no—
238C3×C8—42no—
239C3⋊C8—81no—
240C3⋊C8—81no—
241C24—21no—
242C2×Q8—21no—
243C2×Q8—21no—
244C2×D4—21no—
245C2×D4—42no—
246C2×D4#21712yes—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—61no—
255C22×C4—61no—
256C22×C4—62no—
257C22×C4D3×D411yescharacteristic
258C22×C4D3×D411yescharacteristic
259C22×C4C2×#6711yescharacteristic
260SD16—44no—
261M16—121no—
262M16—121no—
263C2×C8—22no—
264C2×C8—121no—
265C2×C8—121no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—122no—
274C4⋊C4—124no—
275C4⋊C4#21712yes—
276C22⋊C4—22no—
277C22⋊C4—22no—
278C22⋊C4—41no—
279C22⋊C4—124no—
280C22⋊C4—124no—
281C42—32no—
282C42—41no—
283C42—61no—
284C42—61no—
285C42—61no—
286C42—122no—
287C42—122no—
288C42—121no—
289C42—121no—
290C42#23711yescharacteristic
291C42#23711yescharacteristic
292C42C22×Dic311yescharacteristic
293C42C2×#6711yescharacteristic
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—42no—
297C22×C3—42no—
298C22×C3—42no—
299C22×C3—82no—
300C22×C3—82no—
301C22×C3#38611yescharacteristic
302C22×C3#45011yescharacteristic
303C22×C3#45211yescharacteristic
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—42no—
308C3×C4—81no—
309C3×C4—81no—
310C3×C4#42112yes—
311C3×C4C4×SD1611yescharacteristic
312C3×C4#44011yescharacteristic
313C3×C4#46111yescharacteristic
314C3×C4#46311yescharacteristic
315Dic3—42no—
316Dic3—42no—
317Dic3—82no—
318C23—22no—
319C23—42no—
320C23—42no—
321C23—41no—
322C23—41no—
323C23#93711yescharacteristic
324Q8—42no—
325Q8—81no—
326D4—22no—
327D4—22no—
328D4—42no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—62no—
341C2×C4—62no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—121no—
345C2×C4—121no—
346C2×C4—122no—
347C2×C4—122no—
348C2×C4—122no—
349C2×C4D3×SD1611yescharacteristic
350C2×C4#91411yescharacteristic
351C2×C4#91511yescharacteristic
352C2×C4#91611yescharacteristic
353C2×C4D4×Dic311yescharacteristic
354C2×C4D4×Dic311yescharacteristic
355C2×C4#93511yescharacteristic
356C2×C4#93911yescharacteristic
357C2×C4#93911yescharacteristic
358C2×C4C2×#21711yescharacteristic
359C8—42no—
360C8—241no—
361C8—241no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3—82no—
365C2×C3#187711yescharacteristic
366C2×C3#189711yescharacteristic
367C2×C3#190611yescharacteristic
368C22—21no—
369C22—21no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—82no—
374C22—82no—
375C22#516911yescharacteristic
376C22#517311yescharacteristic
377C22#522011yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—42no—
382C4—81no—
383C4—81no—
384C4—122no—
385C4—122no—
386C4—242no—
387C4Dic3×SD1611yescharacteristic
388C4#516211yescharacteristic
389C4#516411yescharacteristic
390C4#516511yescharacteristic
391C4#523612yes—
392C3#923511yescharacteristic, complemented, Sylow
393C2—21no—
394C2—21no—
395C2—82no—
396C2#7171211yescharacteristic
397C2#7204711yescharacteristic
398C2#7211411yescharacteristic
subgroupG/H
centerC2×C4#939
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3101C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenter#3101D3
hypocenterC3#9235
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3101

Complements

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

NH ≅ G/Ntype
3. #71839C2semidirect
4. #71835C2semidirect
6. #67746C2semidirect
7. #67736C2semidirect
13. C3×#476C4semidirect
14. C3×#461C4semidirect
84. #476Dic3semidirect
85. #461Dic3semidirect
149. C4×Dic3SD16semidirect
392. C3#9235semidirect

Maximal subgroups

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

subgroupindex
C3×#31012
#823192
#718392
#718352
#715982
#677462
#677362
#92353