Finite Group Databasebeta

← #10690424

Subgroups — #10690424

Order 768.

500 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3167C211yesmaximal, characteristic
2#83277C211yesmaximal, characteristic
3#71841C211yesmaximal, characteristic, complemented
4#71824C211yesmaximal, characteristic, complemented
5#71811C211yesmaximal, characteristic
6#67656C211yesmaximal, characteristic, complemented
7#67655C211yesmaximal, characteristic, complemented
8#9055—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C3×#532C2211yescharacteristic
11C2×#948—21no—
12C3×#498C411yescharacteristic, complemented
13C3×#492C411yescharacteristic, complemented
14C3×#487C411yescharacteristic, complemented
15C3×#486C411yescharacteristic, complemented
16C3×#465—22no—
17C3×C4×SD16—22no—
18C2×C3×C4×C8C2211yescharacteristic
19#5234—21no—
20#5012—21no—
21#5011—22no—
22D4×C3⋊C8—21no—
23C4×C4⋊Dic3C2211yescharacteristic
24#4925C2211yescharacteristic
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26C4×#208C2211yescharacteristic
27#4537—22no—
28#4535—21no—
29#4535—21no—
30#4532—21no—
31#4532—21no—
32#3167D311yescharacteristic
33#2935—31no—
34#1986—31no—
35#1502—31no—
36#1481—31no—
37#1279—31no—
38#1278—31no—
39C2×C3×#41—21no—
40C2×C3×SD16—22no—
41C22×C3×C8—21no—
42C3×C4⋊Q8C2×C411yescharacteristic
43C3×C4⋊Q8C2×C411yescharacteristic
44C3×C4⋊Q8C2×C411yescharacteristic
45C3×#127C2×C411yescharacteristic
46C3×#125—21no—
47C3×#124—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×#121—21no—
51C3×#121D412yes—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C3×C4×D4—21no—
55C3×C4×D4D412yes—
56C3×#117—21no—
57C2×C3×C42C2311yescharacteristic
58#948—42no—
59#948—42no—
60C2×C4⋊Dic3—21no—
61C2×C4×Dic3—21no—
62C2×#208—22no—
63C2×#208D411yescharacteristic
64C2×#208D411yescharacteristic
65C22×C3⋊C8—22no—
66C22×C3⋊C8—21no—
67C3×#106—22no—
68C3×#103—22no—
69C3×#102—22no—
70C3×C4×C8—21no—
71C3×C4×C8C2×C411yescharacteristic
72C3×C4×C8C2×C411yescharacteristic
73C3⋊#95—21no—
74#830—22no—
75#830—22no—
76#830—21no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)D411yescharacteristic
80C4⋊(C3⋊C8)D411yescharacteristic
81#803—21no—
82C4×C3⋊C8—21no—
83C4×C3⋊C8—21no—
84C2×#131—61no—
85#533C2×D311yescharacteristic
86#532C2×D311yescharacteristic
87#498Dic311yescharacteristic, complemented
88#492Dic311yescharacteristic, complemented
89#487Dic311yescharacteristic, complemented
90#486Dic311yescharacteristic, complemented
91#465—22no—
92C4×SD16—22no—
93#436—61no—
94#435—62no—
95C8×D4—61no—
96#431—31no—
97C2×C4⋊C8—31no—
98#408—61no—
99C4×M16—31no—
100C2×C4×C8C2×D311yescharacteristic
101C4×C4⋊C4—31no—
102#339—61no—
103#339—61no—
104#327—62no—
105Q8⋊C8—61no—
106Q8⋊C8—61no—
107C3×#41—41no—
108C3×#41—41no—
109C3×#41—41no—
110C3×#41—41no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—21no—
114C2×C3×D4—22no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4C22⋊C412yes—
118C22×C3×C4—22no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4#4111yescharacteristic
122C22×Dic3—41no—
123C3×SD16—44no—
124C2×C3×C8—21no—
125C2×C3×C8—21no—
126C2×C3×C8—42no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4C22⋊C412yes—
135C3×C22⋊C4—22no—
136C3×C22⋊C4—22no—
137C3×C22⋊C4—41no—
138C3×C42—41no—
139C3×C42C22×C411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42#4111yescharacteristic
143C4⋊Dic3—22no—
144C4⋊Dic3—41no—
145C4×Dic3—41no—
146C4×Dic3—41no—
147#208—21no—
148#208—21no—
149#208—21no—
150#208—44no—
151#208SD1612yescomplemented
152C2×C3⋊C8—21no—
153C2×C3⋊C8—21no—
154C2×C3⋊C8—21no—
155C2×C3⋊C8—21no—
156C2×C3⋊C8—44no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159C2×C3⋊C8—41no—
160C2×#41—21no—
161C2×SD16—22no—
162#131—122no—
163#131—122no—
164C2×M16—31no—
165C2×M16—31no—
166C2×M16—62no—
167C22×C8—21no—
168C22×C8—62no—
169C22×C8—61no—
170C4⋊Q8C2×Dic311yescharacteristic
171C4⋊Q8C2×Dic311yescharacteristic
172C4⋊Q8C2×Dic311yescharacteristic
173#127C2×Dic311yescharacteristic
174#125—21no—
175#124—21no—
176C22⋊Q8—21no—
177C22⋊Q8—21no—
178#121—21no—
179#121#6712yes—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×D4—21no—
183C4×D4#6712yes—
184#117—21no—
185C2×C4⋊C4—61no—
186C2×C42—61no—
187C2×C42C22×D311yescharacteristic
188#106—22no—
189C4⋊C8—31no—
190C4⋊C8—31no—
191C4⋊C8—61no—
192C4⋊C8—61no—
193#103—22no—
194#102—22no—
195C22⋊C8—62no—
196C22⋊C8—62no—
197C22⋊C8—61no—
198#97—61no—
199C4×C8—21no—
200C4×C8—61no—
201C4×C8—61no—
202C4×C8C2×Dic311yescharacteristic
203C4×C8C2×Dic311yescharacteristic
204#95—61no—
205C23×C3—22no—
206C23×C3#12111yescharacteristic
207C3×Q8—42no—
208C3×Q8—41no—
209C3×Q8—41no—
210C3×Q8—41no—
211C3×Q8—41no—
212C3×D4—42no—
213C3×D4—42no—
214C3×D4—42no—
215C3×D4—41no—
216C3×D4—41no—
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—42no—
224C2×C3×C4—42no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4C2×C22⋊C411yescharacteristic
228C2×C3×C4C4×D411yescharacteristic
229C2×C3×C4C4×D411yescharacteristic
230C2×C3×C4#12011yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12311yescharacteristic
233C2×C3×C4C2×SD1611yescharacteristic
234C2×C3×C4#13511yescharacteristic
235C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
236C2×C3×C4#13711yescharacteristic
237C2×Dic3—42no—
238C2×Dic3—81no—
239C2×Dic3—81no—
240C3×C8—22no—
241C3×C8—41no—
242C3⋊C8—22no—
243C3⋊C8—41no—
244C3⋊C8—41no—
245C3⋊C8—41no—
246C3⋊C8—81no—
247#41—41no—
248#41—41no—
249#41—41no—
250#41—41no—
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—21no—
254C2×D4—22no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4#21712yes—
258C22×C4—22no—
259C22×C4—121no—
260C22×C4D3×D411yescharacteristic
261C22×C4#23711yescharacteristic
262C22×C4C2×#6711yescharacteristic
263SD16—44no—
264M16—32no—
265M16—61no—
266M16—61no—
267M16—61no—
268M16—124no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—42no—
272C2×C8—61no—
273C2×C8—61no—
274C2×C8—61no—
275C2×C8—61no—
276C2×C8—124no—
277C2×C8—121no—
278C2×C8—121no—
279C2×C8—121no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—62no—
288C4⋊C4—121no—
289C4⋊C4#21712yes—
290C22⋊C4—22no—
291C22⋊C4—22no—
292C22⋊C4—41no—
293C42—41no—
294C42—121no—
295C42—121no—
296C42D3×D411yescharacteristic
297C42#23711yescharacteristic
298C42C22×Dic311yescharacteristic
299C42C2×#6711yescharacteristic
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—42no—
303C22×C3—42no—
304C22×C3—42no—
305C22×C3#38611yescharacteristic
306C22×C3#46111yescharacteristic
307C22×C3#46311yescharacteristic
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—42no—
312C3×C4—81no—
313C3×C4—81no—
314C3×C4C4×SD1611yescharacteristic
315C3×C4#44011yescharacteristic
316C3×C4#44311yescharacteristic
317C3×C4#44411yescharacteristic
318C3×C4#45011yescharacteristic
319C3×C4#45211yescharacteristic
320Dic3—82no—
321C23—22no—
322C23#93911yescharacteristic
323Q8—42no—
324Q8—41no—
325Q8—41no—
326Q8—41no—
327Q8—41no—
328D4—42no—
329D4—42no—
330D4—42no—
331D4—41no—
332D4—41no—
333D4—41no—
334D4—41no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—122no—
344C2×C4—241no—
345C2×C4—241no—
346C2×C4D3×SD1611yescharacteristic
347C2×C4#91411yescharacteristic
348C2×C4#91511yescharacteristic
349C2×C4#91611yescharacteristic
350C2×C4D4×Dic311yescharacteristic
351C2×C4D4×Dic311yescharacteristic
352C2×C4#93511yescharacteristic
353C2×C4#93711yescharacteristic
354C2×C4#93911yescharacteristic
355C2×C4C2×#21711yescharacteristic
356C8—22no—
357C8—41no—
358C8—62no—
359C8—121no—
360C8—121no—
361C8—121no—
362C8—241no—
363C2×C3—21no—
364C2×C3—21no—
365C2×C3—82no—
366C2×C3#187611yescharacteristic
367C2×C3#187711yescharacteristic
368C2×C3#188411yescharacteristic
369C22—21no—
370C22—21no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22#516211yescharacteristic
375C22#516511yescharacteristic
376C22#522011yescharacteristic
377C4—21no—
378C4—21no—
379C4—21no—
380C4—42no—
381C4—81no—
382C4—81no—
383C4—242no—
384C4Dic3×SD1611yescharacteristic
385C4#516411yescharacteristic
386C4#516911yescharacteristic
387C4#517311yescharacteristic
388C4#519611yescharacteristic
389C4#519711yescharacteristic
390C3#905511yescharacteristic, complemented, Sylow
391C2—21no—
392C2—21no—
393C2—82no—
394C2#7181711yescharacteristic
395C2#7204711yescharacteristic
396C2#7208711yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3167C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#3167D3
hypocenterC3#9055
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
3. #71841C2semidirect
4. #71824C2semidirect
6. #67656C2semidirect
7. #67655C2semidirect
12. C3×#498C4semidirect
13. C3×#492C4semidirect
14. C3×#487C4semidirect
15. C3×#486C4semidirect
87. #498Dic3semidirect
88. #492Dic3semidirect
89. #487Dic3semidirect
90. #486Dic3semidirect
151. #208SD16semidirect
390. C3#9055semidirect

Maximal subgroups

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

subgroupindex
C3×#31672
#832772
#718412
#718242
#718112
#676562
#676552
#90553