Finite Group Databasebeta

← #10795148

Subgroups — #10795148

Order 768.

1262 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#887C211yesmaximal, characteristic
2#78619C212yesmaximal
3C3×#1821C211yesmaximal, characteristic
4#71562C211yesmaximal, characteristic
5#69990C212yesmaximal
6#68997C214yesmaximal, complemented
7#68973C214yesmaximal, complemented
8#20727—31nomaximal, Sylow
9C23×Dic6C2211yescharacteristic
10C22×C3×C4⋊C4C2211yescharacteristic
11C22×Dic3⋊C4C412yes—
12C22×C4×Dic3C412yes—
13C22×C4×Dic3C2211yescharacteristic
14C2×#887—22no—
15C2×#887—22no—
16C2×#887—22no—
17C2×#887—22no—
18C2×#887C412yes—
19C2×#887C412yes—
20C2×#887C2212yes—
21C2×#887C2212yes—
22C3×#407C2212yes—
23#5207C2212yes—
24C2×#824C2218yes—
25#5118C2214yes—
26Dic3×M16C2214yes—
27#4719—22no—
28#4719—22no—
29#4718—22no—
30#4718—22no—
31C3×#343C2214yes—
32#4530C2214yes—
33C22×C4×Q8—31no—
34#2920—32no—
35#1821—31no—
36#1821D311yescharacteristic
37#1801—34no—
38#1784—32no—
39C4×#101—34no—
40C23×C3×C4C2×C412yes—
41C23×C3×C4C2311yescharacteristic
42C23×Dic3C2×C412yes—
43C23×Dic3C2×C411yescharacteristic
44C23×Dic3C2×C411yescharacteristic
45C22×Dic6—22no—
46C22×Dic6—22no—
47C22×Dic6—28no—
48C22×Dic6C2×C412yes—
49C22×Dic6C2312yes—
50C2×C3×M16C2314yes—
51C2×C3×C4⋊C4—22no—
52C2×C3×C4⋊C4—22no—
53C2×C3×C4⋊C4C2×C412yes—
54C2×C3×C4⋊C4C2312yes—
55C2×Dic3⋊C4—24no—
56C2×Dic3⋊C4—24no—
57C2×Dic3⋊C4C2×C414yes—
58C2×Dic3⋊C4C2×C414yes—
59C2×C4×Dic3—22no—
60C2×C4×Dic3—22no—
61C2×C4×Dic3—22no—
62C2×C4×Dic3—22no—
63C2×C4×Dic3C2×C414yes—
64C2×C4×Dic3C2×C414yes—
65C2×C4×Dic3C2×C411yescharacteristic
66C2×C4×Dic3C2×C411yescharacteristic
67C2×C4×Dic3D412yes—
68C2×C4×Dic3D412yes—
69C2×C4×Dic3D412yes—
70C2×C4×Dic3D412yes—
71C2×C4×Dic3C2311yescharacteristic
72C2×C4×Dic3C2311yescharacteristic
73C2×#208C2314yes—
74#887—24no—
75#887—24no—
76#887—24no—
77#887—24no—
78#887—24no—
79#887—24no—
80#887—24no—
81#887—24no—
82C3×C22⋊C8—22no—
83C3×C22⋊C8—22no—
84#830—22no—
85#830—22no—
86#824—216no—
87#815—24no—
88Dic3⋊C8—24no—
89Dic3⋊C8—24no—
90C8×Dic3—24no—
91C23×Q8—31no—
92C2×C4×Q8—32no—
93C2×C4×Q8—32no—
94C2×C4×Q8—32no—
95C2×C4×Q8—32no—
96C2×C4×Q8—62no—
97C2×C4×Q8—62no—
98C2×C4×Q8—62no—
99C2×C4×Q8—62no—
100C22×C4⋊C4—32no—
101C22×C4⋊C4C2×D311yescharacteristic
102C22×C42—31no—
103C22×C42—32no—
104#432—62no—
105#432—62no—
106#431—62no—
107#431—62no—
108#423—34no—
109C2×#101—38no—
110#407—32no—
111#407C2×D312yes—
112C4×M16—34no—
113#343—34no—
114#343C2×D314yes—
115C24×C3C22×C411yescharacteristic
116C22×C3×C4—24no—
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—44no—
120C22×C3×C4C22×C411yescharacteristic
121C22×C3×C4C22×C414yes—
122C22×C3×C4C2×D412yes—
123C22×C3×C4C2×D412yes—
124C22×C3×C4C2411yescharacteristic
125C22×Dic3—22no—
126C22×Dic3—22no—
127C22×Dic3—24no—
128C22×Dic3—28no—
129C22×Dic3—42no—
130C22×Dic3C22⋊C411yescharacteristic
131C22×Dic3C22⋊C411yescharacteristic
132C22×Dic3C22⋊C411yescharacteristic
133C22×Dic3C22⋊C412yes—
134C22×Dic3C22⋊C412yes—
135C22×Dic3C22⋊C411yescharacteristic
136C22×Dic3C22×C411yescharacteristic
137C22×Dic3C22×C411yescharacteristic
138C22×Dic3C22×C414yes—
139C22×Dic3C22×C411yescharacteristic
140C22×Dic3C22×C411yescharacteristic
141C2×Dic6—216no—
142C2×Dic6—216no—
143C2×Dic6—24no—
144C2×Dic6—24no—
145C2×Dic6—24no—
146C2×Dic6—28no—
147C2×Dic6#4118yes—
148C3×M16—28no—
149C2×C3×C8—24no—
150C3×C4⋊C4—24no—
151C3×C4⋊C4—24no—
152Dic3⋊C4—28no—
153Dic3⋊C4—28no—
154C4×Dic3—24no—
155C4×Dic3—24no—
156C4×Dic3—24no—
157C4×Dic3—24no—
158C4×Dic3C22⋊C412yes—
159C4×Dic3C22⋊C412yes—
160C4×Dic3C22⋊C412yes—
161C4×Dic3C22⋊C412yes—
162C4×Dic3C2×D412yes—
163C4×Dic3C2×D412yes—
164C4×Dic3C2×D412yes—
165C4×Dic3C2×D412yes—
166#208—28no—
167C2×C3⋊C8—24no—
168C22×Q8—32no—
169C22×Q8—32no—
170C22×Q8—68no—
171C22×Q8—62no—
172C22×Q8—62no—
173C23×C4—31no—
174C23×C4—31no—
175C23×C4—32no—
176C23×C4C4×D312yes—
177C23×C4C22×D311yescharacteristic
178C2×M16—34no—
179C2×M16C22×D314yes—
180C4×Q8—64no—
181C4×Q8—64no—
182C4×Q8—64no—
183C4×Q8—64no—
184C4×Q8—64no—
185C4×Q8—64no—
186C4×Q8—64no—
187C4×Q8—64no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—34no—
191C2×C4⋊C4—34no—
192C2×C4⋊C4—64no—
193C2×C4⋊C4—64no—
194C2×C4⋊C4C4×D312yes—
195C2×C4⋊C4C22×D312yes—
196C2×C42—31no—
197C2×C42—31no—
198C2×C42—32no—
199C2×C42—32no—
200C2×C42—32no—
201C2×C42—32no—
202C2×C42—31no—
203C2×C42—31no—
204C2×C42—34no—
205C2×C42—34no—
206C2×C42—62no—
207C2×C42—62no—
208C2×C42—62no—
209C2×C42—62no—
210C4⋊C8—64no—
211C4⋊C8—64no—
212#101—616no—
213C22⋊C8—22no—
214C22⋊C8—22no—
215C22⋊C8—62no—
216C22⋊C8—62no—
217#97—64no—
218C4×C8—64no—
219C23×C3—22no—
220C23×C3—22no—
221C23×C3—22no—
222C23×C3C2×C22⋊C411yescharacteristic
223C23×C3C2×C22⋊C411yescharacteristic
224C23×C3C23×C411yescharacteristic
225C2×C3×C4—22no—
226C2×C3×C4—24no—
227C2×C3×C4—24no—
228C2×C3×C4—24no—
229C2×C3×C4—44no—
230C2×C3×C4—44no—
231C2×C3×C4—44no—
232C2×C3×C4C2×C22⋊C412yes—
233C2×C3×C4C22×D412yes—
234C2×C3×C4C2×#4114yes—
235C2×Dic3—28no—
236C2×Dic3—28no—
237C2×Dic3—24no—
238C2×Dic3—24no—
239C2×Dic3—22no—
240C2×Dic3—24no—
241C2×Dic3—22no—
242C2×Dic3—22no—
243C2×Dic3—22no—
244C2×Dic3—22no—
245C2×Dic3—42no—
246C2×Dic3—41no—
247C2×Dic3—41no—
248C2×Dic3#10114yescomplemented
249C2×Dic3C2×C22⋊C411yescharacteristic
250C2×Dic3C2×C22⋊C411yescharacteristic
251C2×Dic3C2×C22⋊C412yes—
252C2×Dic3C2×C22⋊C412yes—
253C2×Dic3C2×C22⋊C411yescharacteristic
254C2×Dic3C2×C22⋊C411yescharacteristic
255C2×Dic3#11718yes—
256Dic6—216no—
257Dic6—216no—
258C3×C8—44no—
259C3⋊C8—44no—
260C24C2×C4×D311yescharacteristic
261C2×Q8—38no—
262C2×Q8—68no—
263C2×Q8—616no—
264C2×Q8—616no—
265C2×Q8—64no—
266C2×Q8—64no—
267C2×Q8—64no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—31no—
272C22×C4—31no—
273C22×C4—31no—
274C22×C4—31no—
275C22×C4—32no—
276C22×C4—31no—
277C22×C4—31no—
278C22×C4—34no—
279C22×C4—32no—
280C22×C4—31no—
281C22×C4—31no—
282C22×C4—44no—
283C22×C4—68no—
284C22×C4—64no—
285C22×C4—62no—
286C22×C4—62no—
287C22×C4—122no—
288C22×C4C2×C4×D311yescharacteristic
289C22×C4C2×C4×D314yes—
290C22×C4D3×D412yes—
291C22×C4D3×D412yes—
292C22×C4C23×D311yescharacteristic
293M16—28no—
294M16—68no—
295C2×C8—24no—
296C2×C8—64no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—68no—
300C4⋊C4—68no—
301C42—32no—
302C42—32no—
303C42—32no—
304C42—32no—
305C42—32no—
306C42—32no—
307C42—32no—
308C42—32no—
309C42—64no—
310C42—64no—
311C42—64no—
312C42—64no—
313C22×C3—28no—
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3C2×#10112yes—
321C22×C3#41312yes—
322C22×C3C22×C22⋊C411yescharacteristic
323C22×C3C2×#11712yes—
324C3×C4—22no—
325C3×C4—44no—
326C3×C4#52514yes—
327Dic3—21no—
328Dic3—21no—
329Dic3—22no—
330Dic3—22no—
331Dic3—28no—
332Dic3C2×#10112yes—
333Dic3C2×#10112yes—
334C23—22no—
335C23—22no—
336C23—22no—
337C23D3×C22⋊C411yescharacteristic
338C23D3×C22⋊C411yescharacteristic
339C23C22×C4×D311yescharacteristic
340Q8—616no—
341Q8—616no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—31no—
347C2×C4—31no—
348C2×C4—34no—
349C2×C4—31no—
350C2×C4—31no—
351C2×C4—32no—
352C2×C4—32no—
353C2×C4—38no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—62no—
358C2×C4—62no—
359C2×C4—62no—
360C2×C4—62no—
361C2×C4—68no—
362C2×C4—68no—
363C2×C4—64no—
364C2×C4—64no—
365C2×C4—62no—
366C2×C4—64no—
367C2×C4—122no—
368C2×C4—121no—
369C2×C4—121no—
370C2×C4D3×C22⋊C412yes—
371C2×C4C2×D3×D412yes—
372C2×C4D3×#4114yes—
373C8—44no—
374C8—124no—
375C2×C3—22no—
376C2×C3—22no—
377C2×C3—22no—
378C2×C3#228011yescharacteristic
379C2×C3C22×#10111yescharacteristic
380C2×C3C2×#41311yescharacteristic
381C22—28no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22#474512yes—
389C22D3×#10112yes—
390C22C2×D3×C22⋊C411yescharacteristic
391C22D3×#11712yes—
392C4—22no—
393C4—32no—
394C4—32no—
395C4—44no—
396C4—68no—
397C4—62no—
398C4—62no—
399C4—61no—
400C4—61no—
401C4#565614yes—
402C3#2072711yescharacteristic, complemented, Sylow
403C2—22no—
404C2—22no—
405C2—22no—
406C2#7405011yescharacteristic
407C2C2×#474511yescharacteristic
408C2C2×D3×#10111yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1821C2
socleC22×C3C22×C22⋊C4
prime radicalC24×C3C22×C4
cosocleC22×C3×C4C24
hypercenter#1821D3
hypocenterC3#20727
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
6. #68997C2semidirect
7. #68973C2semidirect
248. C2×Dic3#101semidirect
402. C3#20727semidirect

Maximal subgroups

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

subgroupindex
C22×#8872
#786192
C3×#18212
#715622
#699902
#689972
#689732
#207273