Finite Group Databasebeta

← #10692952

Subgroups — #10692952

Order 768.

934 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2408C211yesmaximal, characteristic
2#82301C211yesmaximal, characteristic, complemented
3#72128C212yesmaximal, complemented
4#71619C211yesmaximal, characteristic, complemented
5#67625C212yesmaximal, complemented
6#9468—31nomaximal, Sylow
7C23×C3×D4C2211yescharacteristic
8C2×C3×#121—21no—
9C2×C3×#120—21no—
10C2×C3×C4×D4C2211yescharacteristic
11C3×#394C412yescomplemented
12C3×#392C412yescomplemented
13C3×#390—21no—
14C3×#380C2211yescharacteristic
15C3×#379—21no—
16C2×#834—21no—
17C2×#832—24no—
18#5212C2212yes—
19C2×#830—21no—
20C2×#830C2212yes—
21D4×C3⋊C8—24no—
22#4925—21no—
23#4526—21no—
24#2972—31no—
25#2408D311yescharacteristic
26#1799—31no—
27#1786—32no—
28#1289—32no—
29C25×C3—21no—
30C22×C3×D4—22no—
31C22×C3×D4—22no—
32C22×C3×D4—21no—
33C22×C3×D4—22no—
34C22×C3×D4—44no—
35C22×C3×D4C2×C412yes—
36C22×C3×D4D411yescharacteristic
37C23×C3×C4D411yescharacteristic
38C23×C3×C4C2311yescharacteristic
39C3×#121—44no—
40C3×#120—42no—
41C3×#120—42no—
42C3×C4×D4—22no—
43C3×C4×D4—22no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4C2×C411yescharacteristic
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4—22no—
48C2×C3×C22⋊C4—22no—
49C2×C3×C22⋊C4—22no—
50C2×C3×C22⋊C4C2×C411yescharacteristic
51C2×C3×C22⋊C4D411yescharacteristic
52C2×C3×C42D411yescharacteristic
53C2×#217—22no—
54C2×C4⋊Dic3—22no—
55C22×C3⋊C8—22no—
56C22×C3⋊C8D414yescomplemented
57C3×#95C2×C412yes—
58#834—42no—
59#832—48no—
60C3⋊#95—22no—
61#830—22no—
62#830—22no—
63#830—22no—
64C4⋊(C3⋊C8)—22no—
65C4×C3⋊C8—22no—
66C23×D4C2×D311yescharacteristic
67C2×#121—21no—
68C2×#120—21no—
69C2×C4×D4C2×D311yescharacteristic
70C8×D4—64no—
71#431—61no—
72C2×#102—64no—
73C2×#99—61no—
74C2×C22⋊C8—32no—
75C2×C22⋊C8—61no—
76#394Dic312yescomplemented
77#392Dic312yescomplemented
78#390—21no—
79#380—32no—
80#380C2×D311yescharacteristic
81#379—21no—
82#342—61no—
83C24×C3—22no—
84C24×C3—44no—
85C24×C3—42no—
86C24×C3—41no—
87C24×C3—41no—
88C24×C3—42no—
89C24×C3—82no—
90C24×C3C22⋊C411yescharacteristic
91C24×C3C2×D411yescharacteristic
92C2×C3×D4—22no—
93C2×C3×D4—22no—
94C2×C3×D4—28no—
95C2×C3×D4—24no—
96C2×C3×D4—24no—
97C2×C3×D4—44no—
98C2×C3×D4—42no—
99C2×C3×D4—44no—
100C2×C3×D4—48no—
101C2×C3×D4—44no—
102C2×C3×D4—44no—
103C2×C3×D4—41no—
104C2×C3×D4—41no—
105C2×C3×D4—82no—
106C2×C3×D4D812yes—
107C2×C3×D4SD1612yes—
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4—22no—
111C22×C3×C4—42no—
112C22×C3×C4—42no—
113C22×C3×C4C22⋊C411yescharacteristic
114C22×C3×C4C22⋊C411yescharacteristic
115C22×C3×C4C22⋊C411yescharacteristic
116C22×C3×C4C22×C411yescharacteristic
117C22×C3×C4C2×D412yes—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4#4112yes—
120C22×Dic3—42no—
121C3×C4⋊C4—22no—
122C3×C4⋊C4—42no—
123C3×C22⋊C4—22no—
124C3×C22⋊C4—22no—
125C3×C22⋊C4—44no—
126C3×C22⋊C4—41no—
127C3×C22⋊C4—41no—
128C3×C22⋊C4—44no—
129C3×C22⋊C4—82no—
130C3×C42—21no—
131C3×C42—21no—
132#217—42no—
133#217—42no—
134C4⋊Dic3—44no—
135C2×C3⋊C8—24no—
136C2×C3⋊C8—44no—
137C2×C3⋊C8—82no—
138C25—21no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—21no—
143C22×D4—44no—
144C22×D4C2×Dic312yes—
145C22×D4#6711yescharacteristic
146C23×C4#6711yescharacteristic
147C23×C4C22×D311yescharacteristic
148C22×C8—34no—
149C22×C8—62no—
150#121—44no—
151#120—42no—
152#120—42no—
153C4×D4—22no—
154C4×D4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—62no—
157C2×C4⋊C4C2×Dic311yescharacteristic
158C2×C22⋊C4—21no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—62no—
163C2×C22⋊C4C2×Dic311yescharacteristic
164C2×C22⋊C4#6711yescharacteristic
165C2×C42#6711yescharacteristic
166C4⋊C8—62no—
167#102—128no—
168#99—122no—
169C22⋊C8—62no—
170C22⋊C8—62no—
171C22⋊C8—62no—
172C4×C8—62no—
173#95—62no—
174#95C2×Dic312yes—
175C23×C3—21no—
176C23×C3—21no—
177C23×C3—24no—
178C23×C3—41no—
179C23×C3—41no—
180C23×C3—41no—
181C23×C3—41no—
182C23×C3—42no—
183C23×C3—48no—
184C23×C3—42no—
185C23×C3—42no—
186C23×C3—42no—
187C23×C3—42no—
188C23×C3—42no—
189C23×C3—84no—
190C23×C3—84no—
191C23×C3—84no—
192C23×C3—88no—
193C23×C3—82no—
194C23×C3—82no—
195C23×C3#10212yescomplemented
196C23×C3C2×C22⋊C411yescharacteristic
197C23×C3#12112yes—
198C23×C3C2×D811yescharacteristic
199C23×C3C2×SD1611yescharacteristic
200C23×C3#13512yes—
201C3×D4—24no—
202C3×D4—24no—
203C3×D4—28no—
204C3×D4—44no—
205C3×D4—44no—
206C3×D4—44no—
207C3×D4—88no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—24no—
211C2×C3×C4—22no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—41no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—81no—
223C2×C3×C4#10212yescomplemented
224C2×C3×C4C4×D412yes—
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12311yescharacteristic
227C2×Dic3—82no—
228C2×Dic3—82no—
229C2×Dic3—82no—
230C3⋊C8—44no—
231C24—22no—
232C24—44no—
233C24—42no—
234C24—41no—
235C24—41no—
236C24—42no—
237C24—82no—
238C24#21711yescharacteristic
239C24C2×#6711yescharacteristic
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—28no—
243C2×D4—24no—
244C2×D4—24no—
245C2×D4—42no—
246C2×D4—44no—
247C2×D4—44no—
248C2×D4—48no—
249C2×D4—44no—
250C2×D4—44no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—82no—
254C2×D4#21312yes—
255C2×D4#21412yes—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—122no—
262C22×C4#21711yescharacteristic
263C22×C4#21711yescharacteristic
264C22×C4#21711yescharacteristic
265C22×C4D3×D412yes—
266C22×C4#23712yes—
267C22×C4C22×Dic311yescharacteristic
268C22×C4C2×#6711yescharacteristic
269C2×C8—64no—
270C2×C8—124no—
271C2×C8—242no—
272C4⋊C4—22no—
273C4⋊C4—42no—
274C4⋊C4—124no—
275C22⋊C4—22no—
276C22⋊C4—22no—
277C22⋊C4—44no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—44no—
281C22⋊C4—82no—
282C22⋊C4—122no—
283C22⋊C4—122no—
284C42—21no—
285C42—21no—
286C22×C3—22no—
287C22×C3—21no—
288C22×C3—22no—
289C22×C3—21no—
290C22×C3—22no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3—42no—
294C22×C3—42no—
295C22×C3—44no—
296C22×C3—44no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—44no—
300C22×C3—81no—
301C22×C3—82no—
302C22×C3—82no—
303C22×C3—88no—
304C22×C3—88no—
305C22×C3—82no—
306C22×C3—82no—
307C22×C3#38611yescharacteristic
308C22×C3C2×#10211yescharacteristic
309C22×C3#41611yescharacteristic
310C22×C3#46512yes—
311C22×C3#46612yes—
312C3×C4—21no—
313C3×C4—22no—
314C3×C4—22no—
315C3×C4—41no—
316C3×C4—41no—
317C3×C4—82no—
318C3×C4B(4,𝔽2)12yes—
319Dic3—162no—
320C23—21no—
321C23—21no—
322C23—24no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—42no—
328C23—48no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—84no—
335C23—84no—
336C23—84no—
337C23—88no—
338C23—82no—
339C23—82no—
340C23#83212yescomplemented
341C23C2×#21311yescharacteristic
342C23C2×#21411yescharacteristic
343C23#93912yes—
344C23#95012yes—
345C23C2×#21711yescharacteristic
346D4—24no—
347D4—24no—
348D4—28no—
349D4—44no—
350D4—44no—
351D4—44no—
352D4—88no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—24no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—81no—
368C2×C4—242no—
369C2×C4—242no—
370C2×C4—242no—
371C2×C4#83212yescomplemented
372C2×C4D4×Dic312yes—
373C2×C4#93511yescharacteristic
374C2×C4#93711yescharacteristic
375C8—124no—
376C2×C3—22no—
377C2×C3—21no—
378C2×C3—21no—
379C2×C3—22no—
380C2×C3—42no—
381C2×C3—84no—
382C2×C3#188911yescharacteristic
383C2×C3#189311yescharacteristic
384C2×C3#191011yescharacteristic
385C22—21no—
386C22—22no—
387C22—21no—
388C22—22no—
389C22—22no—
390C22—41no—
391C22—41no—
392C22—42no—
393C22—42no—
394C22—44no—
395C22—44no—
396C22—41no—
397C22—41no—
398C22—44no—
399C22—81no—
400C22—82no—
401C22—82no—
402C22—88no—
403C22—88no—
404C22—82no—
405C22—82no—
406C22#503812yes—
407C22#504112yes—
408C22C2×#83211yescharacteristic
409C22#522011yescharacteristic
410C22#523311yescharacteristic
411C4—22no—
412C4—21no—
413C4—22no—
414C4—41no—
415C4—41no—
416C4—82no—
417C4—482no—
418C4#503212yes—
419C3#946811yescharacteristic, complemented, Sylow
420C2—21no—
421C2—22no—
422C2—21no—
423C2—22no—
424C2—42no—
425C2—84no—
426C2#7166911yescharacteristic
427C2#7168111yescharacteristic
428C2#7185411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#2408C2
socleC22×C3#386
cosocleC23×C3×C4C23
hypercenter#2408D3
hypocenterC3#9468
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2408

Complements

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

NH ≅ G/Ntype
2. #82301C2semidirect
3. #72128C2semidirect
4. #71619C2semidirect
5. #67625C2semidirect
11. C3×#394C4semidirect
12. C3×#392C4semidirect
56. C22×C3⋊C8D4semidirect
76. #394Dic3semidirect
77. #392Dic3semidirect
195. C23×C3#102semidirect
223. C2×C3×C4#102semidirect
340. C23#832semidirect
371. C2×C4#832semidirect
419. C3#9468semidirect

Maximal subgroups

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

subgroupindex
C3×#24082
#823012
#721282
#716192
#676252
#94683