Finite Group Databasebeta

← #10702903

Subgroups — #10702903

Order 768.

890 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2297C211yesmaximal, characteristic
2C2×#5237C211yesmaximal, characteristic
3#84074C211yesmaximal, characteristic
4C2×#4533C212yesmaximal, complemented
5#71527C212yesmaximal, complemented
6#11175—31nomaximal, Sylow
7C22×C3×#41C2211yescharacteristic
8C2×C3×#117C2211yescharacteristic
9C22×C3×C42C2211yescharacteristic
10C22×#208C2212yes—
11C22×#208C2212yes—
12C3×#391—21no—
13C3×#390—21no—
14C3×#386C414yes—
15C3×C4×C22⋊C4—22no—
16#5237—24no—
17#5234—24no—
18C2×#836—21no—
19C2×#833—21no—
20#5207—22no—
21#4533—24no—
22#4524—24no—
23C2×#413—31no—
24#2879—31no—
25#2297D311yescharacteristic
26C2×#344—32no—
27#1728—32no—
28C2×C3×#41—22no—
29C2×C3×#41—22no—
30C2×C3×#41—22no—
31C2×C3×#41—22no—
32C2×C3×#41—22no—
33C2×C3×#41—22no—
34C22×C3×Q8D411yescharacteristic
35C22×C3×D4C2×C412yes—
36C22×C3×D4D411yescharacteristic
37C23×C3×C4C2×C412yes—
38C23×C3×C4D412yes—
39C23×C3×C4C2311yescharacteristic
40C3×#117—22no—
41C3×#117—22no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C22⋊C4—24no—
44C2×C3×C22⋊C4C2×C412yes—
45C2×C3×C42—21no—
46C2×C3×C42—21no—
47C2×C3×C42—21no—
48C2×C3×C42—22no—
49C2×C3×C42—22no—
50C2×#208—22no—
51C2×#208—22no—
52C2×#208—22no—
53C2×#208—22no—
54C2×#208—22no—
55C2×#208—22no—
56C2×#208—22no—
57C2×#208—22no—
58C2×#208—22no—
59C2×#208—22no—
60C2×#208—22no—
61C2×#208D414yes—
62C22×C3⋊C8—22no—
63C22×C3⋊C8D414yes—
64C3×#95—24no—
65#836—24no—
66#833—24no—
67#830—28no—
68C22×#41C2×D311yescharacteristic
69C22×M16—32no—
70C22×M16—32no—
71C2×#117C2×D311yescharacteristic
72C22×C42C2×D311yescharacteristic
73#413—64no—
74C2×#101—61no—
75C2×#100—61no—
76#408—64no—
77#407—62no—
78#391—21no—
79#390—21no—
80#386Dic314yes—
81C4×C22⋊C4—22no—
82#344—64no—
83#336—64no—
84C24×C3C22⋊C412yes—
85C24×C3C22×C411yescharacteristic
86C3×#41—42no—
87C3×#41—42no—
88C3×#41—42no—
89C3×#41—42no—
90C3×#41—42no—
91C3×#41—42no—
92C3×#41—42no—
93C3×#41—42no—
94C2×C3×Q8—21no—
95C2×C3×Q8—22no—
96C2×C3×Q8—22no—
97C2×C3×Q8—21no—
98C2×C3×D4—22no—
99C2×C3×D4—22no—
100C2×C3×D4—24no—
101C2×C3×D4—24no—
102C2×C3×D4—21no—
103C2×C3×D4—22no—
104C2×C3×D4—22no—
105C2×C3×D4—21no—
106C22×C3×C4—22no—
107C22×C3×C4—24no—
108C22×C3×C4—22no—
109C22×C3×C4—24no—
110C22×C3×C4—24no—
111C22×C3×C4—22no—
112C22×C3×C4—22no—
113C22×C3×C4—44no—
114C22×C3×C4C22⋊C412yes—
115C22×C3×C4C2×D412yes—
116C22×C3×C4C2×D412yes—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4#4112yes—
120C22×C3×C4#4112yes—
121C22×C3×C4#4112yes—
122C22×C3×C4#4112yes—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C22⋊C4—24no—
126C3×C22⋊C4—48no—
127C3×C42—41no—
128C3×C42—42no—
129C3×C42—42no—
130C3×C42—41no—
131#208—24no—
132#208—24no—
133#208—42no—
134#208—42no—
135#208—42no—
136#208—42no—
137#208—42no—
138#208—42no—
139C2×C3⋊C8—24no—
140C2×C3⋊C8—24no—
141C2×C3⋊C8—24no—
142C2×C3⋊C8—24no—
143C2×C3⋊C8—42no—
144C2×C3⋊C8—42no—
145C2×#41—22no—
146C2×#41—22no—
147C2×#41—22no—
148C2×#41—22no—
149C2×#41—22no—
150C2×#41—22no—
151C22×Q8#6711yescharacteristic
152C22×D4C2×Dic312yes—
153C22×D4#6711yescharacteristic
154C23×C4C2×Dic312yes—
155C23×C4#6712yes—
156C23×C4C22×D311yescharacteristic
157C2×M16—34no—
158C2×M16—62no—
159C2×M16—62no—
160C2×M16—62no—
161C2×M16—62no—
162C2×M16—62no—
163C2×M16—62no—
164C2×M16—62no—
165C2×M16—62no—
166C2×M16—62no—
167C2×M16—62no—
168C2×M16—62no—
169C22×C8—34no—
170C22×C8—62no—
171#117—22no—
172#117—22no—
173C2×C4⋊C4—21no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4C2×Dic312yes—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181#101—64no—
182#100—64no—
183C22⋊C8—68no—
184#95—24no—
185C23×C3—22no—
186C23×C3—22no—
187C23×C3—22no—
188C23×C3—42no—
189C23×C3—44no—
190C23×C3C2×C22⋊C411yescharacteristic
191C23×C3#11711yescharacteristic
192C23×C3#11711yescharacteristic
193C23×C3C4×D412yes—
194C23×C3C4×D412yes—
195C3×Q8—42no—
196C3×Q8—42no—
197C3×D4—44no—
198C3×D4—44no—
199C3×D4—42no—
200C3×D4—42no—
201C2×C3×C4—24no—
202C2×C3×C4—28no—
203C2×C3×C4—22no—
204C2×C3×C4—22no—
205C2×C3×C4—22no—
206C2×C3×C4—42no—
207C2×C3×C4—44no—
208C2×C3×C4—42no—
209C2×C3×C4—44no—
210C2×C3×C4—44no—
211C2×C3×C4—44no—
212C2×C3×C4—42no—
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12112yes—
216C2×C3×C4#12112yes—
217C2×C3×C4#12112yes—
218C2×C3×C4#12112yes—
219C2×C3×C4#12312yes—
220C2×C3×C4#12312yes—
221C2×C3×C4#12312yes—
222C2×C3×C4#12312yes—
223C2×C3×C4#12311yescharacteristic
224C2×C3×C4#12311yescharacteristic
225C2×C3×C4#12412yes—
226C2×C3×C4#12412yes—
227C2×C3×C4#12612yes—
228C2×C3×C4#12612yes—
229C3⋊C8—44no—
230C3⋊C8—44no—
231C24#21712yes—
232C24C22×Dic311yescharacteristic
233#41—42no—
234#41—42no—
235#41—42no—
236#41—42no—
237#41—42no—
238#41—42no—
239#41—42no—
240#41—42no—
241C2×Q8—22no—
242C2×Q8—21no—
243C2×Q8—21no—
244C2×Q8—22no—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—24no—
248C2×D4—24no—
249C2×D4—22no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—22no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—44no—
261C22×C4#21712yes—
262C22×C4#23512yes—
263C22×C4#23512yes—
264C22×C4D3×D412yes—
265C22×C4D3×D412yes—
266C22×C4#23712yes—
267C22×C4#23712yes—
268C22×C4C2×#6711yescharacteristic
269C22×C4C2×#6711yescharacteristic
270M16—64no—
271M16—64no—
272M16—122no—
273M16—122no—
274M16—122no—
275M16—122no—
276M16—122no—
277M16—122no—
278C2×C8—64no—
279C2×C8—64no—
280C2×C8—64no—
281C2×C8—64no—
282C2×C8—122no—
283C2×C8—122no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C22⋊C4—24no—
287C22⋊C4—48no—
288C42—41no—
289C42—41no—
290C42—42no—
291C42—42no—
292C22×C3—22no—
293C22×C3—24no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—42no—
297C22×C3—44no—
298C22×C3—44no—
299C22×C3—42no—
300C22×C3—82no—
301C22×C3#38111yescharacteristic
302C22×C3#38611yescharacteristic
303C22×C3#38611yescharacteristic
304C22×C3#38812yes—
305C22×C3#38812yes—
306C3×C4—44no—
307C3×C4—42no—
308C3×C4—82no—
309C3×C4#39412yes—
310C3×C4#39412yes—
311C3×C4#39712yes—
312C3×C4#39712yes—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—42no—
317C23—44no—
318C23#92611yescharacteristic
319C23#92611yescharacteristic
320C23D4×Dic312yes—
321C23D4×Dic312yes—
322C23C2×#21711yescharacteristic
323Q8—42no—
324Q8—42no—
325D4—44no—
326D4—44no—
327D4—42no—
328D4—42no—
329C2×C4—24no—
330C2×C4—28no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—42no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4#87912yes—
342C2×C4#87912yes—
343C2×C4#88312yes—
344C2×C4#88312yes—
345C2×C4#88412yes—
346C2×C4#88412yes—
347C2×C4#88512yes—
348C2×C4#88512yes—
349C2×C4#92912yes—
350C2×C4#92912yes—
351C2×C4#93511yescharacteristic
352C2×C4#93511yescharacteristic
353C2×C4#93711yescharacteristic
354C2×C4#93711yescharacteristic
355C2×C4#93912yes—
356C2×C4#93912yes—
357C8—124no—
358C8—124no—
359C2×C3—21no—
360C2×C3—21no—
361C2×C3—22no—
362C2×C3—22no—
363C2×C3—82no—
364C2×C3#144011yescharacteristic
365C2×C3#189112yes—
366C22—22no—
367C22—24no—
368C22—41no—
369C22—41no—
370C22—42no—
371C22—44no—
372C22—44no—
373C22—42no—
374C22—82no—
375C22#495112yes—
376C22#495112yes—
377C22#521111yescharacteristic
378C22#522011yescharacteristic
379C22#522011yescharacteristic
380C4—44no—
381C4—42no—
382C4—82no—
383C4#495212yes—
384C4#495212yes—
385C4#495912yes—
386C4#495912yes—
387C3#1117511yescharacteristic, complemented, Sylow
388C2—21no—
389C2—21no—
390C2—22no—
391C2—22no—
392C2—82no—
393C2#7154811yescharacteristic
394C2#7171412yes—
subgroupG/H
centerC2×C4#937
commutatorC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2297C2
socleC22×C3#386
prime radicalC22×C3×#41C22
cosocleC23×C3×C4C23
hypercenter#2297D3
hypocenterC3#11175
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2297

Complements

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

NH ≅ G/Ntype
4. C2×#4533C2semidirect
5. #71527C2semidirect
387. C3#11175semidirect

Maximal subgroups

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

subgroupindex
C3×#22972
C2×#52372
#840742
C2×#45332
#715272
#111753