Finite Group Databasebeta

← #10702893

Subgroups — #10702893

Order 768.

854 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2311C211yesmaximal, characteristic
2C2×#5237C211yesmaximal, characteristic
3#84074C211yesmaximal, characteristic
4C2×#4533C212yesmaximal, complemented
5C2×#4530C212yesmaximal, complemented
6#11173—31nomaximal, Sylow
7C22×C3×#41C2211yescharacteristic
8C2×C3×#117C2211yescharacteristic
9C22×C3×C4⋊C4C2211yescharacteristic
10C22×#208C2212yes—
11C22×#208C2212yes—
12C3×#391—21no—
13C3×#390—21no—
14C3×#386C414yes—
15C3×#380—22no—
16#5237—24no—
17#5234—24no—
18C2×#836—21no—
19C2×#833—21no—
20#5207—22no—
21#4533—24no—
22#4530—24no—
23C2×#413—31no—
24#2879—31no—
25#2311D311yescharacteristic
26C2×#344—32no—
27C2×#343—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×C4⋊C4—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—22no—
46C2×C3×C4⋊C4—22no—
47C2×C3×C22⋊C4—24no—
48C2×C3×C22⋊C4C2×C412yes—
49C2×C3×C42—21no—
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×#208D414yes—
61C2×#208D414yes—
62C22×C3⋊C8—22no—
63C22×C3⋊C8—22no—
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×C4⋊C4C2×D311yescharacteristic
73#413—64no—
74C2×#101—61no—
75C2×#100—61no—
76#408—64no—
77#407—62no—
78#391—21no—
79#390—21no—
80#386Dic314yes—
81#380—22no—
82#344—64no—
83#343—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—24no—
109C22×C3×C4—22no—
110C22×C3×C4—22no—
111C22×C3×C4—22no—
112C22×C3×C4—24no—
113C22×C3×C4—82no—
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—42no—
125C3×C4⋊C4—42no—
126C3×C4⋊C4—41no—
127C3×C22⋊C4—24no—
128C3×C22⋊C4—84no—
129C3×C42—41no—
130C3×C42—41no—
131#208—24no—
132#208—24no—
133#208—24no—
134#208—24no—
135#208—42no—
136#208—42no—
137#208—42no—
138#208—42no—
139C2×C3⋊C8—24no—
140C2×C3⋊C8—24no—
141C2×C3⋊C8—42no—
142C2×C3⋊C8—42no—
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—34no—
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—62no—
170C22×C8—62no—
171#117—22no—
172#117—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4C2×Dic312yes—
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—41no—
189C23×C3—41no—
190C23×C3—42no—
191C23×C3—42no—
192C23×C3C2×C22⋊C411yescharacteristic
193C23×C3#11711yescharacteristic
194C23×C3#11711yescharacteristic
195C23×C3C4×D412yes—
196C23×C3C4×D412yes—
197C3×Q8—42no—
198C3×Q8—42no—
199C3×D4—44no—
200C3×D4—44no—
201C3×D4—42no—
202C3×D4—42no—
203C2×C3×C4—22no—
204C2×C3×C4—22no—
205C2×C3×C4—22no—
206C2×C3×C4—44no—
207C2×C3×C4—44no—
208C2×C3×C4—42no—
209C2×C3×C4—44no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4—42no—
213C2×C3×C4—84no—
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12112yes—
217C2×C3×C4#12112yes—
218C2×C3×C4#12112yes—
219C2×C3×C4#12112yes—
220C2×C3×C4#12312yes—
221C2×C3×C4#12312yes—
222C2×C3×C4#12312yes—
223C2×C3×C4#12312yes—
224C2×C3×C4#12311yescharacteristic
225C2×C3×C4#12311yescharacteristic
226C2×C3×C4#12412yes—
227C2×C3×C4#12412yes—
228C2×C3×C4#12612yes—
229C2×C3×C4#12612yes—
230C3⋊C8—44no—
231C3⋊C8—44no—
232C24#21712yes—
233C24C22×Dic311yescharacteristic
234#41—42no—
235#41—42no—
236#41—42no—
237#41—42no—
238#41—42no—
239#41—42no—
240#41—42no—
241#41—42no—
242C2×Q8—22no—
243C2×Q8—21no—
244C2×Q8—21no—
245C2×Q8—22no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—24no—
249C2×D4—24no—
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—22no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—82no—
262C22×C4#21712yes—
263C22×C4#23512yes—
264C22×C4#23512yes—
265C22×C4D3×D412yes—
266C22×C4D3×D412yes—
267C22×C4#23712yes—
268C22×C4#23712yes—
269C22×C4C2×#6711yescharacteristic
270C22×C4C2×#6711yescharacteristic
271M16—64no—
272M16—64no—
273M16—64no—
274M16—64no—
275M16—122no—
276M16—122no—
277M16—122no—
278M16—122no—
279C2×C8—64no—
280C2×C8—64no—
281C2×C8—122no—
282C2×C8—122no—
283C2×C8—122no—
284C2×C8—122no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C22⋊C4—24no—
290C22⋊C4—84no—
291C42—41no—
292C42—41no—
293C22×C3—21no—
294C22×C3—22no—
295C22×C3—21no—
296C22×C3—22no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—42no—
300C22×C3—44no—
301C22×C3—42no—
302C22×C3—42no—
303C22×C3—42no—
304C22×C3—82no—
305C22×C3#38111yescharacteristic
306C22×C3#38611yescharacteristic
307C22×C3#38611yescharacteristic
308C22×C3#38812yes—
309C22×C3#38812yes—
310C3×C4—42no—
311C3×C4—82no—
312C3×C4—82no—
313C3×C4#39412yes—
314C3×C4#39412yes—
315C3×C4#39712yes—
316C3×C4#39712yes—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—41no—
321C23—41no—
322C23—42no—
323C23—42no—
324C23#92611yescharacteristic
325C23#92611yescharacteristic
326C23D4×Dic312yes—
327C23D4×Dic312yes—
328C23C2×#21711yescharacteristic
329Q8—42no—
330Q8—42no—
331D4—44no—
332D4—44no—
333D4—42no—
334D4—42no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—84no—
346C2×C4#87912yes—
347C2×C4#87912yes—
348C2×C4#88312yes—
349C2×C4#88312yes—
350C2×C4#88412yes—
351C2×C4#88412yes—
352C2×C4#88512yes—
353C2×C4#88512yes—
354C2×C4#92912yes—
355C2×C4#92912yes—
356C2×C4#93511yescharacteristic
357C2×C4#93511yescharacteristic
358C2×C4#93711yescharacteristic
359C2×C4#93711yescharacteristic
360C2×C4#93912yes—
361C2×C4#93912yes—
362C8—124no—
363C8—124no—
364C2×C3—21no—
365C2×C3—21no—
366C2×C3—22no—
367C2×C3—22no—
368C2×C3—82no—
369C2×C3#144011yescharacteristic
370C2×C3#191411yescharacteristic
371C2×C3#191511yescharacteristic
372C22—21no—
373C22—22no—
374C22—21no—
375C22—22no—
376C22—41no—
377C22—41no—
378C22—42no—
379C22—44no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—82no—
384C22#495112yes—
385C22#495112yes—
386C22#521111yescharacteristic
387C22#522011yescharacteristic
388C22#522011yescharacteristic
389C4—42no—
390C4—82no—
391C4—82no—
392C4#495212yes—
393C4#495212yes—
394C4#495912yes—
395C4#495912yes—
396C3#1117311yescharacteristic, complemented, Sylow
397C2—21no—
398C2—21no—
399C2—22no—
400C2—22no—
401C2—82no—
402C2#7154811yescharacteristic
403C2#7185511yescharacteristic
404C2#7186611yescharacteristic
subgroupG/H
centerC22#5220
commutatorC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2311C2
socleC22×C3#386
prime radicalC22×C3×#41C22
cosocleC23×C3×C4C23
hypercenter#2311D3
hypocenterC3#11173
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2311

Complements

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

NH ≅ G/Ntype
4. C2×#4533C2semidirect
5. C2×#4530C2semidirect
396. C3#11173semidirect

Maximal subgroups

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

subgroupindex
C3×#23112
C2×#52372
#840742
C2×#45332
C2×#45302
#111733