Finite Group Databasebeta

← #10702894

Subgroups — #10702894

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×#2879C211yesmaximal, characteristic
2#84087C211yesmaximal, characteristic
3C2×#5237C211yesmaximal, characteristic
4C2×#4559C212yesmaximal, complemented
5C2×#4554C212yesmaximal, complemented
6#11173—31nomaximal, Sylow
7C22×C3×#41C2211yescharacteristic
8C22×C3×M16C2212yes—
9C2×#926C2211yescharacteristic
10C22×C4⋊Dic3C2211yescharacteristic
11C22×#208C2212yes—
12C3×#408—24no—
13C3×#407—22no—
14#5237—24no—
15#5229—21no—
16C2×#836—21no—
17#5221—21no—
18#5220C414yes—
19C2×#833—21no—
20#5212—22no—
21#4559—24no—
22#4554—24no—
23C2×#413—31no—
24#2879D311yescharacteristic
25#2311—31no—
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×C4D412yes—
38C23×C3×C4C2311yescharacteristic
39C23×Dic3C2×C412yes—
40C2×C3×M16—22no—
41C2×C3×M16—22no—
42C2×C3×M16—22no—
43C2×C3×M16—22no—
44C2×C3×M16—22no—
45C2×C3×M16D414yes—
46C22×C3×C8—22no—
47C2×#217—24no—
48C2×#217C2×C412yes—
49#926—22no—
50#926—22no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3—22no—
55C2×C4⋊Dic3—22no—
56C2×C4×Dic3—21no—
57C2×#208—22no—
58C2×#208—22no—
59C2×#208—22no—
60C2×#208—22no—
61C2×#208—22no—
62C2×#208D414yes—
63C22×C3⋊C8—22no—
64C3×C22⋊C8—28no—
65#836—24no—
66#833—24no—
67C3⋊#95—24no—
68C22×#41C2×D311yescharacteristic
69C22×M16—32no—
70C22×M16C2×D312yes—
71C2×#117—31no—
72C22×C4⋊C4—31no—
73#413—64no—
74C2×#101—61no—
75C2×#100—61no—
76#408—24no—
77#407—22no—
78#391—61no—
79#390—61no—
80#386—34no—
81#380—62no—
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—22no—
109C22×C3×C4—24no—
110C22×C3×C4C22⋊C412yes—
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D412yes—
113C22×C3×C4C2×D412yes—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4#4112yes—
116C22×C3×C4#4112yes—
117C22×C3×C4#4112yes—
118C22×C3×C4#4112yes—
119C22×Dic3—22no—
120C22×Dic3—22no—
121C22×Dic3—24no—
122C22×Dic3—82no—
123C3×M16—24no—
124C3×M16—24no—
125C3×M16—42no—
126C3×M16—42no—
127C2×C3×C8—24no—
128C2×C3×C8—42no—
129C2×C3×C8—42no—
130#217—24no—
131#217—84no—
132C4⋊Dic3—41no—
133C4⋊Dic3—42no—
134C4⋊Dic3—42no—
135C4⋊Dic3—41no—
136C4×Dic3—41no—
137C4×Dic3—41no—
138#208—24no—
139#208—24no—
140#208—42no—
141#208—42no—
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×Q8D1211yescharacteristic
152C22×D4C4×D312yes—
153C22×D4D1211yescharacteristic
154C23×C4—32no—
155C23×C4#6712yes—
156C23×C4C22×D311yescharacteristic
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—34no—
163C2×M16—62no—
164C2×M16—62no—
165C2×M16—62no—
166C2×M16—62no—
167C2×M16—62no—
168C2×M16#6714yes—
169C22×C8—22no—
170C22×C8—62no—
171#117—62no—
172#117—62no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4—62no—
175C2×C4⋊C4—62no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—61no—
178C2×C22⋊C4—32no—
179C2×C22⋊C4—64no—
180C2×C42—61no—
181#101—64no—
182#100—64no—
183C22⋊C8—28no—
184#95—64no—
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#12011yescharacteristic
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4#12112yes—
211C2×C3×C4#12112yes—
212C2×C3×C4#12112yes—
213C2×C3×C4#12112yes—
214C2×C3×C4#12312yes—
215C2×C3×C4#12312yes—
216C2×C3×C4#12311yescharacteristic
217C2×C3×C4#12311yescharacteristic
218C2×C3×C4#12312yes—
219C2×C3×C4#12312yes—
220C2×C3×C4#12412yes—
221C2×C3×C4#12412yes—
222C2×C3×C4#12612yes—
223C2×C3×C4#12612yes—
224C2×Dic3—42no—
225C2×Dic3—44no—
226C2×Dic3—42no—
227C2×Dic3—42no—
228C2×Dic3—42no—
229C2×Dic3—84no—
230C3×C8—44no—
231C3⋊C8—44no—
232C24(C2×D3)⋊C412yes—
233C24C2×C4×D311yescharacteristic
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—21no—
243C2×Q8—22no—
244C2×Q8—21no—
245C2×Q8—22no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—24no—
249C2×D4—24no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—22no—
253C2×D4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—62no—
259C22×C4—62no—
260C22×C4—64no—
261C22×C4—242no—
262C22×C4(C2×D3)⋊C412yes—
263C22×C4C2×D1211yescharacteristic
264C22×C4#23512yes—
265C22×C4D3×D412yes—
266C22×C4#23712yes—
267C22×C4#23712yes—
268C22×C4#23712yes—
269C22×C4C2×#6711yescharacteristic
270C22×C4C2×#6712yes—
271M16—24no—
272M16—24no—
273M16—42no—
274M16—42no—
275M16—64no—
276M16—64no—
277M16—122no—
278M16—122no—
279C2×C8—24no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—64no—
283C2×C8—122no—
284C2×C8—122no—
285C4⋊C4—121no—
286C4⋊C4—122no—
287C4⋊C4—122no—
288C4⋊C4—121no—
289C22⋊C4—64no—
290C22⋊C4—244no—
291C42—121no—
292C42—121no—
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#39412yes—
312C3×C4#39412yes—
313C3×C4#39712yes—
314C3×C4#39712yes—
315Dic3—82no—
316Dic3—82no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—41no—
321C23—41no—
322C23—42no—
323C23—42no—
324C23#87711yescharacteristic
325C23#87711yescharacteristic
326C23#67⋊C412yes—
327C23C2×(C2×D3)⋊C411yescharacteristic
328C23C4×#6712yes—
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—122no—
341C2×C4—124no—
342C2×C4—122no—
343C2×C4—122no—
344C2×C4—122no—
345C2×C4—244no—
346C2×C4#87912yes—
347C2×C4#87912yes—
348C2×C4#88211yescharacteristic
349C2×C4#88312yes—
350C2×C4#88512yes—
351C2×C4#88611yescharacteristic
352C2×C4#88612yes—
353C2×C4#92911yescharacteristic
354C2×C4#93012yes—
355C2×C4#93512yes—
356C2×C4#93512yes—
357C2×C4#93612yes—
358C2×C4#93812yes—
359C2×C4#93912yes—
360C2×C4#93912yes—
361C2×C4#95311yescharacteristic
362C8—44no—
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#494311yescharacteristic
385C22#494412yes—
386C22#494512yes—
387C22#495811yescharacteristic
388C22#521311yescharacteristic
389C4—42no—
390C4—242no—
391C4—242no—
392C4#495212yes—
393C4#495312yes—
394C4#496112yes—
395C4#522212yes—
396C3#1117311yescharacteristic, complemented, Sylow
397C2—21no—
398C2—21no—
399C2—22no—
400C2—22no—
401C2—82no—
402C2#7154911yescharacteristic
403C2#7203011yescharacteristic
404C2#7203111yescharacteristic
subgroupG/H
centerC22#5213
commutatorC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2879C2
socleC22×C3#386
prime radicalC22×C3×#41C22
cosocleC23×C3×C4C23
hypercenter#2879D3
hypocenterC3#11173
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2879

Complements

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

NH ≅ G/Ntype
4. C2×#4559C2semidirect
5. C2×#4554C2semidirect
396. C3#11173semidirect

Maximal subgroups

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

subgroupindex
C3×#28792
#840872
C2×#52372
C2×#45592
C2×#45542
#111733