Finite Group Databasebeta

← #10896386

Subgroups — #10896386

Order 768.

790 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
1#83023C211yesmaximal, characteristic, complemented
2#76298C211yesmaximal, characteristic
3C3×#2025C211yesmaximal, characteristic, complemented
4#71743C211yesmaximal, characteristic, complemented
5#70170C212yesmaximal, complemented
6#70161C212yesmaximal
7#69655C211yesmaximal, characteristic, complemented
8#68894C212yesmaximal, complemented
9#68797C212yesmaximal, complemented
10#68682C211yesmaximal, characteristic, complemented
11#68606C211yesmaximal, characteristic, complemented
12#29265—31nomaximal, Sylow
13C2×C3×#125C2211yescharacteristic
14C2×#940C2211yescharacteristic
15C2×D4×Dic3C2211yescharacteristic
16C2×#893C2211yescharacteristic
17C2×C3×#102C2212yes—
18C3×#390C2211yescharacteristic
19C2×#832C2212yes—
20C2×#821C2212yes—
21C2×#817C2212yes—
22C2×Dic3⋊C8C2212yes—
23#4984—22no—
24#4984C2212yes—
25#4980—22no—
26#4977C2211yescharacteristic
27Dic3×C4⋊C4C2212yes—
28C2×#809C2212yes—
29C2×#807C2212yes—
30#4958—22no—
31#4956—22no—
32#4939C2211yescharacteristic
33#4935C2211yescharacteristic
34#4932C2211yescharacteristic
35C3×#340C2212yes—
36C3×#336C2211yescharacteristic
37#4550C2212yes—
38#4529C2212yes—
39#4524C2211yescharacteristic
40#2330—31no—
41#2328—31no—
42#2094—31no—
43#2093—31no—
44#2025—31no—
45#2025D311yescharacteristic, complemented
46#1982—32no—
47#1949—32no—
48#1864—32no—
49#1862—32no—
50#1832—31no—
51C22×C3×D4C2311yescharacteristic
52C22×#67—22no—
53C23×Dic3—21no—
54C22×D12C2311yescharacteristic
55C22×C4×D3—21no—
56C22×C3×C8C2312yes—
57C3×#125—24no—
58C2×C3×C4⋊C4—22no—
59C2×C3×C4⋊C4C2311yescharacteristic
60C2×C3×C4⋊C4C2312yes—
61C2×C3×C22⋊C4—22no—
62C2×C3×C42C2311yescharacteristic
63C2×#217—21no—
64#940—22no—
65#940Q814yes—
66D4×Dic3—42no—
67C2×(C2×D3)⋊C4—21no—
68C2×(C2×D3)⋊C4—22no—
69C2×C4⋊Dic3—22no—
70C2×C4⋊Dic3—22no—
71C2×C4⋊Dic3C2312yes—
72C2×C4⋊Dic3C2311yescharacteristic
73C2×C4×Dic3—22no—
74C2×C4×Dic3C2311yescharacteristic
75C2×C4×Dic3C2311yescharacteristic
76C22×C3⋊C8C2312yes—
77#893—42no—
78C3×#102—24no—
79#832—42no—
80C3⋊#95—22no—
81C3⋊#95—22no—
82C3⋊#95—22no—
83C3⋊#95D412yes—
84C3⋊#95D412yes—
85#821—42no—
86#817—42no—
87Dic3⋊C8—24no—
88#809—24no—
89#807—42no—
90C2×#127—31no—
91C2×#125C2×D311yescharacteristic
92C2×C4×D4—31no—
93C2×C4×D4—31no—
94C2×#107—32no—
95C2×#106—32no—
96C2×C4⋊C8—32no—
97C2×#102—32no—
98C2×#102—32no—
99C2×#102—32no—
100C2×#102C2×D312yes—
101#390—31no—
102#390C2×D311yescharacteristic
103#389—32no—
104#389—62no—
105#387—62no—
106#386—62no—
107#386—62no—
108#384—31no—
109#382—31no—
110C4×C4⋊C4—32no—
111#376—31no—
112#340—32no—
113#340C2×D312yes—
114#336—32no—
115#336—31no—
116#336C2×D311yescharacteristic
117C24×C3—21no—
118C23×D3—21no—
119C2×C3×D4—22no—
120C2×C3×D4—21no—
121C2×C3×D4—22no—
122C2×C3×D4C2×Q812yes—
123C22×C3×C4—22no—
124C22×C3×C4—21no—
125C22×C3×C4C2×D412yes—
126C22×C3×C4C2411yescharacteristic
127C2×#67—28no—
128C2×#67—82no—
129C2×#67—82no—
130C22×Dic3—22no—
131C22×Dic3—21no—
132C22×Dic3—42no—
133C22×Dic3—82no—
134C22×Dic3C2×D411yescharacteristic
135C22×Dic3C2×D411yescharacteristic
136C22×Dic3C2×D412yes—
137C2×D12—22no—
138C2×D12—22no—
139C2×D12—21no—
140C2×D12C2×Q812yes—
141C2×C4×D3—42no—
142C2×C4×D3—82no—
143C2×C3×C8—22no—
144C2×C3×C8—22no—
145C2×C3×C8—22no—
146C3×C4⋊C4—28no—
147C3×C4⋊C4—22no—
148C3×C4⋊C4—22no—
149C3×C4⋊C4#4114yes—
150C3×C22⋊C4—82no—
151C3×C42—22no—
152#217—42no—
153(C2×D3)⋊C4—42no—
154(C2×D3)⋊C4—82no—
155C4⋊Dic3—22no—
156C4⋊Dic3—22no—
157C4⋊Dic3—22no—
158C4⋊Dic3—82no—
159C4⋊Dic3—82no—
160C4×Dic3—22no—
161C4×Dic3—82no—
162C4×Dic3C2×Q812yes—
163C4×Dic3#4112yes—
164C2×C3⋊C8—22no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—22no—
167C22×D4—31no—
168C22×D4—62no—
169C22×D4C22×D311yescharacteristic
170C23×C4—61no—
171C23×C4—61no—
172C22×C8—32no—
173C22×C8C22×D312yes—
174#127—34no—
175#127—62no—
176#125—24no—
177C4×D4—122no—
178C4×D4—122no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—31no—
181C2×C4⋊C4—32no—
182C2×C4⋊C4—62no—
183C2×C4⋊C4—62no—
184C2×C4⋊C4C22×D311yescharacteristic
185C2×C4⋊C4C22×D312yes—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—61no—
188C2×C22⋊C4—61no—
189C2×C22⋊C4—62no—
190C2×C42—31no—
191C2×C42—31no—
192C2×C42—62no—
193C2×C42C22×D311yescharacteristic
194#107—122no—
195#106—122no—
196C4⋊C8—64no—
197#102—24no—
198#102—64no—
199#102—122no—
200#102—122no—
201#95—32no—
202#95—32no—
203#95—62no—
204#95—62no—
205#95—62no—
206C23×C3—22no—
207C23×C3—41no—
208C23×C3—41no—
209C23×C3—42no—
210C23×C3—42no—
211C23×C3C22×D411yescharacteristic
212C22×D3—22no—
213C22×D3—41no—
214C22×D3—41no—
215C22×D3—42no—
216C22×D3—42no—
217C3×D4—24no—
218C3×D4—42no—
219C2×C3×C4—22no—
220C2×C3×C4—24no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—42no—
224C2×C3×C4—42no—
225C2×C3×C4—41no—
226C2×C3×C4—42no—
227C2×C3×C4C22×Q811yescharacteristic
228C2×C3×C4C2×#4112yes—
229C2×C3×C4C2×#4111yescharacteristic
230C2×C3×C4H(5,𝔽2)12yes—
231#67—88no—
232C2×Dic3—22no—
233C2×Dic3—22no—
234C2×Dic3—21no—
235C2×Dic3—21no—
236C2×Dic3—22no—
237C2×Dic3—22no—
238C2×Dic3—41no—
239C2×Dic3—42no—
240C2×Dic3—42no—
241C2×Dic3—42no—
242C2×Dic3—42no—
243C2×Dic3—84no—
244C2×Dic3C22⋊Q814yes—
245D12—24no—
246D12—42no—
247C4×D3—84no—
248C3×C8—42no—
249C3⋊C8—42no—
250C24—21no—
251C24—61no—
252C2×D4—22no—
253C2×D4—21no—
254C2×D4—22no—
255C2×D4—32no—
256C2×D4—68no—
257C2×D4—62no—
258C2×D4—61no—
259C2×D4—62no—
260C2×D4—242no—
261C2×D4—242no—
262C2×D4D3×Q812yes—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—32no—
266C22×C4—31no—
267C22×C4—31no—
268C22×C4—62no—
269C22×C4—61no—
270C22×C4—122no—
271C22×C4—122no—
272C22×C4—242no—
273C22×C4—242no—
274C22×C4D3×D412yes—
275C22×C4C23×D311yescharacteristic
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—62no—
280C2×C8—62no—
281C2×C8—62no—
282C4⋊C4—28no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—62no—
286C4⋊C4—62no—
287C4⋊C4—62no—
288C4⋊C4—242no—
289C4⋊C4—242no—
290C4⋊C4#23714yes—
291C22⋊C4—82no—
292C22⋊C4—122no—
293C22⋊C4—122no—
294C22⋊C4—242no—
295C42—22no—
296C42—32no—
297C42—32no—
298C42—62no—
299C42—242no—
300C22×C3—44no—
301C22×C3—42no—
302C22×C3—82no—
303C22×C3—81no—
304C22×C3—81no—
305C22×C3C2×C22⋊Q811yescharacteristic
306C22×C3#52511yescharacteristic
307C22×C3#53511yescharacteristic
308C22×C3#57612yes—
309C22×C3#57712yes—
310C2×D3—44no—
311C2×D3—42no—
312C2×D3—82no—
313C2×D3—81no—
314C2×D3—81no—
315C3×C4—42no—
316C3×C4—44no—
317C3×C4—81no—
318C3×C4#55412yes—
319C3×C4#55912yes—
320Dic3—24no—
321Dic3—41no—
322Dic3—41no—
323Dic3—82no—
324Dic3—81no—
325C23—22no—
326C23—41no—
327C23—42no—
328C23—42no—
329C23—41no—
330C23—62no—
331C23—121no—
332C23—122no—
333C23—122no—
334C23—121no—
335C23C2×D3×D411yescharacteristic
336D4—24no—
337D4—42no—
338D4—64no—
339D4—122no—
340D4—248no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—34no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—62no—
351C2×C4—62no—
352C2×C4—61no—
353C2×C4—61no—
354C2×C4—62no—
355C2×C4—62no—
356C2×C4—121no—
357C2×C4—122no—
358C2×C4—122no—
359C2×C4—122no—
360C2×C4—122no—
361C2×C4—244no—
362C2×C4—244no—
363C2×C4C2×#23712yes—
364C2×C4C2×D3×Q811yescharacteristic
365C2×C4D3×#4111yescharacteristic
366C2×C4#100912yes—
367C8—42no—
368C8—122no—
369C2×C3—82no—
370C2×C3#258011yescharacteristic
371C2×C3#308112yes—
372C2×C3#311912yes—
373C2×C3#322011yescharacteristic
374C2×C3#322211yescharacteristic
375D3—82no—
376C22—42no—
377C22—44no—
378C22—81no—
379C22—82no—
380C22—81no—
381C22—122no—
382C22—124no—
383C22—241no—
384C22—242no—
385C22—241no—
386C22#558611yescharacteristic
387C22D3×C22⋊Q811yescharacteristic
388C22#567411yescharacteristic
389C22#577012yes—
390C22#577412yes—
391C4—42no—
392C4—44no—
393C4—64no—
394C4—81no—
395C4—121no—
396C4—121no—
397C4—242no—
398C4—241no—
399C4#563512yes—
400C4#568112yes—
401C3#2926511yescharacteristic, complemented, Sylow
402C2—82no—
403C2—242no—
404C2#7680211yescharacteristic
405C2#8007312yes—
406C2#8116812yes—
407C2#8164611yescharacteristic
408C2#8171911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2025C2
socleC23×C3C22×D4
prime radicalC2×#940C22
hypercenter#2025D3
hypocenterC3#29265
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2025

Complements

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

NH ≅ G/Ntype
1. #83023C2semidirect
3. C3×#2025C2semidirect
4. #71743C2semidirect
5. #70170C2semidirect
7. #69655C2semidirect
8. #68894C2semidirect
9. #68797C2semidirect
10. #68682C2semidirect
11. #68606C2semidirect
45. #2025D3semidirect
401. C3#29265semidirect

Maximal subgroups

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

subgroupindex
#830232
#762982
C3×#20252
#717432
#701702
#701612
#696552
#688942
#687972
#686822
#686062
#292653