Finite Group Databasebeta

← #10896381

Subgroups — #10896381

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
1C3×#2328C211yesmaximal, characteristic
2#73390C211yesmaximal, characteristic, complemented
3#71761C212yesmaximal
4#71607C212yesmaximal, complemented
5#71606C212yesmaximal, complemented
6#69656C211yesmaximal, characteristic, complemented
7#69655C211yesmaximal, characteristic, complemented
8#69639C211yesmaximal, characteristic, complemented
9#69639C211yesmaximal, characteristic, complemented
10#69540C212yesmaximal, complemented
11#69437C211yesmaximal, characteristic, complemented
12#29265—31nomaximal, Sylow
13C2×C3×#127C2211yescharacteristic
14C2×C3×C4×D4C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×#870C2211yescharacteristic
17C3×#390C2211yescharacteristic
18C3×#390C2211yescharacteristic
19C3×#386—22no—
20C3×#386—22no—
21C3×#382C2211yescharacteristic
22C3×#376C2211yescharacteristic
23C2×#832C2212yes—
24C2×#832C2212yes—
25C2×#832C2212yes—
26C2×#832C2212yes—
27C2×#808C2212yes—
28C2×#807C2212yes—
29#4935C2211yescharacteristic
30C4×C4⋊Dic3C2212yes—
31#4928C2212yes—
32C2×C4⋊(C3⋊C8)C2212yes—
33#4655—22no—
34#4647—22no—
35#4529C2212yes—
36#4529C2212yes—
37#4524C2211yescharacteristic
38#4524C2211yescharacteristic
39#4524C2212yes—
40#2330—31no—
41#2328D311yescharacteristic
42#2094—31no—
43#2093—31no—
44#2025—31no—
45#2025—31no—
46#1982—32no—
47#1949—32no—
48#1864—32no—
49#1862—32no—
50#1832—31no—
51C22×C3×D4—22no—
52C22×C3×D4C2311yescharacteristic
53C22×C3×D4C2311yescharacteristic
54C23×C3×C4—21no—
55C23×C3×C4—21no—
56C3×#127—22no—
57C3×#127Q814yes—
58C3×C4×D4—42no—
59C3×C4×D4—42no—
60C2×C3×C4⋊C4C2311yescharacteristic
61C2×C3×C4⋊C4C2311yescharacteristic
62C2×C3×C22⋊C4—21no—
63C2×C3×C22⋊C4—21no—
64C2×C3×C22⋊C4—22no—
65C2×C3×C22⋊C4—22no—
66C2×C3×C42C2311yescharacteristic
67C2×C3×C42C2311yescharacteristic
68C2×C3×C42C2311yescharacteristic
69C2×C4⋊Dic3—22no—
70C2×C4⋊Dic3C2312yes—
71C2×C4⋊Dic3C2312yes—
72C2×Dic3⋊C4—22no—
73C2×Dic3⋊C4—22no—
74C2×C4×Dic3—22no—
75C22×C3⋊C8C2312yes—
76C22×C3⋊C8C2312yes—
77#870—24no—
78C3×#95—22no—
79C3×#95D412yes—
80C3×#95D412yes—
81#832—24no—
82#832—24no—
83#832—42no—
84#832—42no—
85C3⋊#95—22no—
86C3⋊#95—22no—
87#808—42no—
88#807—42no—
89C4⋊(C3⋊C8)—24no—
90C2×#127C2×D311yescharacteristic
91C2×#125—31no—
92C2×C4×D4C2×D311yescharacteristic
93C2×C4×D4C2×D311yescharacteristic
94C2×#107—32no—
95C2×#106—32no—
96C2×C4⋊C8—32no—
97C2×#102—32no—
98C2×#102—32no—
99C2×#102—32no—
100C2×#102—32no—
101#390C2×D311yescharacteristic
102#390C2×D311yescharacteristic
103#389—32no—
104#389—62no—
105#387—62no—
106#386—22no—
107#386—22no—
108#384—31no—
109#382C2×D311yescharacteristic
110C4×C4⋊C4—32no—
111#376C2×D311yescharacteristic
112#340—32no—
113#340—32no—
114#336—31no—
115#336—31no—
116#336—32no—
117C24×C3—21no—
118C24×C3—21no—
119C2×C3×D4—28no—
120C2×C3×D4—22no—
121C2×C3×D4—22no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—22no—
125C2×C3×D4—22no—
126C2×C3×D4—82no—
127C2×C3×D4—82no—
128C2×C3×D4C2×Q812yes—
129C2×C3×D4C2×Q812yes—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—42no—
133C22×C3×C4—42no—
134C22×C3×C4—82no—
135C22×C3×C4—82no—
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D412yes—
139C22×C3×C4C2×D412yes—
140C22×C3×C4C2411yescharacteristic
141C22×Dic3—22no—
142C22×Dic3—22no—
143C3×C4⋊C4—22no—
144C3×C4⋊C4—22no—
145C3×C22⋊C4—42no—
146C3×C22⋊C4—42no—
147C3×C22⋊C4—82no—
148C3×C22⋊C4—82no—
149C3×C42—22no—
150C3×C42—22no—
151C3×C42C2×Q812yes—
152C3×C42#4112yes—
153C4⋊Dic3—22no—
154C4⋊Dic3—22no—
155C4⋊Dic3—22no—
156C4⋊Dic3—82no—
157C4⋊Dic3#4114yes—
158Dic3⋊C4—28no—
159Dic3⋊C4—82no—
160C4×Dic3—82no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—22no—
164C2×C3⋊C8—22no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—22no—
167C22×D4—22no—
168C22×D4C22×D311yescharacteristic
169C22×D4C22×D311yescharacteristic
170C23×C4—21no—
171C23×C4—21no—
172C22×C8—32no—
173C22×C8—32no—
174#127—22no—
175#127Dic614yes—
176#125—64no—
177C4×D4—42no—
178C4×D4—42no—
179C2×C4⋊C4—32no—
180C2×C4⋊C4—32no—
181C2×C4⋊C4—62no—
182C2×C4⋊C4—62no—
183C2×C4⋊C4—62no—
184C2×C4⋊C4C22×D311yescharacteristic
185C2×C4⋊C4C22×D311yescharacteristic
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C42—62no—
191C2×C42C22×D311yescharacteristic
192C2×C42C22×D311yescharacteristic
193C2×C42C22×D311yescharacteristic
194#107—122no—
195#106—122no—
196C4⋊C8—64no—
197#102—64no—
198#102—64no—
199#102—122no—
200#102—122no—
201#95—22no—
202#95—62no—
203#95—62no—
204#95#6712yes—
205#95#6712yes—
206C23×C3—22no—
207C23×C3—22no—
208C23×C3—41no—
209C23×C3—41no—
210C23×C3—41no—
211C23×C3—41no—
212C23×C3—42no—
213C23×C3—42no—
214C23×C3—42no—
215C23×C3—42no—
216C23×C3C22×D411yescharacteristic
217C3×D4—24no—
218C3×D4—24no—
219C3×D4—42no—
220C3×D4—42no—
221C3×D4—88no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4—84no—
236C2×C3×C4—84no—
237C2×C3×C4C22⋊Q814yes—
238C2×C3×C4C22×Q811yescharacteristic
239C2×C3×C4C2×#4111yescharacteristic
240C2×C3×C4C2×#4112yes—
241C2×C3×C4H(5,𝔽2)12yes—
242C2×Dic3—24no—
243C2×Dic3—42no—
244C2×Dic3—42no—
245C2×Dic3—42no—
246C2×Dic3—42no—
247C2×Dic3—42no—
248C3⋊C8—42no—
249C3⋊C8—42no—
250C24—21no—
251C24—21no—
252C2×D4—28no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—82no—
260C2×D4—82no—
261C2×D4C2×Dic612yes—
262C2×D4C2×Dic612yes—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—62no—
268C22×C4—62no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4C2×#6711yescharacteristic
272C22×C4C2×#6712yes—
273C22×C4C2×#6711yescharacteristic
274C22×C4C2×#6712yes—
275C22×C4C23×D311yescharacteristic
276C2×C8—62no—
277C2×C8—62no—
278C2×C8—62no—
279C2×C8—62no—
280C2×C8—62no—
281C2×C8—62no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—34no—
285C4⋊C4—68no—
286C4⋊C4—62no—
287C4⋊C4—62no—
288C4⋊C4—62no—
289C4⋊C4—242no—
290C4⋊C4—242no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—82no—
294C22⋊C4—82no—
295C42—22no—
296C42—22no—
297C42—242no—
298C42C2×Dic612yes—
299C42#23512yes—
300C22×C3—44no—
301C22×C3—44no—
302C22×C3—42no—
303C22×C3—42no—
304C22×C3—82no—
305C22×C3—82no—
306C22×C3—81no—
307C22×C3—81no—
308C22×C3—81no—
309C22×C3—81no—
310C22×C3C2×C22⋊Q811yescharacteristic
311C22×C3#52511yescharacteristic
312C22×C3#53511yescharacteristic
313C22×C3#57612yes—
314C22×C3#57712yes—
315C3×C4—24no—
316C3×C4—42no—
317C3×C4—41no—
318C3×C4—41no—
319C3×C4—81no—
320C3×C4—81no—
321C3×C4#55412yes—
322C3×C4#55912yes—
323Dic3—44no—
324Dic3—82no—
325C23—22no—
326C23—22no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23C22×#6711yescharacteristic
336D4—24no—
337D4—24no—
338D4—42no—
339D4—42no—
340D4—88no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—64no—
355C2×C4—84no—
356C2×C4—84no—
357C2×C4—122no—
358C2×C4—122no—
359C2×C4—122no—
360C2×C4—122no—
361C2×C4—122no—
362C2×C4#92414yes—
363C2×C4C22×Dic611yescharacteristic
364C2×C4C2×#23511yescharacteristic
365C2×C4#100412yes—
366C2×C4D3×#4112yes—
367C8—122no—
368C8—122no—
369C2×C3—82no—
370C2×C3—82no—
371C2×C3#258011yescharacteristic
372C2×C3#308112yes—
373C2×C3#311912yes—
374C2×C3#322011yescharacteristic
375C2×C3#322211yescharacteristic
376C22—44no—
377C22—44no—
378C22—42no—
379C22—42no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—82no—
385C22—82no—
386C22C2×#92411yescharacteristic
387C22#580511yescharacteristic
388C22#582811yescharacteristic
389C22#583512yes—
390C22#583612yes—
391C4—24no—
392C4—41no—
393C4—41no—
394C4—42no—
395C4—81no—
396C4—81no—
397C4—124no—
398C4—242no—
399C4#553912yes—
400C4#555912yes—
401C3#2926511yescharacteristic, complemented, Sylow
402C2—82no—
403C2—82no—
404C2#8238811yescharacteristic
405C2#8253912yes—
406C2#8338811yescharacteristic
407C2#8339311yescharacteristic
408C2#8343412yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2328C2
socleC23×C3C22×D4
prime radicalC2×C3×#127C22
hypercenter#2328D3
hypocenterC3#29265
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ #376 ◁ #2328

Complements

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

NH ≅ G/Ntype
2. #73390C2semidirect
4. #71607C2semidirect
5. #71606C2semidirect
6. #69656C2semidirect
7. #69655C2semidirect
8. #69639C2semidirect
9. #69639C2semidirect
10. #69540C2semidirect
11. #69437C2semidirect
401. C3#29265semidirect

Maximal subgroups

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

subgroupindex
C3×#23282
#733902
#717612
#716072
#716062
#696562
#696552
#696392
#695402
#694372
#292653