Finite Group Databasebeta

← #10896429

Subgroups — #10896429

Order 768.

726 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2326C211yesmaximal, characteristic
2#73390C211yesmaximal, characteristic, complemented
3#71721C212yesmaximal
4#71629C212yesmaximal, complemented
5#71607C212yesmaximal
6#69685C211yesmaximal, characteristic, complemented
7#69669C211yesmaximal, characteristic, complemented
8#69656C211yesmaximal, characteristic
9#69639C211yesmaximal, characteristic
10#69568C212yesmaximal, complemented
11#69437C211yesmaximal, characteristic, complemented
12#29269—31nomaximal, Sylow
13C2×C3×#124C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×#870C2211yescharacteristic
17C3×#391C2211yescharacteristic
18C3×#390C2211yescharacteristic
19C3×#388—22no—
20C3×#385—22no—
21C3×#384C2211yescharacteristic
22C3×#376C2211yescharacteristic
23C2×#835C2212yes—
24C2×#835C2212yes—
25C2×#832C2212yes—
26C2×#832C2212yes—
27C2×#808C2212yes—
28C2×#808C2212yes—
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#2326D311yescharacteristic
42#2096—31no—
43#2094—31no—
44#2026—31no—
45#2025—31no—
46#1983—32no—
47#1948—32no—
48#1865—32no—
49#1864—32no—
50#1832—31no—
51C22×C3×Q8C2311yescharacteristic
52C22×C3×D4C2311yescharacteristic
53C23×C3×C4—21no—
54C3×#124—22no—
55C3×#124Q814yes—
56C3×C4×Q8—42no—
57C3×C4×D4—42no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4—22no—
60C2×C3×C4⋊C4C2311yescharacteristic
61C2×C3×C4⋊C4C2311yescharacteristic
62C2×C3×C22⋊C4—22no—
63C2×C3×C22⋊C4—21no—
64C2×C3×C22⋊C4—22no—
65C2×C3×C42—21no—
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#835—24no—
82#835—42no—
83#832—24no—
84#832—42no—
85C3⋊#95—22no—
86C3⋊#95—22no—
87#808—42no—
88#808—42no—
89C4⋊(C3⋊C8)—24no—
90C2×#125—31no—
91C2×#124C2×D311yescharacteristic
92C2×C4×Q8C2×D311yescharacteristic
93C2×C4×D4C2×D311yescharacteristic
94C2×#106—32no—
95C2×#106—32no—
96C2×C4⋊C8—32no—
97C2×#103—32no—
98C2×#103—32no—
99C2×#102—32no—
100C2×#102—32no—
101#391C2×D311yescharacteristic
102#390C2×D311yescharacteristic
103#389—32no—
104#389—62no—
105#388—22no—
106#387—62no—
107#385—22no—
108#384—31no—
109#384C2×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—
118C2×C3×Q8—21no—
119C2×C3×Q8—22no—
120C2×C3×Q8—22no—
121C2×C3×Q8C2×Q812yes—
122C2×C3×D4—22no—
123C2×C3×D4—21no—
124C2×C3×D4—22no—
125C2×C3×D4C2×Q812yes—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—42no—
130C22×C3×C4—82no—
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D412yes—
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D412yes—
135C22×C3×C4C2411yescharacteristic
136C22×Dic3—22no—
137C22×Dic3—22no—
138C3×C4⋊C4—22no—
139C3×C4⋊C4—22no—
140C3×C4⋊C4—42no—
141C3×C4⋊C4—82no—
142C3×C22⋊C4—28no—
143C3×C22⋊C4—42no—
144C3×C22⋊C4—82no—
145C3×C42—22no—
146C3×C42—22no—
147C3×C42—42no—
148C3×C42C2×Q812yes—
149C3×C42#4112yes—
150C4⋊Dic3—22no—
151C4⋊Dic3—22no—
152C4⋊Dic3—22no—
153C4⋊Dic3—82no—
154C4⋊Dic3#4114yes—
155Dic3⋊C4—28no—
156Dic3⋊C4—82no—
157C4×Dic3—82no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—22no—
164C22×Q8C22×D311yescharacteristic
165C22×D4C22×D311yescharacteristic
166C23×C4—21no—
167C22×C8—32no—
168C22×C8—32no—
169#125—64no—
170#124—22no—
171#124Dic614yes—
172C4×Q8—42no—
173C4×D4—42no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—32no—
177C2×C4⋊C4—32no—
178C2×C4⋊C4—62no—
179C2×C4⋊C4—62no—
180C2×C4⋊C4—62no—
181C2×C4⋊C4C22×D311yescharacteristic
182C2×C4⋊C4C22×D311yescharacteristic
183C2×C22⋊C4—22no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—22no—
186C2×C42—21no—
187C2×C42—62no—
188C2×C42C22×D311yescharacteristic
189C2×C42C22×D311yescharacteristic
190C2×C42C22×D311yescharacteristic
191#106—122no—
192#106—122no—
193C4⋊C8—64no—
194#103—64no—
195#103—122no—
196#102—64no—
197#102—122no—
198#95—22no—
199#95—62no—
200#95—62no—
201#95#6712yes—
202#95#6712yes—
203C23×C3—22no—
204C23×C3—41no—
205C23×C3—41no—
206C23×C3—42no—
207C23×C3—42no—
208C23×C3C22×D411yescharacteristic
209C3×Q8—24no—
210C3×Q8—42no—
211C3×D4—24no—
212C3×D4—42no—
213C2×C3×C4—22no—
214C2×C3×C4—22no—
215C2×C3×C4—22no—
216C2×C3×C4—22no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—22no—
220C2×C3×C4—22no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4—84no—
230C2×C3×C4C22⋊Q814yes—
231C2×C3×C4C22×Q811yescharacteristic
232C2×C3×C4C2×#4111yescharacteristic
233C2×C3×C4C2×#4112yes—
234C2×C3×C4H(5,𝔽2)12yes—
235C2×Dic3—24no—
236C2×Dic3—42no—
237C2×Dic3—42no—
238C2×Dic3—42no—
239C2×Dic3—42no—
240C2×Dic3—42no—
241C3⋊C8—42no—
242C3⋊C8—42no—
243C24—21no—
244C2×Q8—21no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8C2×Dic612yes—
248C2×D4—21no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4C2×Dic612yes—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—42no—
256C22×C4—62no—
257C22×C4—62no—
258C22×C4—82no—
259C22×C4C2×#6712yes—
260C22×C4C2×#6711yescharacteristic
261C22×C4C2×#6711yescharacteristic
262C22×C4C2×#6712yes—
263C22×C4C23×D311yescharacteristic
264C2×C8—62no—
265C2×C8—62no—
266C2×C8—62no—
267C2×C8—62no—
268C2×C8—62no—
269C2×C8—62no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—34no—
273C4⋊C4—42no—
274C4⋊C4—68no—
275C4⋊C4—62no—
276C4⋊C4—62no—
277C4⋊C4—62no—
278C4⋊C4—82no—
279C4⋊C4—242no—
280C4⋊C4—242no—
281C22⋊C4—28no—
282C22⋊C4—42no—
283C22⋊C4—82no—
284C42—22no—
285C42—22no—
286C42—42no—
287C42—242no—
288C42C2×Dic612yes—
289C42#23512yes—
290C22×C3—44no—
291C22×C3—42no—
292C22×C3—82no—
293C22×C3—81no—
294C22×C3—81no—
295C22×C3C2×C22⋊Q811yescharacteristic
296C22×C3#52511yescharacteristic
297C22×C3#53511yescharacteristic
298C22×C3#57712yes—
299C22×C3#57712yes—
300C3×C4—24no—
301C3×C4—42no—
302C3×C4—42no—
303C3×C4—41no—
304C3×C4—41no—
305C3×C4—81no—
306C3×C4—81no—
307C3×C4#55412yes—
308C3×C4#55912yes—
309Dic3—44no—
310Dic3—82no—
311C23—22no—
312C23—41no—
313C23—41no—
314C23—42no—
315C23—42no—
316C23C22×#6711yescharacteristic
317Q8—24no—
318Q8—42no—
319D4—24no—
320D4—42no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—64no—
338C2×C4—84no—
339C2×C4—122no—
340C2×C4—122no—
341C2×C4—122no—
342C2×C4—122no—
343C2×C4—122no—
344C2×C4#92414yes—
345C2×C4C22×Dic611yescharacteristic
346C2×C4C2×#23511yescharacteristic
347C2×C4#100412yes—
348C2×C4D3×#4112yes—
349C8—122no—
350C8—122no—
351C2×C3—82no—
352C2×C3#258011yescharacteristic
353C2×C3#308412yes—
354C2×C3#312512yes—
355C2×C3#322111yescharacteristic
356C2×C3#322211yescharacteristic
357C22—44no—
358C22—42no—
359C22—82no—
360C22—81no—
361C22—81no—
362C22C2×#92411yescharacteristic
363C22#580511yescharacteristic
364C22#582811yescharacteristic
365C22#583612yes—
366C22#583712yes—
367C4—24no—
368C4—42no—
369C4—41no—
370C4—41no—
371C4—42no—
372C4—81no—
373C4—81no—
374C4—124no—
375C4—242no—
376C4#553912yes—
377C4#555912yes—
378C3#2926911yescharacteristic, complemented, Sylow
379C2—82no—
380C2#8238811yescharacteristic
381C2#8256012yes—
382C2#8339311yescharacteristic
383C2#8340911yescharacteristic
384C2#8346212yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2326C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#2326D3
hypocenterC3#29269
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ #376 ◁ #2326

Complements

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

NH ≅ G/Ntype
2. #73390C2semidirect
4. #71629C2semidirect
6. #69685C2semidirect
7. #69669C2semidirect
10. #69568C2semidirect
11. #69437C2semidirect
378. C3#29269semidirect

Maximal subgroups

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

subgroupindex
C3×#23262
#733902
#717212
#716292
#716072
#696852
#696692
#696562
#696392
#695682
#694372
#292693