Finite Group Databasebeta

← #10896392

Subgroups — #10896392

Order 768.

678 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2327C211yesmaximal, characteristic
2#73390C211yesmaximal, characteristic
3#71778C212yesmaximal
4#71629C212yesmaximal
5#71628C212yesmaximal
6#69685C211yesmaximal, characteristic
7#69684C211yesmaximal, characteristic
8#69669C211yesmaximal, characteristic
9#69669C211yesmaximal, characteristic
10#69540C212yesmaximal
11#69437C211yesmaximal, characteristic
12#29266—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C2×#870C2211yescharacteristic
17C3×#391C2211yescharacteristic
18C3×#391C2211yescharacteristic
19C3×#387—22no—
20C3×#387—22no—
21C3×#382C2211yescharacteristic
22C3×#376C2211yescharacteristic
23C2×#835C2212yes—
24C2×#835C2212yes—
25C2×#835C2212yes—
26C2×#835C2212yes—
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#2327D311yescharacteristic
42#2096—31no—
43#2095—31no—
44#2026—31no—
45#2026—31no—
46#1981—32no—
47#1949—32no—
48#1865—32no—
49#1863—32no—
50#1832—31no—
51C22×C3×Q8C2311yescharacteristic
52C22×C3×Q8C2311yescharacteristic
53C3×C4⋊Q8—22no—
54C3×C4⋊Q8Q814yes—
55C3×C4×Q8—42no—
56C3×C4×Q8—42no—
57C2×C3×C4⋊C4—22no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4C2311yescharacteristic
61C2×C3×C4⋊C4C2311yescharacteristic
62C2×C3×C42—21no—
63C2×C3×C42—21no—
64C2×C3×C42C2311yescharacteristic
65C2×C3×C42C2311yescharacteristic
66C2×C3×C42C2311yescharacteristic
67C2×C4⋊Dic3—22no—
68C2×C4⋊Dic3C2312yes—
69C2×C4⋊Dic3C2312yes—
70C2×Dic3⋊C4—22no—
71C2×Dic3⋊C4—22no—
72C2×C4×Dic3—22no—
73C22×C3⋊C8C2312yes—
74C22×C3⋊C8C2312yes—
75#870—24no—
76C3×#95—22no—
77C3×#95—22no—
78C3×#95—22no—
79C3×#95D412yes—
80C3×#95D412yes—
81#835—24no—
82#835—24no—
83#835—42no—
84#835—42no—
85C3⋊#95—22no—
86C3⋊#95—22no—
87#808—42no—
88#807—42no—
89C4⋊(C3⋊C8)—24no—
90C2×C4⋊Q8C2×D311yescharacteristic
91C2×#125—31no—
92C2×C4×Q8C2×D311yescharacteristic
93C2×C4×Q8C2×D311yescharacteristic
94C2×#107—32no—
95C2×#106—32no—
96C2×C4⋊C8—32no—
97C2×#103—32no—
98C2×#103—32no—
99C2×#103—32no—
100C2×#103—32no—
101#391C2×D311yescharacteristic
102#391C2×D311yescharacteristic
103#389—32no—
104#389—62no—
105#387—22no—
106#387—22no—
107#387—62no—
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—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—22no—
120C2×C3×Q8—22no—
121C2×C3×Q8—22no—
122C2×C3×Q8—22no—
123C2×C3×Q8C2×Q812yes—
124C2×C3×Q8C2×Q812yes—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D412yes—
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D412yes—
133C22×C3×C4C2411yescharacteristic
134C22×Dic3—22no—
135C22×Dic3—22no—
136C3×C4⋊C4—28no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4—22no—
139C3×C4⋊C4—42no—
140C3×C4⋊C4—42no—
141C3×C42—22no—
142C3×C42—22no—
143C3×C42—42no—
144C3×C42—42no—
145C3×C42C2×Q812yes—
146C3×C42#4112yes—
147C4⋊Dic3—22no—
148C4⋊Dic3—22no—
149C4⋊Dic3—22no—
150C4⋊Dic3—82no—
151C4⋊Dic3#4114yes—
152Dic3⋊C4—28no—
153Dic3⋊C4—82no—
154C4×Dic3—82no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—22no—
161C22×Q8C22×D311yescharacteristic
162C22×Q8C22×D311yescharacteristic
163C22×C8—32no—
164C22×C8—32no—
165C4⋊Q8—22no—
166C4⋊Q8Dic614yes—
167#125—64no—
168C4×Q8—42no—
169C4×Q8—42no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—32no—
174C2×C4⋊C4—32no—
175C2×C4⋊C4—62no—
176C2×C4⋊C4—62no—
177C2×C4⋊C4—62no—
178C2×C4⋊C4C22×D311yescharacteristic
179C2×C4⋊C4C22×D311yescharacteristic
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—62no—
183C2×C42C22×D311yescharacteristic
184C2×C42C22×D311yescharacteristic
185C2×C42C22×D311yescharacteristic
186#107—122no—
187#106—122no—
188C4⋊C8—64no—
189#103—64no—
190#103—64no—
191#103—122no—
192#103—122no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—62no—
197#95—62no—
198#95#6712yes—
199#95#6712yes—
200C23×C3C22×D411yescharacteristic
201C3×Q8—24no—
202C3×Q8—24no—
203C3×Q8—42no—
204C3×Q8—42no—
205C2×C3×C4—22no—
206C2×C3×C4—22no—
207C2×C3×C4—22no—
208C2×C3×C4—22no—
209C2×C3×C4—22no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—22no—
215C2×C3×C4—22no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—42no—
223C2×C3×C4—42no—
224C2×C3×C4C22⋊Q814yes—
225C2×C3×C4C22×Q811yescharacteristic
226C2×C3×C4C2×#4111yescharacteristic
227C2×C3×C4C2×#4112yes—
228C2×C3×C4H(5,𝔽2)12yes—
229C2×Dic3—24no—
230C2×Dic3—42no—
231C2×Dic3—42no—
232C2×Dic3—42no—
233C2×Dic3—42no—
234C2×Dic3—42no—
235C3⋊C8—42no—
236C3⋊C8—42no—
237C2×Q8—21no—
238C2×Q8—21no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8C2×Dic612yes—
244C2×Q8C2×Dic612yes—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—62no—
250C22×C4—62no—
251C22×C4C2×#6712yes—
252C22×C4C2×#6711yescharacteristic
253C22×C4C2×#6711yescharacteristic
254C22×C4C2×#6712yes—
255C22×C4C23×D311yescharacteristic
256C2×C8—62no—
257C2×C8—62no—
258C2×C8—62no—
259C2×C8—62no—
260C2×C8—62no—
261C2×C8—62no—
262C4⋊C4—28no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—34no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—68no—
269C4⋊C4—62no—
270C4⋊C4—62no—
271C4⋊C4—62no—
272C4⋊C4—242no—
273C4⋊C4—242no—
274C42—22no—
275C42—22no—
276C42—42no—
277C42—42no—
278C42—242no—
279C42C2×Dic612yes—
280C42#23512yes—
281C22×C3C2×C22⋊Q811yescharacteristic
282C22×C3#52511yescharacteristic
283C22×C3#53511yescharacteristic
284C22×C3#57712yes—
285C22×C3#57812yes—
286C3×C4—24no—
287C3×C4—42no—
288C3×C4—42no—
289C3×C4—42no—
290C3×C4—41no—
291C3×C4—41no—
292C3×C4—81no—
293C3×C4—81no—
294C3×C4#55412yes—
295C3×C4#55912yes—
296Dic3—44no—
297Dic3—82no—
298C23C22×#6711yescharacteristic
299Q8—24no—
300Q8—24no—
301Q8—42no—
302Q8—42no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—64no—
323C2×C4—122no—
324C2×C4—122no—
325C2×C4—122no—
326C2×C4—122no—
327C2×C4—122no—
328C2×C4#92414yes—
329C2×C4C22×Dic611yescharacteristic
330C2×C4C2×#23511yescharacteristic
331C2×C4#100412yes—
332C2×C4D3×#4112yes—
333C8—122no—
334C8—122no—
335C2×C3#258011yescharacteristic
336C2×C3#308212yes—
337C2×C3#312012yes—
338C2×C3#322111yescharacteristic
339C2×C3#322311yescharacteristic
340C22C2×#92411yescharacteristic
341C22#580511yescharacteristic
342C22#582811yescharacteristic
343C22#583712yes—
344C22#583812yes—
345C4—24no—
346C4—41no—
347C4—42no—
348C4—42no—
349C4—41no—
350C4—42no—
351C4—81no—
352C4—81no—
353C4—124no—
354C4—242no—
355C4#553912yes—
356C4#555912yes—
357C3#2926611yescharacteristic, complemented, Sylow
358C2#8238811yescharacteristic
359C2#8255012yes—
360C2#8340911yescharacteristic
361C2#8341411yescharacteristic
362C2#8344612yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2327C2
socle, prime radicalC23×C3C22×D4
hypercenter#2327D3
hypocenterC3#29266
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ #376 ◁ #2327

Complements

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

NH ≅ G/Ntype
357. C3#29266semidirect

Maximal subgroups

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

subgroupindex
C3×#23272
#733902
#717782
#716292
#716282
#696852
#696842
#696692
#695402
#694372
#292663