Finite Group Databasebeta

← #10896399

Subgroups — #10896399

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
1#76485C211yesmaximal, characteristic
2#74322C211yesmaximal, characteristic
3C3×#2096C211yesmaximal, characteristic
4#70588C212yesmaximal
5#70564C212yesmaximal
6#70235C212yesmaximal
7#70229C212yesmaximal
8#69669C211yesmaximal, characteristic
9#69590C211yesmaximal, characteristic
10#68690C211yesmaximal, characteristic
11#68606C211yesmaximal, characteristic
12#29266—31nomaximal, Sylow
13C2×Q8×Dic3C2211yescharacteristic
14C2×#944C2211yescharacteristic
15C2×#887C2211yescharacteristic
16C2×#870C2211yescharacteristic
17C2×C3×#106C2212yes—
18C2×C3×#103C2212yes—
19C3×#391C2211yescharacteristic
20C3×#384C2211yescharacteristic
21C2×#835C2212yes—
22C2×#816C2212yes—
23C2×Dic3⋊C8C2212yes—
24#4978—22no—
25#4977C2211yescharacteristic
26#4976—22no—
27Dic3×C4⋊C4C2212yes—
28#4972C2212yes—
29C2×#810C2212yes—
30C2×#807C2212yes—
31#4932C2211yescharacteristic
32#4929C2211yescharacteristic
33#4655—22no—
34#4652—22no—
35C3×#336C2211yescharacteristic
36#4551C2212yes—
37#4550C2212yes—
38#4529C2212yes—
39#4524C2211yescharacteristic
40#2330—31no—
41#2327—31no—
42#2096D311yescharacteristic
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×Dic6C2311yescharacteristic
53C22×C3×C8C2312yes—
54C2×C3×C4⋊C4—22no—
55C2×C3×C4⋊C4C2312yes—
56C2×C3×C4⋊C4C2311yescharacteristic
57C2×C3×C42C2311yescharacteristic
58Q8×Dic3—42no—
59#944—22no—
60#944Q814yes—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3C2311yescharacteristic
63C2×C4⋊Dic3C2312yes—
64C2×Dic3⋊C4—22no—
65C2×Dic3⋊C4—22no—
66C2×Dic3⋊C4—21no—
67C2×Dic3⋊C4—22no—
68C2×C4×Dic3—21no—
69C2×C4×Dic3—21no—
70C2×C4×Dic3—22no—
71C2×C4×Dic3C2311yescharacteristic
72C2×C4×Dic3C2311yescharacteristic
73C22×C3⋊C8C2312yes—
74#887—42no—
75#870—24no—
76C3×#106—42no—
77C3×#103—42no—
78C3×#95—22no—
79#835—24no—
80C3⋊#95—22no—
81C3⋊#95—22no—
82C3⋊#95—22no—
83C3⋊#95—22no—
84C3⋊#95D412yes—
85C3⋊#95D412yes—
86#816—24no—
87Dic3⋊C8—24no—
88#810—42no—
89#807—42no—
90C2×C4⋊Q8—31no—
91C2×#125—31no—
92C2×C4×Q8—31no—
93C2×C4×Q8—31no—
94C2×#107—32no—
95C2×#106C2×D312yes—
96C2×C4⋊C8—32no—
97C2×#103—32no—
98C2×#103—32no—
99C2×#103—32no—
100C2×#103C2×D312yes—
101#391—31no—
102#391C2×D311yescharacteristic
103#389—32no—
104#389—62no—
105#387—62no—
106#387—62no—
107#387—62no—
108#384C2×D311yescharacteristic
109#382—31no—
110C4×C4⋊C4—32no—
111#376—31no—
112#340—32no—
113#340—32no—
114#336—32no—
115#336—31no—
116#336C2×D311yescharacteristic
117C2×C3×Q8—21no—
118C2×C3×Q8—22no—
119C2×C3×Q8—22no—
120C2×C3×Q8C2×Q812yes—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—22no—
124C22×C3×C4C2×D412yes—
125C22×C3×C4C2411yescharacteristic
126C22×Dic3—22no—
127C22×Dic3—21no—
128C22×Dic3—21no—
129C22×Dic3C2×D411yescharacteristic
130C22×Dic3C2×D411yescharacteristic
131C22×Dic3C2×D412yes—
132C2×Dic6—22no—
133C2×Dic6—22no—
134C2×Dic6—21no—
135C2×Dic6C2×Q812yes—
136C2×C3×C8—22no—
137C2×C3×C8—22no—
138C2×C3×C8—22no—
139C3×C4⋊C4—22no—
140C3×C4⋊C4—22no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—82no—
143C3×C42—22no—
144C4⋊Dic3—22no—
145C4⋊Dic3—22no—
146C4⋊Dic3—42no—
147C4⋊Dic3#4114yes—
148Dic3⋊C4—28no—
149Dic3⋊C4—28no—
150Dic3⋊C4—42no—
151Dic3⋊C4—82no—
152C4×Dic3—22no—
153C4×Dic3—42no—
154C4×Dic3—42no—
155C4×Dic3—82no—
156C4×Dic3C2×Q812yes—
157C4×Dic3#4112yes—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—22no—
161C22×Q8—31no—
162C22×Q8C22×D311yescharacteristic
163C22×C8—32no—
164C22×C8C22×D312yes—
165C4⋊Q8—34no—
166C4⋊Q8—62no—
167#125—64no—
168C4×Q8—122no—
169C4×Q8—122no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—32no—
172C2×C4⋊C4—31no—
173C2×C4⋊C4—62no—
174C2×C4⋊C4—62no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—62no—
178C2×C4⋊C4C22×D311yescharacteristic
179C2×C4⋊C4C22×D312yes—
180C2×C42—31no—
181C2×C42—31no—
182C2×C42—61no—
183C2×C42—61no—
184C2×C42—62no—
185C2×C42C22×D311yescharacteristic
186#107—122no—
187#106—42no—
188C4⋊C8—64no—
189#103—42no—
190#103—64no—
191#103—64no—
192#103—122no—
193#95—22no—
194#95—32no—
195#95—32no—
196#95—62no—
197#95—62no—
198#95—62no—
199#95—62no—
200C23×C3C22×D411yescharacteristic
201C3×Q8—24no—
202C3×Q8—42no—
203C2×C3×C4—22no—
204C2×C3×C4—22no—
205C2×C3×C4—22no—
206C2×C3×C4—22no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4C22×Q811yescharacteristic
215C2×C3×C4C2×#4112yes—
216C2×C3×C4C2×#4111yescharacteristic
217C2×C3×C4H(5,𝔽2)12yes—
218C2×Dic3—22no—
219C2×Dic3—22no—
220C2×Dic3—22no—
221C2×Dic3—24no—
222C2×Dic3—21no—
223C2×Dic3—21no—
224C2×Dic3—22no—
225C2×Dic3—22no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—42no—
229C2×Dic3—42no—
230C2×Dic3—41no—
231C2×Dic3—42no—
232C2×Dic3C22⋊Q814yes—
233Dic6—24no—
234Dic6—42no—
235C3×C8—42no—
236C3⋊C8—42no—
237C2×Q8—21no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—32no—
241C2×Q8—61no—
242C2×Q8—62no—
243C2×Q8—62no—
244C2×Q8D3×Q812yes—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—32no—
249C22×C4—31no—
250C22×C4—31no—
251C22×C4—62no—
252C22×C4—61no—
253C22×C4—61no—
254C22×C4D3×D412yes—
255C22×C4C23×D311yescharacteristic
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—22no—
259C2×C8—62no—
260C2×C8—62no—
261C2×C8—62no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—34no—
266C4⋊C4—68no—
267C4⋊C4—68no—
268C4⋊C4—62no—
269C4⋊C4—62no—
270C4⋊C4—82no—
271C4⋊C4—122no—
272C4⋊C4—122no—
273C4⋊C4—242no—
274C42—22no—
275C42—32no—
276C42—32no—
277C42—62no—
278C42—122no—
279C42—122no—
280C42—242no—
281C22×C3C2×C22⋊Q811yescharacteristic
282C22×C3#52511yescharacteristic
283C22×C3#53511yescharacteristic
284C22×C3#57712yes—
285C22×C3#57812yes—
286C3×C4—42no—
287C3×C4—42no—
288C3×C4—82no—
289C3×C4—81no—
290C3×C4#55412yes—
291C3×C4#55912yes—
292Dic3—24no—
293Dic3—42no—
294Dic3—44no—
295Dic3—41no—
296Dic3—41no—
297Dic3—81no—
298C23C2×D3×D411yescharacteristic
299Q8—24no—
300Q8—42no—
301Q8—64no—
302Q8—122no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—34no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—62no—
316C2×C4—62no—
317C2×C4—64no—
318C2×C4—61no—
319C2×C4—61no—
320C2×C4—62no—
321C2×C4—62no—
322C2×C4—62no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4—122no—
326C2×C4—122no—
327C2×C4—121no—
328C2×C4—122no—
329C2×C4#100412yes—
330C2×C4C2×D3×Q811yescharacteristic
331C2×C4D3×#4112yes—
332C2×C4D3×#4111yescharacteristic
333C8—42no—
334C8—122no—
335C2×C3#258011yescharacteristic
336C2×C3#308212yes—
337C2×C3#312012yes—
338C2×C3#322111yescharacteristic
339C2×C3#322311yescharacteristic
340C22D3×C22⋊Q811yescharacteristic
341C22#565611yescharacteristic
342C22#571511yescharacteristic
343C22#576212yes—
344C22#577912yes—
345C4—42no—
346C4—42no—
347C4—64no—
348C4—82no—
349C4—81no—
350C4—124no—
351C4—121no—
352C4—121no—
353C4—122no—
354C4—241no—
355C4#563612yes—
356C4#566412yes—
357C3#2926611yescharacteristic, complemented, Sylow
358C2#7804111yescharacteristic
359C2#8101312yes—
360C2#8141612yes—
361C2#8176811yescharacteristic
362C2#8187911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2096C2
socle, prime radicalC23×C3C22×D4
hypercenter#2096D3
hypocenterC3#29266
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2096

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
#764852
#743222
C3×#20962
#705882
#705642
#702352
#702292
#696692
#695902
#686902
#686062
#292663