Finite Group Databasebeta

← #10896430

Subgroups — #10896430

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
1#76484C211yesmaximal, characteristic
2#74322C211yesmaximal, characteristic, complemented
3C3×#2096C211yesmaximal, characteristic, complemented
4#70584C212yesmaximal
5#70563C212yesmaximal, complemented
6#70237C212yesmaximal, complemented
7#70230C212yesmaximal
8#69669C211yesmaximal, characteristic, complemented
9#69587C211yesmaximal, characteristic
10#68679C211yesmaximal, characteristic
11#68606C211yesmaximal, characteristic, complemented
12#29269—31nomaximal, Sylow
13C2×#947C2211yescharacteristic
14C2×Q8×Dic3C2211yescharacteristic
15C2×#893C2211yescharacteristic
16C2×#870C2211yescharacteristic
17C2×C3×#106C2212yes—
18C2×C3×#103C2212yes—
19C3×#391C2211yescharacteristic
20C3×#384C2211yescharacteristic
21C2×#835C2212yes—
22C2×#821C2212yes—
23C2×Dic3⋊C8C2212yes—
24#4978—22no—
25#4976—22no—
26#4975C2211yescharacteristic
27Dic3×C4⋊C4C2212yes—
28#4972C2212yes—
29C2×#809C2212yes—
30C2×#808C2212yes—
31#4939C2211yescharacteristic
32#4932C2211yescharacteristic
33#4669—22no—
34#4668—22no—
35C3×#336C2211yescharacteristic
36#4551C2212yes—
37#4550C2212yes—
38#4529C2212yes—
39#4524C2211yescharacteristic
40#2330—31no—
41#2326—31no—
42#2096D311yescharacteristic, complemented
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×D12C2311yescharacteristic
53C22×C4×D3—21no—
54C22×C3×C8C2312yes—
55C2×C3×C4⋊C4—22no—
56C2×C3×C4⋊C4C2312yes—
57C2×C3×C4⋊C4C2311yescharacteristic
58C2×C3×C42C2311yescharacteristic
59#947—22no—
60#947Q814yes—
61Q8×Dic3—42no—
62C2×(C2×D3)⋊C4—22no—
63C2×(C2×D3)⋊C4—21no—
64C2×(C2×D3)⋊C4—22no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3C2311yescharacteristic
67C2×C4⋊Dic3C2312yes—
68C2×Dic3⋊C4—22no—
69C2×Dic3⋊C4—22no—
70C2×Dic3⋊C4—22no—
71C2×C4×Dic3—21no—
72C2×C4×Dic3—22no—
73C2×C4×Dic3C2311yescharacteristic
74C2×C4×Dic3C2311yescharacteristic
75C22×C3⋊C8C2312yes—
76#893—42no—
77#870—24no—
78C3×#106—42no—
79C3×#103—42no—
80C3×#95—22no—
81#835—24no—
82C3⋊#95—22no—
83C3⋊#95—22no—
84C3⋊#95D412yes—
85C3⋊#95D412yes—
86#821—24no—
87Dic3⋊C8—24no—
88#809—42no—
89#808—42no—
90C2×#125—31no—
91C2×#124—31no—
92C2×C4×Q8—31no—
93C2×C4×D4—31no—
94C2×#106—32no—
95C2×#106C2×D312yes—
96C2×C4⋊C8—32no—
97C2×#103—32no—
98C2×#103C2×D312yes—
99C2×#102—32no—
100C2×#102—32no—
101#391C2×D311yescharacteristic
102#390—31no—
103#389—32no—
104#389—62no—
105#388—62no—
106#387—62no—
107#385—62no—
108#384—31no—
109#384C2×D311yescharacteristic
110C4×C4⋊C4—32no—
111#376—31no—
112#340—32no—
113#340—32no—
114#336—32no—
115#336—31no—
116#336C2×D311yescharacteristic
117C23×D3—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—22no—
120C2×C3×Q8—22no—
121C2×C3×Q8C2×Q812yes—
122C22×C3×C4—21no—
123C22×C3×C4—22no—
124C22×C3×C4—21no—
125C22×C3×C4C2×D412yes—
126C22×C3×C4C2411yescharacteristic
127C22×Dic3—22no—
128C22×Dic3—21no—
129C22×Dic3C2×D411yescharacteristic
130C22×Dic3C2×D411yescharacteristic
131C22×Dic3C2×D412yes—
132C2×D12—22no—
133C2×D12—22no—
134C2×D12—21no—
135C2×D12C2×Q812yes—
136C2×C4×D3—42no—
137C2×C4×D3—82no—
138C2×C3×C8—22no—
139C2×C3×C8—22no—
140C2×C3×C8—22no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—22no—
143C3×C4⋊C4—22no—
144C3×C4⋊C4—82no—
145C3×C42—22no—
146(C2×D3)⋊C4—28no—
147(C2×D3)⋊C4—42no—
148(C2×D3)⋊C4—82no—
149C4⋊Dic3—22no—
150C4⋊Dic3—22no—
151C4⋊Dic3—42no—
152C4⋊Dic3#4114yes—
153Dic3⋊C4—28no—
154Dic3⋊C4—82no—
155Dic3⋊C4—82no—
156C4×Dic3—22no—
157C4×Dic3—42no—
158C4×Dic3—82no—
159C4×Dic3C2×Q812yes—
160C4×Dic3#4112yes—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—22no—
164C22×Q8C22×D311yescharacteristic
165C22×D4—31no—
166C23×C4—61no—
167C22×C8—32no—
168C22×C8C22×D312yes—
169#125—64no—
170#124—34no—
171#124—62no—
172C4×Q8—122no—
173C4×D4—122no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—32no—
176C2×C4⋊C4—31no—
177C2×C4⋊C4—62no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—62no—
180C2×C4⋊C4—62no—
181C2×C4⋊C4C22×D311yescharacteristic
182C2×C4⋊C4C22×D312yes—
183C2×C22⋊C4—62no—
184C2×C22⋊C4—61no—
185C2×C22⋊C4—62no—
186C2×C42—31no—
187C2×C42—31no—
188C2×C42—61no—
189C2×C42—62no—
190C2×C42C22×D311yescharacteristic
191#106—42no—
192#106—122no—
193C4⋊C8—64no—
194#103—42no—
195#103—64no—
196#102—64no—
197#102—122no—
198#95—22no—
199#95—32no—
200#95—32no—
201#95—62no—
202#95—62no—
203C23×C3C22×D411yescharacteristic
204C22×D3—22no—
205C22×D3—41no—
206C22×D3—41no—
207C22×D3—42no—
208C22×D3—42no—
209C3×Q8—24no—
210C3×Q8—42no—
211C2×C3×C4—22no—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—22no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—42no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4C22×Q811yescharacteristic
223C2×C3×C4C2×#4112yes—
224C2×C3×C4C2×#4111yescharacteristic
225C2×C3×C4H(5,𝔽2)12yes—
226C2×Dic3—22no—
227C2×Dic3—22no—
228C2×Dic3—24no—
229C2×Dic3—21no—
230C2×Dic3—21no—
231C2×Dic3—22no—
232C2×Dic3—22no—
233C2×Dic3—42no—
234C2×Dic3—42no—
235C2×Dic3—41no—
236C2×Dic3—42no—
237C2×Dic3C22⋊Q814yes—
238D12—24no—
239D12—42no—
240C4×D3—84no—
241C3×C8—42no—
242C3⋊C8—42no—
243C24—61no—
244C2×Q8—21no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8D3×Q812yes—
248C2×D4—32no—
249C2×D4—62no—
250C2×D4—62no—
251C2×D4—61no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—32no—
256C22×C4—31no—
257C22×C4—31no—
258C22×C4—62no—
259C22×C4—61no—
260C22×C4—122no—
261C22×C4—242no—
262C22×C4D3×D412yes—
263C22×C4C23×D311yescharacteristic
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—62no—
268C2×C8—62no—
269C2×C8—62no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—34no—
274C4⋊C4—68no—
275C4⋊C4—62no—
276C4⋊C4—62no—
277C4⋊C4—82no—
278C4⋊C4—122no—
279C4⋊C4—242no—
280C4⋊C4—242no—
281C22⋊C4—68no—
282C22⋊C4—122no—
283C22⋊C4—242no—
284C42—22no—
285C42—32no—
286C42—32no—
287C42—62no—
288C42—122no—
289C42—242no—
290C22×C3C2×C22⋊Q811yescharacteristic
291C22×C3#52511yescharacteristic
292C22×C3#53511yescharacteristic
293C22×C3#57712yes—
294C22×C3#57712yes—
295C2×D3—44no—
296C2×D3—42no—
297C2×D3—82no—
298C2×D3—81no—
299C2×D3—81no—
300C3×C4—42no—
301C3×C4—42no—
302C3×C4—82no—
303C3×C4—81no—
304C3×C4#55412yes—
305C3×C4#55912yes—
306Dic3—24no—
307Dic3—44no—
308Dic3—41no—
309Dic3—41no—
310Dic3—81no—
311C23—62no—
312C23—121no—
313C23—121no—
314C23—122no—
315C23—122no—
316C23C2×D3×D411yescharacteristic
317Q8—24no—
318Q8—42no—
319D4—64no—
320D4—122no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—34no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—62no—
334C2×C4—62no—
335C2×C4—64no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—62no—
339C2×C4—62no—
340C2×C4—122no—
341C2×C4—122no—
342C2×C4—121no—
343C2×C4—122no—
344C2×C4—244no—
345C2×C4#100412yes—
346C2×C4C2×D3×Q811yescharacteristic
347C2×C4D3×#4112yes—
348C2×C4D3×#4111yescharacteristic
349C8—42no—
350C8—122no—
351C2×C3#258011yescharacteristic
352C2×C3#308412yes—
353C2×C3#312512yes—
354C2×C3#322111yescharacteristic
355C2×C3#322211yescharacteristic
356D3—82no—
357C22—124no—
358C22—122no—
359C22—242no—
360C22—241no—
361C22—241no—
362C22D3×C22⋊Q811yescharacteristic
363C22#565611yescharacteristic
364C22#571511yescharacteristic
365C22#576212yes—
366C22#577812yes—
367C4—42no—
368C4—42no—
369C4—64no—
370C4—82no—
371C4—81no—
372C4—124no—
373C4—121no—
374C4—121no—
375C4—241no—
376C4#563612yes—
377C4#566412yes—
378C3#2926911yescharacteristic, complemented, Sylow
379C2—242no—
380C2#7804111yescharacteristic
381C2#8101412yes—
382C2#8141712yes—
383C2#8176811yescharacteristic
384C2#8187811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2096C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2096D3
hypocenterC3#29269
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
2. #74322C2semidirect
3. C3×#2096C2semidirect
5. #70563C2semidirect
6. #70237C2semidirect
8. #69669C2semidirect
11. #68606C2semidirect
42. #2096D3semidirect
378. C3#29269semidirect

Maximal subgroups

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

subgroupindex
#764842
#743222
C3×#20962
#705842
#705632
#702372
#702302
#696692
#695872
#686792
#686062
#292693