Finite Group Databasebeta

← #10896433

Subgroups — #10896433

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×#2330C211yesmaximal, characteristic, complemented
2#73387C211yesmaximal, characteristic
3#71744C211yesmaximal, characteristic, complemented
4#71743C211yesmaximal, characteristic
5#69632C212yesmaximal, complemented
6#69626C212yesmaximal
7#69591C211yesmaximal, characteristic, complemented
8#69587C211yesmaximal, characteristic
9#69488C212yesmaximal
10#69467C212yesmaximal, complemented
11#69437C211yesmaximal, characteristic, complemented
12#29269—31nomaximal, Sylow
13C2×C3×#125C2211yescharacteristic
14C2×#875C2211yescharacteristic
15C2×C4×D12C2211yescharacteristic
16C2×C4×Dic6C2211yescharacteristic
17C3×#389—22no—
18C3×#389C2212yes—
19C3×#387—22no—
20C3×#384C2211yescharacteristic
21C3×C4×C4⋊C4C2212yes—
22C3×#376C2211yescharacteristic
23C2×#810C2212yes—
24C2×#810C2212yes—
25C2×#809C2212yes—
26C2×#809C2212yes—
27C2×#808C2212yes—
28C2×#808C2212yes—
29#4939C2211yescharacteristic
30#4935C2211yescharacteristic
31#4929C2211yescharacteristic
32C2×C4⋊(C3⋊C8)C2212yes—
33#4670—22no—
34#4668—22no—
35#4529C2212yes—
36#4529C2212yes—
37#4524C2212yes—
38#4524C2211yescharacteristic
39#4524C2211yescharacteristic
40#2330D311yescharacteristic, complemented
41#2326—31no—
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×D12C2311yescharacteristic
52C22×C4×D3—21no—
53C22×Dic6C2311yescharacteristic
54C3×#125—24no—
55C2×C3×C4⋊C4—22no—
56C2×C3×C4⋊C4—22no—
57C2×C3×C4⋊C4—22no—
58C2×C3×C4⋊C4C2312yes—
59C2×C3×C4⋊C4C2312yes—
60C2×C3×C42—22no—
61C2×C3×C42C2311yescharacteristic
62C2×C3×C42C2311yescharacteristic
63C2×C3×C42C2311yescharacteristic
64C2×(C2×D3)⋊C4—22no—
65C2×(C2×D3)⋊C4—21no—
66C2×(C2×D3)⋊C4—22no—
67C2×C4⋊Dic3—22no—
68C2×C4⋊Dic3C2311yescharacteristic
69C2×C4⋊Dic3C2311yescharacteristic
70C2×Dic3⋊C4—21no—
71C2×C4×Dic3—21no—
72C22×C3⋊C8C2312yes—
73C22×C3⋊C8C2312yes—
74#875—22no—
75#875Q814yes—
76C4×D12—42no—
77C4×Dic6—42no—
78C3×#95—22no—
79C3×#95—22no—
80C3×#95D412yes—
81C3×#95D412yes—
82C3⋊#95—22no—
83#810—24no—
84#810—42no—
85#809—24no—
86#809—42no—
87#808—42no—
88#808—42no—
89C4⋊(C3⋊C8)—24no—
90C2×#125C2×D311yescharacteristic
91C2×#124—31no—
92C2×C4×Q8—31no—
93C2×C4×D4—31no—
94C2×#106—32no—
95C2×#106—32no—
96C2×C4⋊C8—32no—
97C2×#103—32no—
98C2×#103—32no—
99C2×#102—32no—
100C2×#102—32no—
101#391—31no—
102#390—31no—
103#389—22no—
104#389C2×D312yes—
105#388—62no—
106#387—22no—
107#385—62no—
108#384—31no—
109#384C2×D311yescharacteristic
110C4×C4⋊C4C2×D312yes—
111#376C2×D311yescharacteristic
112#340—32no—
113#340—32no—
114#336—31no—
115#336—31no—
116#336—32no—
117C23×D3—21no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4C2×D412yes—
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D412yes—
124C22×C3×C4C2411yescharacteristic
125C22×Dic3—21no—
126C22×Dic3—21no—
127C22×Dic3—21no—
128C2×D12—22no—
129C2×D12—22no—
130C2×D12—21no—
131C2×D12C2×Q812yes—
132C2×C4×D3—42no—
133C2×C4×D3—82no—
134C2×Dic6—22no—
135C2×Dic6—22no—
136C2×Dic6—21no—
137C2×Dic6C2×Q812yes—
138C3×C4⋊C4—28no—
139C3×C4⋊C4—22no—
140C3×C4⋊C4—22no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—82no—
143C3×C4⋊C4—82no—
144C3×C4⋊C4#4114yes—
145C3×C42—22no—
146C3×C42—22no—
147C3×C42—82no—
148C3×C42C2×Q812yes—
149C3×C42#4112yes—
150(C2×D3)⋊C4—28no—
151(C2×D3)⋊C4—42no—
152(C2×D3)⋊C4—82no—
153C4⋊Dic3—22no—
154C4⋊Dic3—22no—
155C4⋊Dic3—82no—
156Dic3⋊C4—42no—
157C4×Dic3—42no—
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×Q8—31no—
165C22×D4—31no—
166C23×C4—61no—
167C22×C8—32no—
168C22×C8—32no—
169#125—24no—
170#124—34no—
171#124—62no—
172C4×Q8—122no—
173C4×D4—122no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—62no—
181C2×C4⋊C4C22×D312yes—
182C2×C4⋊C4C22×D312yes—
183C2×C22⋊C4—62no—
184C2×C22⋊C4—61no—
185C2×C22⋊C4—62no—
186C2×C42—22no—
187C2×C42—61no—
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—22no—
200#95—62no—
201#95#6712yes—
202#95#6712yes—
203C23×C3C22×D411yescharacteristic
204C22×D3—22no—
205C22×D3—41no—
206C22×D3—41no—
207C22×D3—42no—
208C22×D3—42no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—24no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—22no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4—22no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—42no—
223C2×C3×C4—42no—
224C2×C3×C4C22⋊Q814yes—
225C2×C3×C4C22×Q811yescharacteristic
226C2×C3×C4C2×#4112yes—
227C2×C3×C4C2×#4111yescharacteristic
228C2×C3×C4H(5,𝔽2)12yes—
229C2×Dic3—22no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3—42no—
235C2×Dic3—42no—
236D12—24no—
237D12—42no—
238C4×D3—84no—
239Dic6—24no—
240Dic6—42no—
241C3⋊C8—42no—
242C3⋊C8—42no—
243C24—61no—
244C2×Q8—32no—
245C2×Q8—61no—
246C2×Q8—62no—
247C2×Q8—62no—
248C2×D4—32no—
249C2×D4—62no—
250C2×D4—61no—
251C2×D4—62no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—61no—
255C22×C4—61no—
256C22×C4—61no—
257C22×C4—122no—
258C22×C4—242no—
259C22×C4C2×#6711yescharacteristic
260C22×C4C2×#6712yes—
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—28no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—62no—
275C4⋊C4—62no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C4⋊C4—122no—
279C4⋊C4—242no—
280C4⋊C4#23514yes—
281C22⋊C4—68no—
282C22⋊C4—122no—
283C22⋊C4—242no—
284C42—22no—
285C42—22no—
286C42—82no—
287C42—122no—
288C42D3×Q812yes—
289C42#23912yes—
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—24no—
301C3×C4—42no—
302C3×C4—44no—
303C3×C4—41no—
304C3×C4—41no—
305C3×C4—82no—
306C3×C4#55412yes—
307C3×C4#55912yes—
308Dic3—42no—
309Dic3—81no—
310Dic3—81no—
311C23—62no—
312C23—121no—
313C23—121no—
314C23—122no—
315C23—122no—
316C23C22×#6711yescharacteristic
317Q8—64no—
318Q8—122no—
319D4—64no—
320D4—122no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—24no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—62no—
337C2×C4—121no—
338C2×C4—121no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—122no—
342C2×C4—122no—
343C2×C4—244no—
344C2×C4#94614yes—
345C2×C4C2×#23512yes—
346C2×C4C2×D3×Q811yescharacteristic
347C2×C4C2×#23911yescharacteristic
348C2×C4#100912yes—
349C8—122no—
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—
362C22#579111yescharacteristic
363C22C2×#94611yescharacteristic
364C22#583011yescharacteristic
365C22#583612yes—
366C22#583712yes—
367C4—24no—
368C4—42no—
369C4—44no—
370C4—41no—
371C4—41no—
372C4—82no—
373C4—122no—
374C4—241no—
375C4—241no—
376C4#557912yes—
377C4#558512yes—
378C3#2926911yescharacteristic, complemented, Sylow
379C2—242no—
380C2#8250111yescharacteristic
381C2#8257612yes—
382C2#8344011yescharacteristic
383C2#8344811yescharacteristic
384C2#8347612yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2330C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2330D3
hypocenterC3#29269
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ #376 ◁ #2330

Complements

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

NH ≅ G/Ntype
1. C3×#2330C2semidirect
3. #71744C2semidirect
5. #69632C2semidirect
7. #69591C2semidirect
10. #69467C2semidirect
11. #69437C2semidirect
40. #2330D3semidirect
378. C3#29269semidirect

Maximal subgroups

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

subgroupindex
C3×#23302
#733872
#717442
#717432
#696322
#696262
#695912
#695872
#694882
#694672
#694372
#292693