Finite Group Databasebeta

← #10896385

Subgroups — #10896385

Order 768.

790 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2330C211yesmaximal, characteristic, complemented
2#73389C211yesmaximal, characteristic
3#71743C211yesmaximal, characteristic, complemented
4#71743C211yesmaximal, characteristic, complemented
5#69626C212yesmaximal, complemented
6#69625C212yesmaximal, complemented
7#69587C211yesmaximal, characteristic, complemented
8#69586C211yesmaximal, characteristic, complemented
9#69487C212yesmaximal
10#69468C212yesmaximal, complemented
11#69437C211yesmaximal, characteristic, complemented
12#29265—31nomaximal, Sylow
13C2×C3×#125C2211yescharacteristic
14C2×#874C2211yescharacteristic
15C2×C4×D12C2211yescharacteristic
16C2×C4×D12C2211yescharacteristic
17C3×#389—22no—
18C3×#389C2212yes—
19C3×#387—22no—
20C3×#384C2211yescharacteristic
21C3×C4×C4⋊C4C2212yes—
22C3×#376C2211yescharacteristic
23C2×#809C2212yes—
24C2×#809C2212yes—
25C2×#809C2212yes—
26C2×#809C2212yes—
27C2×#808C2212yes—
28C2×#807C2212yes—
29#4939C2211yescharacteristic
30#4939C2211yescharacteristic
31#4936C2211yescharacteristic
32C2×C4⋊(C3⋊C8)C2212yes—
33#4665—22no—
34#4665—22no—
35#4529C2212yes—
36#4529C2212yes—
37#4524C2211yescharacteristic
38#4524C2211yescharacteristic
39#4524C2212yes—
40#2330D311yescharacteristic, complemented
41#2328—31no—
42#2094—31no—
43#2093—31no—
44#2025—31no—
45#2025—31no—
46#1982—32no—
47#1949—32no—
48#1864—32no—
49#1862—32no—
50#1832—31no—
51C22×D12—22no—
52C22×D12C2311yescharacteristic
53C22×D12C2311yescharacteristic
54C22×C4×D3—21no—
55C22×C4×D3—21no—
56C3×#125—24no—
57C2×C3×C4⋊C4—22no—
58C2×C3×C4⋊C4—22no—
59C2×C3×C4⋊C4—22no—
60C2×C3×C4⋊C4C2312yes—
61C2×C3×C4⋊C4C2312yes—
62C2×C3×C42—22no—
63C2×C3×C42C2311yescharacteristic
64C2×C3×C42C2311yescharacteristic
65C2×C3×C42C2311yescharacteristic
66C2×(C2×D3)⋊C4—21no—
67C2×(C2×D3)⋊C4—21no—
68C2×(C2×D3)⋊C4—22no—
69C2×(C2×D3)⋊C4—22no—
70C2×C4⋊Dic3C2311yescharacteristic
71C2×C4⋊Dic3C2311yescharacteristic
72C22×C3⋊C8C2312yes—
73C22×C3⋊C8C2312yes—
74#874—22no—
75#874Q814yes—
76C4×D12—42no—
77C4×D12—42no—
78C3×#95—22no—
79C3×#95—22no—
80C3×#95D412yes—
81C3×#95D412yes—
82C3⋊#95—22no—
83#809—24no—
84#809—24no—
85#809—42no—
86#809—42no—
87#808—42no—
88#807—42no—
89C4⋊(C3⋊C8)—24no—
90C2×#127—31no—
91C2×#125C2×D311yescharacteristic
92C2×C4×D4—31no—
93C2×C4×D4—31no—
94C2×#107—32no—
95C2×#106—32no—
96C2×C4⋊C8—32no—
97C2×#102—32no—
98C2×#102—32no—
99C2×#102—32no—
100C2×#102—32no—
101#390—31no—
102#390—31no—
103#389—22no—
104#389C2×D312yes—
105#387—22no—
106#386—62no—
107#386—62no—
108#384C2×D311yescharacteristic
109#382—31no—
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—
118C23×D3—21no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D412yes—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D412yes—
125C22×C3×C4C2411yescharacteristic
126C22×Dic3—21no—
127C22×Dic3—21no—
128C2×D12—22no—
129C2×D12—22no—
130C2×D12—22no—
131C2×D12—22no—
132C2×D12—21no—
133C2×D12—21no—
134C2×D12—28no—
135C2×D12—82no—
136C2×D12—82no—
137C2×D12C2×Q812yes—
138C2×D12C2×Q812yes—
139C2×C4×D3—42no—
140C2×C4×D3—42no—
141C2×C4×D3—82no—
142C2×C4×D3—82no—
143C3×C4⋊C4—28no—
144C3×C4⋊C4—22no—
145C3×C4⋊C4—22no—
146C3×C4⋊C4—22no—
147C3×C4⋊C4—82no—
148C3×C4⋊C4—82no—
149C3×C4⋊C4#4114yes—
150C3×C42—22no—
151C3×C42—22no—
152C3×C42—82no—
153C3×C42C2×Q812yes—
154C3×C42#4112yes—
155(C2×D3)⋊C4—42no—
156(C2×D3)⋊C4—42no—
157(C2×D3)⋊C4—82no—
158(C2×D3)⋊C4—82no—
159C4⋊Dic3—22no—
160C4⋊Dic3—22no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—22no—
164C2×C3⋊C8—22no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—22no—
167C22×D4—31no—
168C22×D4—31no—
169C22×D4—62no—
170C23×C4—61no—
171C23×C4—61no—
172C22×C8—32no—
173C22×C8—32no—
174#127—34no—
175#127—62no—
176#125—24no—
177C4×D4—122no—
178C4×D4—122no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—31no—
183C2×C4⋊C4—31no—
184C2×C4⋊C4C22×D312yes—
185C2×C4⋊C4C22×D312yes—
186C2×C22⋊C4—61no—
187C2×C22⋊C4—61no—
188C2×C22⋊C4—62no—
189C2×C22⋊C4—62no—
190C2×C42—22no—
191C2×C42C22×D311yescharacteristic
192C2×C42C22×D311yescharacteristic
193C2×C42C22×D311yescharacteristic
194#107—122no—
195#106—122no—
196C4⋊C8—64no—
197#102—64no—
198#102—64no—
199#102—122no—
200#102—122no—
201#95—22no—
202#95—22no—
203#95—62no—
204#95#6712yes—
205#95#6712yes—
206C23×C3C22×D411yescharacteristic
207C22×D3—22no—
208C22×D3—22no—
209C22×D3—41no—
210C22×D3—41no—
211C22×D3—41no—
212C22×D3—41no—
213C22×D3—42no—
214C22×D3—42no—
215C22×D3—42no—
216C22×D3—42no—
217C2×C3×C4—22no—
218C2×C3×C4—22no—
219C2×C3×C4—22no—
220C2×C3×C4—24no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—22no—
226C2×C3×C4—22no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4C22⋊Q814yes—
233C2×C3×C4C22×Q811yescharacteristic
234C2×C3×C4C2×#4112yes—
235C2×C3×C4C2×#4111yescharacteristic
236C2×C3×C4H(5,𝔽2)12yes—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3—42no—
240C2×Dic3—42no—
241D12—24no—
242D12—24no—
243D12—42no—
244D12—42no—
245D12—88no—
246C4×D3—84no—
247C4×D3—84no—
248C3⋊C8—42no—
249C3⋊C8—42no—
250C24—61no—
251C24—61no—
252C2×D4—32no—
253C2×D4—32no—
254C2×D4—68no—
255C2×D4—62no—
256C2×D4—62no—
257C2×D4—61no—
258C2×D4—61no—
259C2×D4—62no—
260C2×D4—62no—
261C2×D4—242no—
262C2×D4—242no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—61no—
266C22×C4—61no—
267C22×C4—122no—
268C22×C4—122no—
269C22×C4—242no—
270C22×C4—242no—
271C22×C4C2×#6711yescharacteristic
272C22×C4C2×#6712yes—
273C22×C4C2×#6711yescharacteristic
274C22×C4C2×#6712yes—
275C22×C4C23×D311yescharacteristic
276C2×C8—62no—
277C2×C8—62no—
278C2×C8—62no—
279C2×C8—62no—
280C2×C8—62no—
281C2×C8—62no—
282C4⋊C4—28no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—62no—
287C4⋊C4—62no—
288C4⋊C4—82no—
289C4⋊C4—82no—
290C4⋊C4#23514yes—
291C22⋊C4—122no—
292C22⋊C4—122no—
293C22⋊C4—242no—
294C22⋊C4—242no—
295C42—22no—
296C42—22no—
297C42—82no—
298C42D3×Q812yes—
299C42#23912yes—
300C22×C3C2×C22⋊Q811yescharacteristic
301C22×C3#52511yescharacteristic
302C22×C3#53511yescharacteristic
303C22×C3#57612yes—
304C22×C3#57712yes—
305C2×D3—44no—
306C2×D3—44no—
307C2×D3—42no—
308C2×D3—42no—
309C2×D3—82no—
310C2×D3—82no—
311C2×D3—81no—
312C2×D3—81no—
313C2×D3—81no—
314C2×D3—81no—
315C3×C4—24no—
316C3×C4—42no—
317C3×C4—44no—
318C3×C4—41no—
319C3×C4—41no—
320C3×C4—82no—
321C3×C4#55412yes—
322C3×C4#55912yes—
323Dic3—81no—
324Dic3—81no—
325C23—62no—
326C23—62no—
327C23—121no—
328C23—121no—
329C23—121no—
330C23—121no—
331C23—122no—
332C23—122no—
333C23—122no—
334C23—122no—
335C23C22×#6711yescharacteristic
336D4—64no—
337D4—64no—
338D4—122no—
339D4—122no—
340D4—248no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—121no—
357C2×C4—121no—
358C2×C4—122no—
359C2×C4—122no—
360C2×C4—244no—
361C2×C4—244no—
362C2×C4#94614yes—
363C2×C4C2×#23512yes—
364C2×C4C2×D3×Q811yescharacteristic
365C2×C4C2×#23911yescharacteristic
366C2×C4#100912yes—
367C8—122no—
368C8—122no—
369C2×C3#258011yescharacteristic
370C2×C3#308112yes—
371C2×C3#311912yes—
372C2×C3#322011yescharacteristic
373C2×C3#322211yescharacteristic
374D3—82no—
375D3—82no—
376C22—124no—
377C22—124no—
378C22—122no—
379C22—122no—
380C22—242no—
381C22—242no—
382C22—241no—
383C22—241no—
384C22—241no—
385C22—241no—
386C22#579111yescharacteristic
387C22C2×#94611yescharacteristic
388C22#583011yescharacteristic
389C22#583512yes—
390C22#583712yes—
391C4—24no—
392C4—42no—
393C4—44no—
394C4—41no—
395C4—41no—
396C4—82no—
397C4—241no—
398C4—241no—
399C4#557912yes—
400C4#558512yes—
401C3#2926511yescharacteristic, complemented, Sylow
402C2—242no—
403C2—242no—
404C2#8250111yescharacteristic
405C2#8256812yes—
406C2#8343611yescharacteristic
407C2#8344811yescharacteristic
408C2#8347412yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2330C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenter#2330D3
hypocenterC3#29265
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. #71743C2semidirect
4. #71743C2semidirect
5. #69626C2semidirect
6. #69625C2semidirect
7. #69587C2semidirect
8. #69586C2semidirect
10. #69468C2semidirect
11. #69437C2semidirect
40. #2330D3semidirect
401. C3#29265semidirect

Maximal subgroups

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

subgroupindex
C3×#23302
#733892
#717432
#696262
#696252
#695872
#695862
#694872
#694682
#694372
#292653