Finite Group Databasebeta

← #10828559

Subgroups — #10828559

Order 768.

1130 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5535C211yesmaximal, characteristic
2#82232C211yesmaximal, characteristic, complemented
3C2×#4737C212yesmaximal, complemented
4C3×#1818C211yesmaximal, characteristic, complemented
5#69932C212yesmaximal, complemented
6#68671C212yesmaximal, complemented
7#68670C212yesmaximal, complemented
8#68634C212yesmaximal, complemented
9#68629C212yesmaximal, complemented
10#24070—31nomaximal, Sylow
11C2×C3×#117C2211yescharacteristic
12C2×#930C2212yes—
13C2×#926C2211yescharacteristic
14C22×C4⋊Dic3C2211yescharacteristic
15#5535—28no—
16C2×#894—22no—
17C2×#884—22no—
18C2×#874C2211yescharacteristic
19C2×C4×D12C2212yes—
20C2×#870C2211yescharacteristic
21C2×C3×C22⋊C8C2212yes—
22C2×#821C2212yes—
23C2×#821C2212yes—
24C2×#821C2212yes—
25C2×#821C2212yes—
26C2×#818C2212yes—
27C2×#817C2212yes—
28#4972—22no—
29#4942—22no—
30#4928C2212yes—
31#4928C2212yes—
32#4737—28no—
33C3×#336C2212yes—
34C3×#336C2211yescharacteristic
35C3×#336C2211yescharacteristic
36#4551C2212yes—
37#4551C2212yes—
38C2×#540—31no—
39C2×#482—32no—
40#2316—31no—
41#2059—32no—
42#2041—32no—
43#1874—32no—
44#1864—32no—
45#1862—32no—
46#1818D311yescharacteristic, complemented
47C23×C3×C4C2311yescharacteristic
48C22×#67—22no—
49C23×Dic3—21no—
50C22×D12—22no—
51C22×D12C2312yes—
52C22×C4×D3—22no—
53C22×C3×C8C2312yes—
54C22×C3×C8C2312yes—
55C3×#117—24no—
56C2×C3×C4⋊C4D412yes—
57C2×C3×C22⋊C4D412yes—
58C2×C3×C42C2311yescharacteristic
59C2×C3×C42C2311yescharacteristic
60C2×#217—21no—
61#930—24no—
62#930D418yes—
63C2×(C2×D3)⋊C4—22no—
64C2×(C2×D3)⋊C4—22no—
65#926—24no—
66C2×C4⋊Dic3—21no—
67C2×C4⋊Dic3—24no—
68C2×C4⋊Dic3C2312yes—
69C2×C4⋊Dic3C2312yes—
70C2×C4⋊Dic3C2312yes—
71C2×Dic3⋊C4—22no—
72C2×Dic3⋊C4—22no—
73C2×Dic3⋊C4—22no—
74C2×C4×Dic3—21no—
75#894—216no—
76#884—216no—
77#874—24no—
78C4×D12—24no—
79C4×D12—24no—
80#870—24no—
81C3×C22⋊C8—24no—
82C3⋊#95—22no—
83C3⋊#95—22no—
84#821—24no—
85#821—24no—
86#821—24no—
87#821—24no—
88#818—24no—
89#817—24no—
90#540—68no—
91C2×#127—31no—
92C2×#125—31no—
93C2×#123—62no—
94C2×#121—32no—
95C2×#121—62no—
96C2×C4×D4—32no—
97C2×#117—31no—
98C2×#117C2×D311yescharacteristic
99C22×C4⋊C4—31no—
100#482—68no—
101C2×#107—32no—
102C2×#106—32no—
103C2×#102—32no—
104C2×#102—32no—
105C2×#102—32no—
106C2×#102—32no—
107C2×C22⋊C8C2×D312yes—
108#389—32no—
109#389—32no—
110#389—62no—
111#380—62no—
112#340—32no—
113#340—32no—
114#336C2×D312yes—
115#336C2×D311yescharacteristic
116#336C2×D311yescharacteristic
117C24×C3C2×D411yescharacteristic
118C23×D3—22no—
119C22×C3×C4—22no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4C2×D412yes—
123C22×C3×C4C2×D412yes—
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D414yes—
126C22×C3×C4C2411yescharacteristic
127C2×#67—28no—
128C2×#67—82no—
129C2×#67—82no—
130C22×Dic3—22no—
131C22×Dic3—22no—
132C22×Dic3—24no—
133C22×Dic3—22no—
134C22×Dic3—82no—
135C2×D12—22no—
136C2×D12—22no—
137C2×D12—22no—
138C2×D12—22no—
139C2×D12—22no—
140C2×D12—28no—
141C2×D12—82no—
142C2×D12—82no—
143C2×D12C2×D414yes—
144C2×C4×D3—28no—
145C2×C4×D3—82no—
146C2×C4×D3—82no—
147C2×C3×C8—22no—
148C2×C3×C8—22no—
149C2×C3×C8—22no—
150C2×C3×C8—22no—
151C2×C3×C8—22no—
152C2×C3×C8—22no—
153C3×C4⋊C4—24no—
154C3×C22⋊C4—24no—
155C3×C42—22no—
156C3×C42—22no—
157#217—42no—
158(C2×D3)⋊C4—28no—
159(C2×D3)⋊C4—28no—
160C4⋊Dic3—24no—
161C4⋊Dic3—22no—
162C4⋊Dic3—22no—
163C4⋊Dic3—22no—
164C4⋊Dic3—22no—
165C4⋊Dic3C2×D414yes—
166C4⋊Dic3#4114yes—
167Dic3⋊C4—28no—
168Dic3⋊C4—82no—
169Dic3⋊C4—82no—
170C4×Dic3—42no—
171C22×D4—32no—
172C22×D4—62no—
173C22×D4—62no—
174C23×C4—61no—
175C23×C4—62no—
176C23×C4C22×D311yescharacteristic
177C22×C8C22×D312yes—
178C22×C8C22×D312yes—
179#127—64no—
180#125—64no—
181#123—616no—
182#121—38no—
183#121—616no—
184#121—64no—
185C4×D4—64no—
186C4×D4—64no—
187#117—24no—
188#117—64no—
189C2×C4⋊C4—32no—
190C2×C4⋊C4—32no—
191C2×C4⋊C4—32no—
192C2×C4⋊C4—64no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4—62no—
195C2×C4⋊C4—62no—
196C2×C4⋊C4—62no—
197C2×C4⋊C4D1212yes—
198C2×C22⋊C4—62no—
199C2×C22⋊C4—62no—
200C2×C22⋊C4—61no—
201C2×C22⋊C4D1212yes—
202C2×C42—61no—
203C2×C42C22×D311yescharacteristic
204C2×C42C22×D311yescharacteristic
205#107—64no—
206#106—64no—
207#102—64no—
208#102—64no—
209#102—64no—
210#102—64no—
211C22⋊C8—24no—
212#95—62no—
213#95—62no—
214C23×C3—21no—
215C23×C3—21no—
216C23×C3—22no—
217C23×C3—22no—
218C23×C3#12012yes—
219C23×C3C22×D411yescharacteristic
220C22×D3—24no—
221C22×D3—42no—
222C22×D3—42no—
223C22×D3—42no—
224C22×D3—42no—
225C22×D3—42no—
226C22×D3—42no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—24no—
230C2×C3×C4—24no—
231C2×C3×C4—22no—
232C2×C3×C4—22no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4#12012yes—
238C2×C3×C4C22×D412yes—
239C2×C3×C4C2×#4112yes—
240C2×C3×C42−1+411yescharacteristic
241C2×C3×C42−1+411yescharacteristic
242#67—88no—
243C2×Dic3—24no—
244C2×Dic3—24no—
245C2×Dic3—42no—
246C2×Dic3—42no—
247C2×Dic3—42no—
248C2×Dic3—42no—
249C2×Dic3—42no—
250C2×Dic3—42no—
251C2×Dic3—42no—
252C2×Dic3—84no—
253D12—24no—
254D12—24no—
255D12—42no—
256D12—42no—
257D12—88no—
258C4×D3—88no—
259C3×C8—42no—
260C3×C8—42no—
261C24—62no—
262C24C2×D1211yescharacteristic
263C2×D4—34no—
264C2×D4—68no—
265C2×D4—68no—
266C2×D4—62no—
267C2×D4—62no—
268C2×D4—62no—
269C2×D4—62no—
270C2×D4—62no—
271C2×D4—242no—
272C2×D4—242no—
273C2×D4—242no—
274C2×D4—242no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—68no—
279C22×C4—62no—
280C22×C4—62no—
281C22×C4—64no—
282C22×C4—62no—
283C22×C4—242no—
284C22×C4—242no—
285C22×C4—242no—
286C22×C4C2×D1212yes—
287C22×C4C2×D1212yes—
288C22×C4C2×D1211yescharacteristic
289C22×C4D3×D414yes—
290C22×C4C23×D311yescharacteristic
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C4⋊C4—24no—
298C4⋊C4—34no—
299C4⋊C4—34no—
300C4⋊C4—64no—
301C4⋊C4—68no—
302C4⋊C4—62no—
303C4⋊C4—62no—
304C4⋊C4—62no—
305C4⋊C4—62no—
306C4⋊C4—242no—
307C4⋊C4—242no—
308C22⋊C4—24no—
309C22⋊C4—68no—
310C22⋊C4—68no—
311C22⋊C4—122no—
312C42—22no—
313C42—22no—
314C42—122no—
315C22×C3—22no—
316C22×C3—24no—
317C22×C3—41no—
318C22×C3—41no—
319C22×C3—42no—
320C22×C3C2×#12011yescharacteristic
321C22×C3#52511yescharacteristic
322C22×C3#53611yescharacteristic
323C22×C3#57612yes—
324C22×C3#57712yes—
325C2×D3—44no—
326C2×D3—44no—
327C2×D3—44no—
328C2×D3—82no—
329C2×D3—82no—
330C2×D3—82no—
331C2×D3—82no—
332C3×C4—22no—
333C3×C4—42no—
334C3×C4—42no—
335C3×C4#54712yes—
336C3×C4#54712yes—
337Dic3—44no—
338Dic3—44no—
339Dic3—82no—
340C23—22no—
341C23—22no—
342C23—21no—
343C23—21no—
344C23—64no—
345C23—122no—
346C23—122no—
347C23—122no—
348C23—122no—
349C23—122no—
350C23—122no—
351C23#88212yes—
352C23C22×D1211yescharacteristic
353D4—64no—
354D4—64no—
355D4—122no—
356D4—122no—
357D4—248no—
358D4—248no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—64no—
370C2×C4—64no—
371C2×C4—122no—
372C2×C4—122no—
373C2×C4—122no—
374C2×C4—122no—
375C2×C4—122no—
376C2×C4—122no—
377C2×C4—122no—
378C2×C4—248no—
379C2×C4—244no—
380C2×C4#88212yes—
381C2×C4C2×D3×D412yes—
382C2×C4D3×#4112yes—
383C2×C4#101011yescharacteristic
384C2×C4#101011yescharacteristic
385C8—42no—
386C8—42no—
387C2×C3—42no—
388C2×C3#240611yescharacteristic
389C2×C3#301212yes—
390C2×C3#312211yescharacteristic
391C2×C3#312611yescharacteristic
392C2×C3#324112yes—
393D3—84no—
394C22—22no—
395C22—24no—
396C22—42no—
397C22—41no—
398C22—41no—
399C22—124no—
400C22—124no—
401C22—124no—
402C22—242no—
403C22—242no—
404C22—242no—
405C22—242no—
406C22C2×#88211yescharacteristic
407C22#552411yescharacteristic
408C22#552611yescharacteristic
409C22#575112yes—
410C22#575212yes—
411C4—22no—
412C4—42no—
413C4—42no—
414C4—124no—
415C4—124no—
416C4—242no—
417C4#565512yes—
418C4#565512yes—
419C3#2407011yescharacteristic, complemented, Sylow
420C2—42no—
421C2—244no—
422C2#7406311yescharacteristic
423C2#7867212yes—
424C2#7936212yes—
425C2#7938611yescharacteristic
426C2#7939511yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1818C2
socleC23×C3C22×D4
prime radicalC2×#5535C2
hypercenter#1818D3
hypocenterC3#24070
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1818

Complements

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

NH ≅ G/Ntype
2. #82232C2semidirect
3. C2×#4737C2semidirect
4. C3×#1818C2semidirect
5. #69932C2semidirect
6. #68671C2semidirect
7. #68670C2semidirect
8. #68634C2semidirect
9. #68629C2semidirect
46. #1818D3semidirect
419. C3#24070semidirect

Maximal subgroups

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

subgroupindex
C2×#55352
#822322
C2×#47372
C3×#18182
#699322
#686712
#686702
#686342
#686292
#240703