Finite Group Databasebeta

← #10876253

Subgroups — #10876253

Order 768.

842 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2314C211yesmaximal, characteristic, complemented
2#82270C211yesmaximal, characteristic
3#71584C211yesmaximal, characteristic, complemented
4#69590C211yesmaximal, characteristic, complemented
5#69586C211yesmaximal, characteristic, complemented
6#69560C211yesmaximal, characteristic, complemented
7#69557C211yesmaximal, characteristic, complemented
8#69522C212yesmaximal
9#69504C212yesmaximal, complemented
10#69465C212yesmaximal, complemented
11#69462C212yesmaximal, complemented
12#27595—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C22×C4⋊Dic3C2211yescharacteristic
16C3×#388—22no—
17C3×#387—22no—
18C3×#384C2211yescharacteristic
19C3×#382C2211yescharacteristic
20C3×#380C2212yes—
21C3×C4×C4⋊C4C2212yes—
22C2×#830C2212yes—
23#4987—22no—
24C2×#810C2212yes—
25C2×#810C2212yes—
26C2×#809C2212yes—
27C2×#809C2212yes—
28C2×#807C2212yes—
29C2×#807C2212yes—
30#4958—22no—
31#4939C2211yescharacteristic
32#4939C2211yescharacteristic
33#4929C2211yescharacteristic
34#4929C2211yescharacteristic
35#4529C2212yes—
36#4529C2212yes—
37#4524C2212yes—
38#4524C2211yescharacteristic
39#4524C2211yescharacteristic
40#2314D311yescharacteristic, complemented
41#2311—31no—
42#2095—31no—
43#2093—31no—
44#2074—31no—
45#2073—31no—
46#1928—32no—
47#1925—32no—
48#1852—32no—
49#1818—31no—
50#1788—32no—
51C23×C3×C4C2311yescharacteristic
52C22×#67—21no—
53C23×Dic3—21no—
54C2×#235—22no—
55C2×#235—24no—
56C2×#235—22no—
57C2×#235—42no—
58C22×D12C2311yescharacteristic
59C22×C4×D3—21no—
60C22×Dic6C2311yescharacteristic
61C3×#117—24no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4D412yes—
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C22⋊C4—22no—
67C2×C3×C22⋊C4D412yes—
68C2×C3×C42—22no—
69C2×C3×C42C2311yescharacteristic
70C2×C3×C42C2311yescharacteristic
71C2×(C2×D3)⋊C4—22no—
72C2×(C2×D3)⋊C4—22no—
73C2×C4⋊Dic3—21no—
74C2×C4⋊Dic3—42no—
75C2×C4⋊Dic3C2311yescharacteristic
76C2×C4⋊Dic3C2311yescharacteristic
77C22×C3⋊C8C2312yes—
78C22×C3⋊C8C2312yes—
79C3×#95—22no—
80C3×#95—22no—
81C3×#95—22no—
82C3⋊#95—22no—
83C3⋊#95—22no—
84#830—24no—
85#810—24no—
86#810—42no—
87#809—24no—
88#809—42no—
89#807—42no—
90#807—42no—
91C22×#41—31no—
92C2×#117C2×D311yescharacteristic
93C22×C4⋊C4—31no—
94C2×#107—32no—
95C2×#107—32no—
96C2×#103—32no—
97C2×#103—32no—
98C2×#102—32no—
99C2×#102—32no—
100C2×C22⋊C8—32no—
101#391—31no—
102#391—31no—
103#390—31no—
104#390—31no—
105#388—22no—
106#387—22no—
107#386—62no—
108#384C2×D311yescharacteristic
109#382C2×D311yescharacteristic
110#380—62no—
111#380C2×D312yes—
112C4×C4⋊C4C2×D312yes—
113#340—32no—
114#340—32no—
115#336—31no—
116#336—31no—
117#336—32no—
118C24×C3C2×D411yescharacteristic
119C23×D3—21no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4C2×D412yes—
126C22×C3×C4C2×D412yes—
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4#4114yes—
129C22×C3×C4C2411yescharacteristic
130C2×#67—24no—
131C2×#67—42no—
132C2×#67—42no—
133C22×Dic3—21no—
134C22×Dic3—21no—
135C22×Dic3—21no—
136C22×Dic3—42no—
137C22×Dic3—82no—
138#235—48no—
139#235—48no—
140C2×D12—22no—
141C2×D12—22no—
142C2×D12—21no—
143C2×D12#4112yes—
144C2×C4×D3—24no—
145C2×C4×D3—42no—
146C2×C4×D3—42no—
147C2×Dic6—22no—
148C2×Dic6—22no—
149C2×Dic6—21no—
150C2×Dic6#4112yes—
151C3×C4⋊C4—24no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4—82no—
157C3×C22⋊C4—24no—
158C3×C22⋊C4—82no—
159C3×C42—22no—
160C3×C42—22no—
161C3×C42—82no—
162(C2×D3)⋊C4—82no—
163(C2×D3)⋊C4—82no—
164C4⋊Dic3—22no—
165C4⋊Dic3—22no—
166C4⋊Dic3—42no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—22no—
169C2×C3⋊C8—22no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8—22no—
172C2×C3⋊C8—22no—
173C2×#41—62no—
174C2×#41—62no—
175C2×#41—64no—
176C2×#41—122no—
177C22×Q8—31no—
178C22×D4—31no—
179C22×D4—61no—
180C23×C4—61no—
181C23×C4—61no—
182C23×C4C22×D311yescharacteristic
183C22×C8—32no—
184C22×C8—32no—
185#117—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—31no—
188C2×C4⋊C4—31no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—122no—
191C2×C4⋊C4#6712yes—
192C2×C4⋊C4C22×D312yes—
193C2×C4⋊C4C22×D312yes—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—62no—
196C2×C22⋊C4—62no—
197C2×C22⋊C4#6712yes—
198C2×C42—22no—
199C2×C42C22×D311yescharacteristic
200C2×C42C22×D311yescharacteristic
201#107—122no—
202#107—122no—
203#103—64no—
204#103—122no—
205#102—64no—
206#102—122no—
207C22⋊C8—64no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—62no—
212#95—62no—
213C23×C3—21no—
214C23×C3—21no—
215C23×C3—22no—
216C23×C3—22no—
217C23×C3#12312yes—
218C23×C3C22×D411yescharacteristic
219C22×D3—22no—
220C22×D3—41no—
221C22×D3—41no—
222C22×D3—42no—
223C22×D3—42no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—24no—
227C2×C3×C4—24no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—22no—
233C2×C3×C4—22no—
234C2×C3×C4—42no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4#12312yes—
241C2×C3×C4C2×#4112yes—
242C2×C3×C4C2×#4111yescharacteristic
243C2×C3×C4C2×#4111yescharacteristic
244C2×C3×C4H(5,𝔽2)12yes—
245#67—48no—
246C2×Dic3—22no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251C2×Dic3—42no—
252C2×Dic3—42no—
253C2×Dic3—84no—
254D12—24no—
255D12—42no—
256C4×D3—48no—
257Dic6—24no—
258Dic6—42no—
259C3⋊C8—42no—
260C3⋊C8—42no—
261C24—61no—
262C24C2×#6711yescharacteristic
263#41—128no—
264#41—128no—
265C2×Q8—32no—
266C2×Q8—61no—
267C2×Q8—62no—
268C2×Q8—62no—
269C2×D4—32no—
270C2×D4—64no—
271C2×D4—62no—
272C2×D4—61no—
273C2×D4—62no—
274C2×D4—122no—
275C2×D4—122no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—64no—
282C22×C4—61no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4—122no—
286C22×C4—122no—
287C22×C4—122no—
288C22×C4—242no—
289C22×C4#23514yes—
290C22×C4C2×#6712yes—
291C22×C4C2×#6712yes—
292C22×C4C2×#6711yescharacteristic
293C22×C4C23×D311yescharacteristic
294C2×C8—62no—
295C2×C8—62no—
296C2×C8—62no—
297C2×C8—62no—
298C2×C8—62no—
299C2×C8—62no—
300C4⋊C4—24no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—62no—
306C4⋊C4—62no—
307C4⋊C4—82no—
308C4⋊C4—122no—
309C22⋊C4—24no—
310C22⋊C4—82no—
311C22⋊C4—242no—
312C22⋊C4—242no—
313C42—22no—
314C42—22no—
315C42—82no—
316C22×C3—24no—
317C22×C3—22no—
318C22×C3—42no—
319C22×C3—41no—
320C22×C3—41no—
321C22×C3C2×#12311yescharacteristic
322C22×C3#52511yescharacteristic
323C22×C3#53511yescharacteristic
324C22×C3#57612yes—
325C22×C3#57812yes—
326C2×D3—44no—
327C2×D3—42no—
328C2×D3—82no—
329C2×D3—81no—
330C2×D3—81no—
331C3×C4—22no—
332C3×C4—42no—
333C3×C4—42no—
334C3×C4—82no—
335C3×C4—82no—
336C3×C4#55112yes—
337C3×C4#55112yes—
338Dic3—42no—
339Dic3—81no—
340Dic3—81no—
341C23—21no—
342C23—21no—
343C23—22no—
344C23—22no—
345C23—62no—
346C23—121no—
347C23—121no—
348C23—122no—
349C23—122no—
350C23#92912yes—
351C23C22×#6711yescharacteristic
352Q8—64no—
353Q8—122no—
354D4—64no—
355D4—128no—
356D4—122no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—62no—
374C2×C4—128no—
375C2×C4—121no—
376C2×C4—121no—
377C2×C4—121no—
378C2×C4—121no—
379C2×C4—122no—
380C2×C4—122no—
381C2×C4—244no—
382C2×C4#92912yes—
383C2×C4C2×#23512yes—
384C2×C4D3×#4111yescharacteristic
385C2×C4D3×#4111yescharacteristic
386C2×C4#100912yes—
387C8—122no—
388C8—122no—
389C2×C3—42no—
390C2×C3#248211yescharacteristic
391C2×C3#309812yes—
392C2×C3#312412yes—
393C2×C3#322011yescharacteristic
394C2×C3#322311yescharacteristic
395D3—82no—
396C22—22no—
397C22—24no—
398C22—42no—
399C22—41no—
400C22—41no—
401C22—124no—
402C22—122no—
403C22—242no—
404C22—241no—
405C22—241no—
406C22C2×#92911yescharacteristic
407C22#582811yescharacteristic
408C22#583011yescharacteristic
409C22#583512yes—
410C22#583812yes—
411C4—22no—
412C4—42no—
413C4—42no—
414C4—82no—
415C4—82no—
416C4—122no—
417C4—241no—
418C4—241no—
419C4#552712yes—
420C4#552712yes—
421C3#2759511yescharacteristic, complemented, Sylow
422C2—42no—
423C2—242no—
424C2#8228211yescharacteristic
425C2#8256712yes—
426C2#8343611yescharacteristic
427C2#8345211yescharacteristic
428C2#8346112yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2314C2
socleC23×C3C22×D4
prime radicalC22×#235C22
hypercenter#2314D3
hypocenterC3#27595
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2314

Complements

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

NH ≅ G/Ntype
1. C3×#2314C2semidirect
3. #71584C2semidirect
4. #69590C2semidirect
5. #69586C2semidirect
6. #69560C2semidirect
7. #69557C2semidirect
9. #69504C2semidirect
10. #69465C2semidirect
11. #69462C2semidirect
40. #2314D3semidirect
421. C3#27595semidirect

Maximal subgroups

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

subgroupindex
C3×#23142
#822702
#715842
#695902
#695862
#695602
#695572
#695222
#695042
#694652
#694622
#275953