Finite Group Databasebeta

← #10804162

Subgroups — #10804162

Order 768.

1426 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#946C211yesmaximal, characteristic
2D3×#407C212yesmaximal
3C3×#1784C211yesmaximal, characteristic, complemented
4#72133C211yesmaximal, characteristic, complemented
5#70115C218yesmaximal, complemented
6#69973C212yesmaximal, complemented
7#21753—31nomaximal, Sylow
8C23×C3×Q8C2211yescharacteristic
9C23×C4×D3C2211yescharacteristic
10C2×#946—22no—
11C2×#946—22no—
12C2×#946—22no—
13C2×#946—22no—
14C2×#946C414yes—
15C2×#946C2214yes—
16C22×(C2×D3)⋊C4C412yes—
17C22×C4⋊Dic3C2211yescharacteristic
18C22×Dic3⋊C4C412yes—
19C2×D3×M16C2214yes—
20C2×C3×#101C2214yescomplemented
21C3×#407C2212yes—
22C2×#836C2214yescomplemented
23#5207C2212yes—
24#5126C2214yes—
25#4726—22no—
26#4726—22no—
27D3×C22⋊C8—22no—
28D3×C22⋊C8—22no—
29#4554C2218yescomplemented
30C22×C22⋊Q8—31no—
31C2×#407—32no—
32#1901—38no—
33#1821—32no—
34#1784—31no—
35#1784D311yescharacteristic, complemented
36C24×D3C2×C411yescharacteristic
37C22×C3×Q8—22no—
38C22×C3×Q8—22no—
39C22×C3×Q8—44no—
40C22×C3×Q8C2×C412yes—
41C22×C3×Q8C2312yes—
42C23×C3×C4C2×C412yes—
43C23×C3×C4C2311yescharacteristic
44C23×Dic3C2×C412yes—
45C23×Dic3C2×C411yescharacteristic
46C22×C4×D3—44no—
47C22×C4×D3C2×C412yes—
48C22×C4×D3D412yes—
49C22×C4×D3D412yes—
50C22×C4×D3D412yes—
51C22×C4×D3D412yes—
52C22×C4×D3C2312yes—
53C2×C3×M16C2314yes—
54#946—24no—
55#946—24no—
56#946—24no—
57#946—24no—
58#946—24no—
59#946—24no—
60#946—24no—
61#946—24no—
62C2×(C2×D3)⋊C4—24no—
63C2×(C2×D3)⋊C4—22no—
64C2×(C2×D3)⋊C4—22no—
65C2×(C2×D3)⋊C4C2×C418yes—
66C2×C4⋊Dic3—22no—
67C2×C4⋊Dic3—22no—
68C2×C4⋊Dic3C2×C412yes—
69C2×C4⋊Dic3C2312yes—
70C2×Dic3⋊C4—24no—
71C2×Dic3⋊C4—22no—
72C2×Dic3⋊C4—22no—
73C2×Dic3⋊C4C2×C418yes—
74C2×#208C2314yes—
75D3×M16—48no—
76C2×ΓL(1,𝔽25)—24no—
77C2×D3×C8—24no—
78C3×#101—44no—
79C3×C22⋊C8—22no—
80C3×C22⋊C8—22no—
81#836—44no—
82#830—22no—
83#830—22no—
84(C2×D3)⋊C8—24no—
85(C2×D3)⋊C8—24no—
86C23×Q8C2×D311yescharacteristic
87C24×C4—31no—
88C22×M16—34no—
89C2×C22⋊Q8—34no—
90C2×C22⋊Q8—34no—
91C2×C22⋊Q8—62no—
92C2×C22⋊Q8—62no—
93C2×C22⋊Q8—62no—
94C2×C22⋊Q8—62no—
95C22×C4⋊C4—32no—
96C22×C4⋊C4—31no—
97C22×C22⋊C4—32no—
98C2×#101—34no—
99C2×#101C2×D314yescomplemented
100#407—32no—
101#407—34no—
102#407—62no—
103#407—62no—
104#407C2×D312yes—
105C2×C22⋊C8—62no—
106C2×C22⋊C8—62no—
107#343—38no—
108C24×C3C22×C411yescharacteristic
109C23×D3—24no—
110C23×D3—42no—
111C23×D3—42no—
112C23×D3C22⋊C412yes—
113C23×D3C22⋊C411yescharacteristic
114C23×D3C22⋊C411yescharacteristic
115C23×D3C22×C412yes—
116C2×C3×Q8—24no—
117C2×C3×Q8—24no—
118C2×C3×Q8—24no—
119C2×C3×Q8—24no—
120C2×C3×Q8—44no—
121C2×C3×Q8—48no—
122C2×C3×Q8—48no—
123C22×C3×C4—24no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—44no—
127C22×C3×C4C22×C414yes—
128C22×C3×C4C22×C411yescharacteristic
129C22×C3×C4C2×D412yes—
130C22×C3×C4C2×D412yes—
131C22×C3×C4C2411yescharacteristic
132C22×Dic3—22no—
133C22×Dic3—22no—
134C22×Dic3—42no—
135C22×Dic3—44no—
136C22×Dic3C22⋊C412yes—
137C22×Dic3C22⋊C411yescharacteristic
138C22×Dic3C22⋊C411yescharacteristic
139C22×Dic3C22×C412yes—
140C22×Dic3C22×C414yes—
141C2×C4×D3—28no—
142C2×C4×D3—44no—
143C2×C4×D3—44no—
144C2×C4×D3—44no—
145C2×C4×D3—44no—
146C2×C4×D3—44no—
147C2×C4×D3C22⋊C412yes—
148C2×C4×D3C22⋊C412yes—
149C2×C4×D3C22⋊C412yes—
150C2×C4×D3C22⋊C412yes—
151C2×C4×D3C2×D412yes—
152C2×C4×D3C2×D412yes—
153C2×C4×D3C2×D412yes—
154C2×C4×D3C2×D412yes—
155C3×M16—28no—
156C2×C3×C8—24no—
157(C2×D3)⋊C4—28no—
158(C2×D3)⋊C4—28no—
159C4⋊Dic3—24no—
160C4⋊Dic3—24no—
161Dic3⋊C4—28no—
162Dic3⋊C4—28no—
163#208—28no—
164C2×C3⋊C8—24no—
165ΓL(1,𝔽25)—48no—
166D3×C8—48no—
167C25—31no—
168C22×Q8—22no—
169C22×Q8—22no—
170C22×Q8—44no—
171C22×Q8C4×D312yes—
172C22×Q8C22×D312yes—
173C23×C4—31no—
174C23×C4—32no—
175C23×C4—32no—
176C23×C4—32no—
177C23×C4—32no—
178C23×C4—32no—
179C23×C4—32no—
180C23×C4—32no—
181C23×C4—124no—
182C23×C4C4×D312yes—
183C23×C4C22×D311yescharacteristic
184C2×M16—34no—
185C2×M16—64no—
186C2×M16—128no—
187C2×M16C22×D314yes—
188C22×C8—64no—
189C22⋊Q8—64no—
190C22⋊Q8—64no—
191C22⋊Q8—64no—
192C22⋊Q8—64no—
193C22⋊Q8—64no—
194C22⋊Q8—64no—
195C22⋊Q8—64no—
196C22⋊Q8—64no—
197C2×C4⋊C4—38no—
198C2×C4⋊C4—32no—
199C2×C4⋊C4—32no—
200C2×C4⋊C4—64no—
201C2×C4⋊C4—62no—
202C2×C4⋊C4—62no—
203C2×C4⋊C4—62no—
204C2×C4⋊C4—62no—
205C2×C22⋊C4—38no—
206C2×C22⋊C4—62no—
207C2×C22⋊C4—64no—
208C2×C22⋊C4—62no—
209#101—44no—
210#101—124no—
211C22⋊C8—22no—
212C22⋊C8—22no—
213C22⋊C8—64no—
214C22⋊C8—64no—
215C22⋊C8—62no—
216C22⋊C8—62no—
217C23×C3—22no—
218C23×C3—22no—
219C23×C3—22no—
220C23×C3C2×C22⋊C411yescharacteristic
221C23×C3C2×C22⋊C411yescharacteristic
222C23×C3C23×C411yescharacteristic
223C22×D3—24no—
224C22×D3—22no—
225C22×D3—24no—
226C22×D3—24no—
227C22×D3—24no—
228C22×D3—41no—
229C22×D3—42no—
230C22×D3—41no—
231C22×D3—41no—
232C22×D3—41no—
233C22×D3—42no—
234C22×D3—416no—
235C22×D3#10114yescomplemented
236C22×D3C2×C22⋊C412yes—
237C22×D3C2×C22⋊C412yes—
238C3×Q8—48no—
239C3×Q8—48no—
240C2×C3×C4—22no—
241C2×C3×C4—24no—
242C2×C3×C4—24no—
243C2×C3×C4—24no—
244C2×C3×C4—24no—
245C2×C3×C4—24no—
246C2×C3×C4—44no—
247C2×C3×C4—44no—
248C2×C3×C4C2×C22⋊C412yes—
249C2×C3×C4C22×D412yes—
250C2×C3×C4H(5,𝔽2)14yes—
251C2×Dic3—24no—
252C2×Dic3—24no—
253C2×Dic3—41no—
254C2×Dic3—42no—
255C2×Dic3—41no—
256C2×Dic3—44no—
257C2×Dic3—44no—
258C2×Dic3—44no—
259C2×Dic3C2×C22⋊C412yes—
260C2×Dic3C2×C22⋊C412yes—
261C4×D3—24no—
262C4×D3—24no—
263C4×D3—24no—
264C4×D3—24no—
265C4×D3—42no—
266C4×D3—42no—
267C4×D3—42no—
268C4×D3—42no—
269C3×C8—44no—
270C3⋊C8—44no—
271C24—31no—
272C24—32no—
273C24—32no—
274C24—31no—
275C24—64no—
276C24—122no—
277C24—122no—
278C24C2×C4×D311yescharacteristic
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—24no—
282C2×Q8—24no—
283C2×Q8—44no—
284C2×Q8—48no—
285C2×Q8—48no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—32no—
290C22×C4—32no—
291C22×C4—31no—
292C22×C4—31no—
293C22×C4—32no—
294C22×C4—32no—
295C22×C4—32no—
296C22×C4—32no—
297C22×C4—32no—
298C22×C4—32no—
299C22×C4—32no—
300C22×C4—32no—
301C22×C4—34no—
302C22×C4—44no—
303C22×C4—68no—
304C22×C4—62no—
305C22×C4—62no—
306C22×C4—124no—
307C22×C4—124no—
308C22×C4—124no—
309C22×C4—124no—
310C22×C4—124no—
311C22×C4—122no—
312C22×C4—124no—
313C22×C4C2×C4×D314yes—
314C22×C4C2×C4×D311yescharacteristic
315C22×C4D3×D412yes—
316C22×C4D3×D412yes—
317C22×C4C23×D311yescharacteristic
318M16—28no—
319M16—68no—
320M16—128no—
321C2×C8—24no—
322C2×C8—64no—
323C2×C8—128no—
324C4⋊C4—68no—
325C4⋊C4—68no—
326C4⋊C4—64no—
327C4⋊C4—64no—
328C22⋊C4—68no—
329C22⋊C4—68no—
330C22×C3—28no—
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3C2×#10112yes—
338C22×C3#41312yes—
339C22×C3C22×C22⋊C411yescharacteristic
340C22×C3#51812yes—
341C2×D3—22no—
342C2×D3—22no—
343C2×D3—22no—
344C2×D3—22no—
345C2×D3—22no—
346C2×D3—44no—
347C2×D3—48no—
348C2×D3—48no—
349C2×D3—44no—
350C2×D3C2×#10114yes—
351C3×C4—22no—
352C3×C4—44no—
353C3×C4#53414yes—
354Dic3—22no—
355Dic3—22no—
356Dic3—44no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—32no—
361C23—34no—
362C23—32no—
363C23—64no—
364C23—64no—
365C23—64no—
366C23—62no—
367C23—64no—
368C23—121no—
369C23—122no—
370C23—121no—
371C23—121no—
372C23—121no—
373C23—122no—
374C23—1216no—
375C23D3×C22⋊C411yescharacteristic
376C23D3×C22⋊C411yescharacteristic
377C23C22×C4×D311yescharacteristic
378Q8—48no—
379Q8—48no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—32no—
387C2×C4—32no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—64no—
391C2×C4—64no—
392C2×C4—64no—
393C2×C4—64no—
394C2×C4—64no—
395C2×C4—64no—
396C2×C4—122no—
397C2×C4—122no—
398C2×C4—122no—
399C2×C4—122no—
400C2×C4—121no—
401C2×C4—122no—
402C2×C4—121no—
403C2×C4—124no—
404C2×C4—124no—
405C2×C4—124no—
406C2×C4D3×C22⋊C412yes—
407C2×C4C2×D3×D412yes—
408C2×C4#100414yes—
409C8—44no—
410C8—124no—
411C2×C3—22no—
412C2×C3—22no—
413C2×C3—22no—
414C2×C3#230811yescharacteristic
415C2×C3C22×#10111yescharacteristic
416C2×C3C2×#41311yescharacteristic
417D3—24no—
418D3—42no—
419C22—28no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—34no—
427C22—62no—
428C22—62no—
429C22—62no—
430C22—62no—
431C22—62no—
432C22—124no—
433C22—128no—
434C22—128no—
435C22—124no—
436C22D3×#10112yes—
437C22#475112yes—
438C22C2×D3×C22⋊C411yescharacteristic
439C22#548612yes—
440C4—22no—
441C4—44no—
442C4—62no—
443C4—62no—
444C4—124no—
445C4#558914yes—
446C3#2175311yescharacteristic, complemented, Sylow
447C2—22no—
448C2—22no—
449C2—22no—
450C2—64no—
451C2—122no—
452C2#7384911yescharacteristic
453C2C2×D3×#10111yescharacteristic
454C2C2×#475111yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1784C2
socleC22×C3C22×C22⋊C4
prime radicalC24×D3C2×C4
cosocleC22×C3×C4C24
hypercenter#1784D3
hypocenterC3#21753
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1784

Complements

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

NH ≅ G/Ntype
3. C3×#1784C2semidirect
4. #72133C2semidirect
5. #70115C2semidirect
6. #69973C2semidirect
20. C2×C3×#101C22semidirect
22. C2×#836C22semidirect
29. #4554C22semidirect
35. #1784D3semidirect
99. C2×#101C2×D3semidirect
235. C22×D3#101semidirect
446. C3#21753semidirect

Maximal subgroups

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

subgroupindex
C22×#9462
D3×#4072
C3×#17842
#721332
#701152
#699732
#217533