Finite Group Databasebeta

← #10804163

Subgroups — #10804163

Order 768.

1426 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#897C211yesmaximal, characteristic
2D3×#407C212yesmaximal
3C3×#1821C211yesmaximal, characteristic, complemented
4#71562C211yesmaximal, characteristic, complemented
5#69990C212yesmaximal, complemented
6#68993C218yesmaximal, complemented
7#21753—31nomaximal, Sylow
8C23×C4×D3C2211yescharacteristic
9C23×Dic6C2211yescharacteristic
10C22×C3×C4⋊C4C2211yescharacteristic
11C22×(C2×D3)⋊C4C412yes—
12C22×C4⋊Dic3C412yes—
13C2×D3×M16C2214yes—
14C2×#897—22no—
15C2×#897—22no—
16C2×#897—22no—
17C2×#897—22no—
18C2×#897C414yes—
19C2×#897C2214yes—
20C3×#407C2212yes—
21#5207C2212yes—
22C2×#824C2218yescomplemented
23#5126C2214yes—
24#4726—22no—
25#4726—22no—
26D3×C22⋊C8—22no—
27D3×C22⋊C8—22no—
28C3×#343C2214yescomplemented
29#4530C2214yescomplemented
30C22×C22⋊Q8—31no—
31C2×#407—32no—
32#1901—38no—
33#1821—31no—
34#1821D311yescharacteristic, complemented
35#1784—32no—
36C24×D3C2×C411yescharacteristic
37C23×C3×C4C2×C412yes—
38C23×C3×C4C2311yescharacteristic
39C23×Dic3C2×C412yes—
40C23×Dic3C2×C411yescharacteristic
41C22×C4×D3—44no—
42C22×C4×D3C2×C412yes—
43C22×C4×D3D412yes—
44C22×C4×D3D412yes—
45C22×C4×D3D412yes—
46C22×C4×D3D412yes—
47C22×C4×D3C2312yes—
48C22×Dic6—22no—
49C22×Dic6—22no—
50C22×Dic6—44no—
51C22×Dic6C2×C412yes—
52C22×Dic6C2312yes—
53C2×C3×M16C2314yes—
54C2×C3×C4⋊C4—22no—
55C2×C3×C4⋊C4—22no—
56C2×C3×C4⋊C4C2×C412yes—
57C2×C3×C4⋊C4C2312yes—
58C2×(C2×D3)⋊C4—24no—
59C2×(C2×D3)⋊C4—22no—
60C2×(C2×D3)⋊C4—22no—
61C2×(C2×D3)⋊C4C2×C418yes—
62C2×C4⋊Dic3—24no—
63C2×C4⋊Dic3—22no—
64C2×C4⋊Dic3—22no—
65C2×C4⋊Dic3C2×C418yes—
66C2×#208C2314yes—
67D3×M16—48no—
68C2×ΓL(1,𝔽25)—24no—
69C2×D3×C8—24no—
70#897—24no—
71#897—24no—
72#897—24no—
73#897—24no—
74#897—24no—
75#897—24no—
76#897—24no—
77#897—24no—
78C3×C22⋊C8—22no—
79C3×C22⋊C8—22no—
80#830—22no—
81#830—22no—
82#824—48no—
83(C2×D3)⋊C8—24no—
84(C2×D3)⋊C8—24no—
85C23×Q8—31no—
86C24×C4—31no—
87C22×M16—34no—
88C2×C22⋊Q8—34no—
89C2×C22⋊Q8—34no—
90C2×C22⋊Q8—62no—
91C2×C22⋊Q8—62no—
92C2×C22⋊Q8—62no—
93C2×C22⋊Q8—62no—
94C22×C4⋊C4—32no—
95C22×C4⋊C4C2×D311yescharacteristic
96C22×C22⋊C4—32no—
97C2×#101—38no—
98#407—32no—
99#407—34no—
100#407—62no—
101#407—62no—
102#407C2×D312yes—
103C2×C22⋊C8—62no—
104C2×C22⋊C8—62no—
105#343—34no—
106#343C2×D314yescomplemented
107C24×C3C22×C411yescharacteristic
108C23×D3—24no—
109C23×D3—42no—
110C23×D3—42no—
111C23×D3C22⋊C412yes—
112C23×D3C22⋊C411yescharacteristic
113C23×D3C22⋊C411yescharacteristic
114C23×D3C22×C412yes—
115C22×C3×C4—24no—
116C22×C3×C4—22no—
117C22×C3×C4—22no—
118C22×C3×C4—44no—
119C22×C3×C4C22×C411yescharacteristic
120C22×C3×C4C22×C414yes—
121C22×C3×C4C2×D412yes—
122C22×C3×C4C2×D412yes—
123C22×C3×C4C2411yescharacteristic
124C22×Dic3—22no—
125C22×Dic3—22no—
126C22×Dic3—44no—
127C22×Dic3—42no—
128C22×Dic3C22⋊C412yes—
129C22×Dic3C22⋊C411yescharacteristic
130C22×Dic3C22⋊C411yescharacteristic
131C22×Dic3C22×C414yes—
132C22×Dic3C22×C412yes—
133C2×C4×D3—28no—
134C2×C4×D3—44no—
135C2×C4×D3—44no—
136C2×C4×D3—44no—
137C2×C4×D3—44no—
138C2×C4×D3—44no—
139C2×C4×D3C22⋊C412yes—
140C2×C4×D3C22⋊C412yes—
141C2×C4×D3C22⋊C412yes—
142C2×C4×D3C22⋊C412yes—
143C2×C4×D3C2×D412yes—
144C2×C4×D3C2×D412yes—
145C2×C4×D3C2×D412yes—
146C2×C4×D3C2×D412yes—
147C2×Dic6—24no—
148C2×Dic6—24no—
149C2×Dic6—28no—
150C2×Dic6—44no—
151C2×Dic6—48no—
152C2×Dic6—48no—
153C3×M16—28no—
154C2×C3×C8—24no—
155C3×C4⋊C4—24no—
156C3×C4⋊C4—24no—
157(C2×D3)⋊C4—28no—
158(C2×D3)⋊C4—28no—
159C4⋊Dic3—28no—
160C4⋊Dic3—28no—
161#208—28no—
162C2×C3⋊C8—24no—
163ΓL(1,𝔽25)—48no—
164D3×C8—48no—
165C25—31no—
166C22×Q8—32no—
167C22×Q8—32no—
168C22×Q8—62no—
169C22×Q8—62no—
170C22×Q8—124no—
171C23×C4—31no—
172C23×C4—32no—
173C23×C4—32no—
174C23×C4—32no—
175C23×C4—32no—
176C23×C4—32no—
177C23×C4—32no—
178C23×C4—32no—
179C23×C4—124no—
180C23×C4C4×D312yes—
181C23×C4C22×D311yescharacteristic
182C2×M16—34no—
183C2×M16—64no—
184C2×M16—128no—
185C2×M16C22×D314yes—
186C22×C8—64no—
187C22⋊Q8—64no—
188C22⋊Q8—64no—
189C22⋊Q8—64no—
190C22⋊Q8—64no—
191C22⋊Q8—64no—
192C22⋊Q8—64no—
193C22⋊Q8—64no—
194C22⋊Q8—64no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—38no—
198C2×C4⋊C4—64no—
199C2×C4⋊C4—62no—
200C2×C4⋊C4—62no—
201C2×C4⋊C4C4×D312yes—
202C2×C4⋊C4C22×D312yes—
203C2×C22⋊C4—38no—
204C2×C22⋊C4—62no—
205C2×C22⋊C4—64no—
206C2×C22⋊C4—62no—
207#101—128no—
208C22⋊C8—22no—
209C22⋊C8—22no—
210C22⋊C8—64no—
211C22⋊C8—64no—
212C22⋊C8—62no—
213C22⋊C8—62no—
214C23×C3—22no—
215C23×C3—22no—
216C23×C3—22no—
217C23×C3C2×C22⋊C411yescharacteristic
218C23×C3C2×C22⋊C411yescharacteristic
219C23×C3C23×C411yescharacteristic
220C22×D3—24no—
221C22×D3—22no—
222C22×D3—24no—
223C22×D3—24no—
224C22×D3—24no—
225C22×D3—41no—
226C22×D3—42no—
227C22×D3—41no—
228C22×D3—41no—
229C22×D3—41no—
230C22×D3—42no—
231C22×D3—416no—
232C22×D3#10114yescomplemented
233C22×D3C2×C22⋊C412yes—
234C22×D3C2×C22⋊C412yes—
235C2×C3×C4—22no—
236C2×C3×C4—24no—
237C2×C3×C4—24no—
238C2×C3×C4—24no—
239C2×C3×C4—44no—
240C2×C3×C4—44no—
241C2×C3×C4—44no—
242C2×C3×C4C2×C22⋊C412yes—
243C2×C3×C4C22×D412yes—
244C2×C3×C4H(5,𝔽2)14yes—
245C2×Dic3—24no—
246C2×Dic3—24no—
247C2×Dic3—28no—
248C2×Dic3—41no—
249C2×Dic3—42no—
250C2×Dic3—41no—
251C2×Dic3—44no—
252C2×Dic3—44no—
253C2×Dic3C2×C22⋊C412yes—
254C2×Dic3C2×C22⋊C412yes—
255C4×D3—24no—
256C4×D3—24no—
257C4×D3—24no—
258C4×D3—24no—
259C4×D3—42no—
260C4×D3—42no—
261C4×D3—42no—
262C4×D3—42no—
263Dic6—48no—
264Dic6—48no—
265C3×C8—44no—
266C3⋊C8—44no—
267C24—31no—
268C24—32no—
269C24—32no—
270C24—31no—
271C24—64no—
272C24—122no—
273C24—122no—
274C24C2×C4×D311yescharacteristic
275C2×Q8—64no—
276C2×Q8—64no—
277C2×Q8—68no—
278C2×Q8—124no—
279C2×Q8—128no—
280C2×Q8—128no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—32no—
285C22×C4—31no—
286C22×C4—31no—
287C22×C4—32no—
288C22×C4—32no—
289C22×C4—32no—
290C22×C4—32no—
291C22×C4—32no—
292C22×C4—32no—
293C22×C4—32no—
294C22×C4—32no—
295C22×C4—32no—
296C22×C4—34no—
297C22×C4—44no—
298C22×C4—68no—
299C22×C4—62no—
300C22×C4—62no—
301C22×C4—124no—
302C22×C4—124no—
303C22×C4—124no—
304C22×C4—124no—
305C22×C4—124no—
306C22×C4—122no—
307C22×C4—124no—
308C22×C4C2×C4×D311yescharacteristic
309C22×C4C2×C4×D314yes—
310C22×C4D3×D412yes—
311C22×C4D3×D412yes—
312C22×C4C23×D311yescharacteristic
313M16—28no—
314M16—68no—
315M16—128no—
316C2×C8—24no—
317C2×C8—64no—
318C2×C8—128no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—68no—
322C4⋊C4—68no—
323C22⋊C4—68no—
324C22⋊C4—68no—
325C22×C3—28no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3C2×#10112yes—
333C22×C3#41312yes—
334C22×C3C22×C22⋊C411yescharacteristic
335C22×C3#51812yes—
336C2×D3—22no—
337C2×D3—22no—
338C2×D3—22no—
339C2×D3—22no—
340C2×D3—22no—
341C2×D3—44no—
342C2×D3—48no—
343C2×D3—48no—
344C2×D3—44no—
345C2×D3C2×#10114yes—
346C3×C4—22no—
347C3×C4—44no—
348C3×C4#53414yes—
349Dic3—22no—
350Dic3—22no—
351Dic3—44no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—32no—
356C23—34no—
357C23—32no—
358C23—64no—
359C23—64no—
360C23—64no—
361C23—62no—
362C23—64no—
363C23—121no—
364C23—122no—
365C23—121no—
366C23—121no—
367C23—121no—
368C23—122no—
369C23—1216no—
370C23D3×C22⋊C411yescharacteristic
371C23D3×C22⋊C411yescharacteristic
372C23C22×C4×D311yescharacteristic
373Q8—128no—
374Q8—128no—
375C2×C4—22no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—32no—
380C2×C4—32no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—64no—
385C2×C4—64no—
386C2×C4—64no—
387C2×C4—64no—
388C2×C4—64no—
389C2×C4—64no—
390C2×C4—68no—
391C2×C4—122no—
392C2×C4—122no—
393C2×C4—122no—
394C2×C4—122no—
395C2×C4—121no—
396C2×C4—122no—
397C2×C4—121no—
398C2×C4—124no—
399C2×C4—124no—
400C2×C4D3×C22⋊C412yes—
401C2×C4C2×D3×D412yes—
402C2×C4#100914yes—
403C8—44no—
404C8—124no—
405C2×C3—22no—
406C2×C3—22no—
407C2×C3—22no—
408C2×C3#230811yescharacteristic
409C2×C3C22×#10111yescharacteristic
410C2×C3C2×#41311yescharacteristic
411D3—24no—
412D3—42no—
413C22—28no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—34no—
421C22—62no—
422C22—62no—
423C22—62no—
424C22—62no—
425C22—62no—
426C22—124no—
427C22—128no—
428C22—128no—
429C22—124no—
430C22#474512yes—
431C22D3×#10112yes—
432C22C2×D3×C22⋊C411yescharacteristic
433C22#552112yes—
434C4—22no—
435C4—44no—
436C4—62no—
437C4—62no—
438C4—124no—
439C4#565314yes—
440C3#2175311yescharacteristic, complemented, Sylow
441C2—22no—
442C2—22no—
443C2—22no—
444C2—64no—
445C2—122no—
446C2#7411911yescharacteristic
447C2C2×#474511yescharacteristic
448C2C2×D3×#10111yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1821C2
socleC22×C3C22×C22⋊C4
prime radicalC24×D3C2×C4
cosocleC22×C3×C4C24
hypercenter#1821D3
hypocenterC3#21753
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
3. C3×#1821C2semidirect
4. #71562C2semidirect
5. #69990C2semidirect
6. #68993C2semidirect
22. C2×#824C22semidirect
28. C3×#343C22semidirect
29. #4530C22semidirect
34. #1821D3semidirect
106. #343C2×D3semidirect
232. C22×D3#101semidirect
440. C3#21753semidirect

Maximal subgroups

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

subgroupindex
C22×#8972
D3×#4072
C3×#18212
#715622
#699902
#689932
#217533