Finite Group Databasebeta

← #10794590

Subgroups — #10794590

Order 768.

1690 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5798C211yesmaximal, characteristic
2C22×C3×#99C211yesmaximal, characteristic
3C22×#834C211yesmaximal, characteristic
4#82095C212yesmaximal, complemented
5C2×#4527C212yesmaximal, complemented
6#70028C218yesmaximal, complemented
7#20633—31nomaximal, Sylow
8C23×C3×D4C2211yescharacteristic
9C22×C3×C22⋊C4C2212yes—
10C22×#217C412yescomplemented
11C22×#217C2212yes—
12C22×#217C2211yescharacteristic
13C22×#217C2211yescharacteristic
14#5798—24no—
15#5798—24no—
16C2×D4×Dic3C414yescomplemented
17C2×D4×Dic3C2214yes—
18C2×#926C412yescomplemented
19C2×C3×#99—24no—
20C2×C3×#99C2214yes—
21C2×#834—24no—
22C2×#834C2214yes—
23#5210—24no—
24#4946—24no—
25#4942C2218yescomplemented
26#4527C2218yescomplemented
27C2×#518—31no—
28C22×#99—31no—
29C22×#99D311yescharacteristic
30#2315—32no—
31#1799—38no—
32C2×#342—32no—
33C25×C3C2×C411yescharacteristic
34C25×C3C2311yescharacteristic
35C22×C3×D4—22no—
36C22×C3×D4—22no—
37C22×C3×D4—24no—
38C22×C3×D4C2×C412yes—
39C22×C3×D4D414yes—
40C22×C3×D4Q814yes—
41C22×C3×D4C2312yes—
42C23×C3×C4—22no—
43C23×C3×C4C2×C411yescharacteristic
44C23×Dic3—22no—
45C23×Dic3C2×C412yes—
46C23×Dic3C2×C414yescomplemented
47C23×Dic3C2312yes—
48C2×C3×C22⋊C4—24no—
49C2×C3×C22⋊C4—24no—
50C2×C3×C22⋊C4—22no—
51C2×C3×C22⋊C4C2314yes—
52C2×#217—22no—
53C2×#217—22no—
54C2×#217—24no—
55C2×#217—24no—
56C2×#217—24no—
57C2×#217—24no—
58C2×#217—22no—
59C2×#217C2×C412yes—
60C2×#217C2×C412yes—
61C2×#217C2×C412yes—
62C2×#217C2×C414yescomplemented
63C2×#217D414yes—
64C2×#217D414yes—
65C2×#217C2314yes—
66C2×#217C2312yes—
67D4×Dic3—216no—
68D4×Dic3—216no—
69#926—24no—
70#926—24no—
71C2×C4⋊Dic3C2×C414yescomplemented
72C2×Dic3⋊C4—28no—
73C2×C4×Dic3C2×C414yescomplemented
74C3×#99—28no—
75#834—28no—
76C3⋊#95—28no—
77C23×D4C2×D311yescharacteristic
78#518—64no—
79#518—64no—
80C2×C4×D4—34no—
81C2×C4×D4—34no—
82C2×#117—32no—
83C22×C22⋊C4—31no—
84C22×C22⋊C4—32no—
85C22×C22⋊C4—31no—
86C22×C22⋊C4—32no—
87C22×C22⋊C4C2×D312yes—
88C2×#99—24no—
89C2×#99—34no—
90C2×#99—64no—
91C2×#99C2×D314yes—
92#385—64no—
93#385—64no—
94#380—38no—
95#342—38no—
96C24×C3—21no—
97C24×C3—21no—
98C24×C3—22no—
99C24×C3—82no—
100C24×C3—82no—
101C24×C3C4⋊C414yescomplemented
102C24×C3C22×C412yes—
103C24×C3C22×C411yescharacteristic
104C24×C3C22×C411yescharacteristic
105C24×C3C2×D412yes—
106C24×C3C2×D412yes—
107C24×C3C2×D412yes—
108C24×C3C2×Q812yes—
109C24×C3#4114yes—
110C24×C3C2411yescharacteristic
111C2×C3×D4—24no—
112C2×C3×D4—24no—
113C2×C3×D4—28no—
114C2×C3×D4—28no—
115C2×C3×D4—28no—
116C2×C3×D4—28no—
117C2×C3×D4—24no—
118C2×C3×D4—24no—
119C2×C3×D4C4⋊C418yescomplemented
120C2×C3×D4C2×D414yes—
121C2×C3×D4C2×Q814yes—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—22no—
125C22×C3×C4—24no—
126C22×C3×C4—44no—
127C22×C3×C4—84no—
128C22×C3×C4C4⋊C414yescomplemented
129C22×C3×C4C22×C412yes—
130C22×Dic3—24no—
131C22×Dic3—28no—
132C22×Dic3—24no—
133C22×Dic3—44no—
134C22×Dic3—82no—
135C22×Dic3—84no—
136C22×Dic3—82no—
137C22×Dic3—84no—
138C22×Dic3C22×C414yes—
139C22×Dic3C22×C414yes—
140C22×Dic3C2×D418yes—
141C3×C22⋊C4—24no—
142C3×C22⋊C4—24no—
143C3×C22⋊C4—44no—
144#217—22no—
145#217—22no—
146#217—24no—
147#217—28no—
148#217—24no—
149#217—24no—
150#217—44no—
151#217C22⋊C414yes—
152#217C22⋊C414yes—
153#217C22⋊C418yes—
154C4⋊Dic3—28no—
155Dic3⋊C4—88no—
156C4×Dic3—28no—
157C25C4×D311yescharacteristic
158C25C22×D311yescharacteristic
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—24no—
162C22×D4Dic614yes—
163C22×D4C4×D312yes—
164C22×D4#6714yes—
165C22×D4C22×D312yes—
166C23×C4—22no—
167C23×C4—32no—
168C23×C4—32no—
169C23×C4—34no—
170C23×C4—62no—
171C23×C4C4×D311yescharacteristic
172C4×D4—616no—
173C4×D4—616no—
174#117—64no—
175#117—64no—
176C2×C4⋊C4—34no—
177C2×C4⋊C4—68no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—32no—
182C2×C22⋊C4—32no—
183C2×C22⋊C4—34no—
184C2×C22⋊C4—34no—
185C2×C22⋊C4—32no—
186C2×C22⋊C4—34no—
187C2×C22⋊C4—32no—
188C2×C22⋊C4—34no—
189C2×C22⋊C4—64no—
190C2×C22⋊C4—64no—
191C2×C22⋊C4—62no—
192C2×C22⋊C4—62no—
193C2×C22⋊C4—64no—
194C2×C22⋊C4—64no—
195C2×C22⋊C4—62no—
196C2×C22⋊C4C22×D314yes—
197C2×C42—34no—
198#99—28no—
199#99—68no—
200#95—68no—
201C23×C3—22no—
202C23×C3—22no—
203C23×C3—24no—
204C23×C3—24no—
205C23×C3—42no—
206C23×C3—44no—
207C23×C3—42no—
208C23×C3—44no—
209C23×C3—42no—
210C23×C3—44no—
211C23×C3—82no—
212C23×C3—82no—
213C23×C3—82no—
214C23×C3—82no—
215C23×C3—84no—
216C23×C3—84no—
217C23×C3—84no—
218C23×C3C2×C22⋊C411yescharacteristic
219C23×C3C2×C22⋊C411yescharacteristic
220C23×C3C2×C22⋊C412yes—
221C23×C3C2×C4⋊C414yes—
222C23×C3C2×C4⋊C412yes—
223C23×C3C2×C4⋊C412yes—
224C23×C3#11712yes—
225C23×C3#11712yes—
226C23×C3#12114yes—
227C23×C3#12114yes—
228C23×C3C22⋊Q814yes—
229C23×C3C22⋊Q814yes—
230C23×C3C23×C411yescharacteristic
231C23×C3C22×D412yes—
232C23×C3C22×D411yescharacteristic
233C23×C3C22×Q811yescharacteristic
234C23×C3C2×#4112yes—
235C3×D4—28no—
236C3×D4—28no—
237C3×D4—216no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—24no—
241C2×C3×C4—24no—
242C2×C3×C4—48no—
243C2×C3×C4—44no—
244C2×C3×C4—88no—
245C2×C3×C4C2×C4⋊C414yes—
246C2×Dic3—24no—
247C2×Dic3—48no—
248C2×Dic3—44no—
249C2×Dic3—44no—
250C2×Dic3—48no—
251C2×Dic3—44no—
252C2×Dic3—88no—
253C2×Dic3—84no—
254C2×Dic3—88no—
255C2×Dic3C2×C22⋊C418yes—
256C24—22no—
257C24—21no—
258C24—21no—
259C24—82no—
260C24—82no—
261C24Dic3⋊C414yescomplemented
262C24C2×Dic612yes—
263C24C2×C4×D312yes—
264C24C2×C4×D311yescharacteristic
265C24C2×C4×D311yescharacteristic
266C24D3×D412yes—
267C24D3×D412yes—
268C24#23714yes—
269C24C2×#6712yes—
270C24C23×D311yescharacteristic
271C2×D4—24no—
272C2×D4—28no—
273C2×D4—28no—
274C2×D4—24no—
275C2×D4—28no—
276C2×D4—28no—
277C2×D4—24no—
278C2×D4—24no—
279C2×D4Dic3⋊C418yescomplemented
280C2×D4C2×Dic614yes—
281C2×D4C2×#6714yes—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—24no—
286C22×C4—34no—
287C22×C4—38no—
288C22×C4—34no—
289C22×C4—44no—
290C22×C4—68no—
291C22×C4—64no—
292C22×C4—64no—
293C22×C4—84no—
294C22×C4—124no—
295C22×C4—242no—
296C22×C4—244no—
297C22×C4—244no—
298C22×C4—242no—
299C22×C4Dic3⋊C414yescomplemented
300C22×C4C2×C4×D312yes—
301C4⋊C4—68no—
302C4⋊C4—248no—
303C22⋊C4—24no—
304C22⋊C4—24no—
305C22⋊C4—34no—
306C22⋊C4—34no—
307C22⋊C4—38no—
308C22⋊C4—44no—
309C22⋊C4—68no—
310C22⋊C4—62no—
311C22⋊C4—62no—
312C22⋊C4—64no—
313C22⋊C4—64no—
314C22⋊C4—64no—
315C22⋊C4—124no—
316C42—68no—
317C22×C3—24no—
318C22×C3—22no—
319C22×C3—24no—
320C22×C3—22no—
321C22×C3—22no—
322C22×C3—24no—
323C22×C3—44no—
324C22×C3—44no—
325C22×C3—42no—
326C22×C3—42no—
327C22×C3—42no—
328C22×C3—42no—
329C22×C3—44no—
330C22×C3—44no—
331C22×C3—41no—
332C22×C3—41no—
333C22×C3—42no—
334C22×C3—44no—
335C22×C3—44no—
336C22×C3—44no—
337C22×C3—84no—
338C22×C3—84no—
339C22×C3#38012yes—
340C22×C3#38012yes—
341C22×C3#38014yes—
342C22×C3C22×C22⋊C411yescharacteristic
343C22×C3C22×C4⋊C411yescharacteristic
344C22×C3C2×#11711yescharacteristic
345C22×C3C2×#12112yes—
346C22×C3C2×C22⋊Q812yes—
347C3×C4—24no—
348C3×C4—84no—
349Dic3—28no—
350Dic3—44no—
351Dic3—84no—
352C23—22no—
353C23—22no—
354C23—24no—
355C23—24no—
356C23—44no—
357C23—42no—
358C23—44no—
359C23—42no—
360C23—42no—
361C23—44no—
362C23—82no—
363C23—82no—
364C23—82no—
365C23—84no—
366C23—82no—
367C23—84no—
368C23—84no—
369C23#87712yes—
370C23#87712yes—
371C23#87814yes—
372C23#87814yes—
373C23D3×C22⋊C411yescharacteristic
374C23D3×C22⋊C411yescharacteristic
375C23D3×C22⋊C412yes—
376C23C2×Dic3⋊C414yes—
377C23C2×Dic3⋊C412yes—
378C23C2×Dic3⋊C412yes—
379C23#93914yes—
380C23#93914yes—
381C23C22×Dic611yescharacteristic
382C23C22×C4×D311yescharacteristic
383C23C2×D3×D412yes—
384C23C2×#23712yes—
385C23C22×#6711yescharacteristic
386D4—216no—
387D4—28no—
388D4—28no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—38no—
394C2×C4—48no—
395C2×C4—44no—
396C2×C4—64no—
397C2×C4—88no—
398C2×C4—128no—
399C2×C4—124no—
400C2×C4—124no—
401C2×C4—128no—
402C2×C4—124no—
403C2×C4—248no—
404C2×C4—244no—
405C2×C4—248no—
406C2×C4C2×Dic3⋊C414yes—
407C2×C3—22no—
408C2×C3—21no—
409C2×C3—21no—
410C2×C3—22no—
411C2×C3—22no—
412C2×C3—22no—
413C2×C3—24no—
414C2×C3—44no—
415C2×C3C2×#38011yescharacteristic
416C2×C3#288312yes—
417C22—24no—
418C22—22no—
419C22—24no—
420C22—22no—
421C22—22no—
422C22—24no—
423C22—44no—
424C22—44no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22—44no—
429C22—44no—
430C22—42no—
431C22—41no—
432C22—42no—
433C22—41no—
434C22—44no—
435C22—44no—
436C22—44no—
437C22—84no—
438C22—84no—
439C22#494212yes—
440C22#494212yes—
441C22#494214yes—
442C22C2×#87711yescharacteristic
443C22C2×#87812yes—
444C22C2×D3×C22⋊C411yescharacteristic
445C22C22×Dic3⋊C411yescharacteristic
446C22C2×#93912yes—
447C4—24no—
448C4—68no—
449C4—84no—
450C4—124no—
451C4—244no—
452C3#2063311yescharacteristic, complemented, Sylow
453C2—22no—
454C2—21no—
455C2—21no—
456C2—22no—
457C2—22no—
458C2—24no—
459C2—22no—
460C2—44no—
461C2#7868812yes—
462C2C2×#494211yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC22×C3×#99C2
socleC22×C3C22×C22⋊C4
prime radicalC23×C3×D4C22
cosocleC24×C3C24
hypercenterC22×#99D3
hypocenterC3#20633
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ C22×#99

Complements

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

NH ≅ G/Ntype
4. #82095C2semidirect
5. C2×#4527C2semidirect
6. #70028C2semidirect
10. C22×#217C4semidirect
16. C2×D4×Dic3C4semidirect
18. C2×#926C4semidirect
25. #4942C22semidirect
26. #4527C22semidirect
46. C23×Dic3C2×C4semidirect
62. C2×#217C2×C4semidirect
71. C2×C4⋊Dic3C2×C4semidirect
73. C2×C4×Dic3C2×C4semidirect
101. C24×C3C4⋊C4semidirect
119. C2×C3×D4C4⋊C4semidirect
128. C22×C3×C4C4⋊C4semidirect
261. C24Dic3⋊C4semidirect
279. C2×D4Dic3⋊C4semidirect
299. C22×C4Dic3⋊C4semidirect
452. C3#20633semidirect

Maximal subgroups

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

subgroupindex
C2×#57982
C22×C3×#992
C22×#8342
#820952
C2×#45272
#700282
#206333