Finite Group Databasebeta

← #10944018

Subgroups — #10944018

Order 768.

1436 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic6C211yesmaximal, characteristic
2#82076C211yesmaximal, characteristic
3C2×#4849C212yesmaximal, complemented
4C2×#4739C212yesmaximal
5C3×#1797C211yesmaximal, characteristic
6#70485C212yesmaximal, complemented
7#69930C212yesmaximal
8#68685C214yesmaximal, complemented
9#32906—31nomaximal, Sylow
10C23×Dic6C2211yescharacteristic
11C22×C3×C42C2211yescharacteristic
12C22×C4⋊Dic3C2211yescharacteristic
13C2×#924C2212yes—
14C22×Dic3⋊C4—21no—
15C22×C4×Dic3—21no—
16C22×Dic12C2214yes—
17C2×#869C2211yescharacteristic
18C2×C4×Dic6—21no—
19C2×C4×Dic6—21no—
20C2×C4×Dic6—28no—
21C2×C4×Dic6C2212yes—
22C2×C4×Dic6C2212yes—
23C2×C3×C4⋊C8C2212yes—
24C2×C3×C22⋊C8C2212yes—
25#5212—21no—
26#5212—21no—
27C2×#816C2214yes—
28C2×#816C2214yes—
29#4935C2211yescharacteristic
30#4929C2212yes—
31#4928—21no—
32#4928—21no—
33#4849—28no—
34#4739—28no—
35C3×#336C2212yes—
36#4551C2214yes—
37C22×C4×Q8—31no—
38C2×#462—32no—
39C2×#451—32no—
40#2857—31no—
41#1878—34no—
42#1872—32no—
43#1803—32no—
44#1797D311yescharacteristic
45C23×C3×C4D411yescharacteristic
46C23×C3×C4D411yescharacteristic
47C23×C3×C4C2311yescharacteristic
48C23×Dic3—21no—
49C23×Dic3—21no—
50C22×Dic6—21no—
51C22×Dic6—22no—
52C22×Dic6—24no—
53C22×Dic6—24no—
54C22×Dic6D414yes—
55C22×Dic6C2312yes—
56C22×Dic6C2312yes—
57C22×C3×C8C2314yes—
58C2×C3×C42—24no—
59C2×C3×C42D411yescharacteristic
60C2×C3×C42D411yescharacteristic
61C2×C3×C42C2311yescharacteristic
62C2×C3×C42C2311yescharacteristic
63C2×#217—22no—
64C2×C4⋊Dic3—21no—
65C2×C4⋊Dic3—24no—
66C2×C4⋊Dic3—21no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3—21no—
70C2×C4⋊Dic3C2312yes—
71C2×C4⋊Dic3C2312yes—
72#924—24no—
73#924—24no—
74C2×Dic3⋊C4—28no—
75C2×Dic3⋊C4—22no—
76C2×Dic3⋊C4—22no—
77C2×Dic3⋊C4—22no—
78C2×C4×Dic3—28no—
79C2×C4×Dic3—22no—
80C2×C4×Dic3—22no—
81C2×Dic12—28no—
82C2×Dic12—28no—
83C2×Dic12—28no—
84#869—24no—
85C4×Dic6—28no—
86C4×Dic6—28no—
87C4×Dic6—24no—
88C4×Dic6—24no—
89C4×Dic6—24no—
90C4×Dic6D418yes—
91C3×C4⋊C8—24no—
92C3×C22⋊C8—24no—
93C3⋊#95—22no—
94C3⋊#95—22no—
95#816—28no—
96#816—28no—
97C23×Q8—31no—
98C22×Q16—34no—
99C2×C4⋊Q8—31no—
100C2×C22⋊Q8—32no—
101C2×C4×Q8—32no—
102C2×C4×Q8—32no—
103C2×C4×Q8—68no—
104C2×C4×Q8—61no—
105C2×C4×Q8—61no—
106C22×C4⋊C4—31no—
107C22×C4⋊C4—61no—
108C22×C42—61no—
109C22×C42C2×D311yescharacteristic
110#462—68no—
111#451—68no—
112C2×C4⋊C8C2×D312yes—
113C2×#103—34no—
114C2×#103—34no—
115C2×C22⋊C8C2×D312yes—
116#391—32no—
117#389—61no—
118#389—61no—
119#384—31no—
120#380—61no—
121#380—61no—
122#340—34no—
123#336C2×D312yes—
124C24×C3C2×D411yescharacteristic
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—41no—
132C22×C3×C4Q1614yescomplemented
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D412yes—
139C22×C3×C4#4112yes—
140C22×C3×C4C2411yescharacteristic
141C22×Dic3—24no—
142C22×Dic3—24no—
143C22×Dic3—24no—
144C22×Dic3—22no—
145C22×Dic3—24no—
146C22×Dic3—22no—
147C22×Dic3—22no—
148C22×Dic3—22no—
149C22×Dic3—42no—
150C22×Dic3—42no—
151C2×Dic6—24no—
152C2×Dic6—24no—
153C2×Dic6—216no—
154C2×Dic6—24no—
155C2×Dic6—22no—
156C2×Dic6—22no—
157C2×Dic6—22no—
158C2×Dic6—216no—
159C2×Dic6—24no—
160C2×Dic6—22no—
161C2×Dic6—24no—
162C2×Dic6—22no—
163C2×Dic6C2×D414yes—
164C2×Dic6C2×D414yes—
165C2×Dic6#4114yes—
166C2×C3×C8—24no—
167C2×C3×C8—24no—
168C2×C3×C8—24no—
169C3×C42—22no—
170C3×C42—22no—
171C3×C42—22no—
172C3×C42C2×D414yes—
173#217—44no—
174C4⋊Dic3—24no—
175C4⋊Dic3—22no—
176C4⋊Dic3—22no—
177C4⋊Dic3—22no—
178C4⋊Dic3—42no—
179C4⋊Dic3—42no—
180C4⋊Dic3—42no—
181C4⋊Dic3—42no—
182C4⋊Dic3C2×D414yes—
183Dic3⋊C4—28no—
184Dic3⋊C4—28no—
185Dic3⋊C4—44no—
186C4×Dic3—28no—
187C4×Dic3—28no—
188Dic12—416no—
189C22×Q8—34no—
190C22×Q8—32no—
191C22×Q8—32no—
192C22×Q8—64no—
193C22×Q8—64no—
194C22×Q8—61no—
195C22×Q8—62no—
196C23×C4—61no—
197C23×C4—61no—
198C23×C4D1211yescharacteristic
199C23×C4D1211yescharacteristic
200C23×C4C22×D311yescharacteristic
201C2×Q16—68no—
202C2×Q16—68no—
203C2×Q16—68no—
204C22×C8C22×D314yes—
205C4⋊Q8—64no—
206C22⋊Q8—64no—
207C22⋊Q8—64no—
208C4×Q8—38no—
209C4×Q8—64no—
210C4×Q8—64no—
211C4×Q8—68no—
212C4×Q8—68no—
213C4×Q8—64no—
214C2×C4⋊C4—32no—
215C2×C4⋊C4—32no—
216C2×C4⋊C4—64no—
217C2×C4⋊C4—68no—
218C2×C4⋊C4—61no—
219C2×C4⋊C4—62no—
220C2×C4⋊C4—62no—
221C2×C4⋊C4—62no—
222C2×C4⋊C4—61no—
223C2×C4⋊C4—61no—
224C2×C4⋊C4—61no—
225C2×C4⋊C4—61no—
226C2×C22⋊C4—62no—
227C2×C42—24no—
228C2×C42—68no—
229C2×C42—62no—
230C2×C42—62no—
231C2×C42D1211yescharacteristic
232C2×C42D1211yescharacteristic
233C2×C42C22×D311yescharacteristic
234C2×C42C22×D311yescharacteristic
235C4⋊C8—24no—
236#103—68no—
237#103—68no—
238C22⋊C8—24no—
239#95—62no—
240#95—62no—
241C23×C3—21no—
242C23×C3—21no—
243C23×C3—21no—
244C23×C3—21no—
245C23×C3#12112yes—
246C23×C3C2×Q1612yes—
247C23×C3#13712yes—
248C23×C3C22×D411yescharacteristic
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—24no—
256C2×C3×C4—24no—
257C2×C3×C4—28no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—44no—
262C2×C3×C4#12012yes—
263C2×C3×C4#12012yes—
264C2×C3×C4#12112yes—
265C2×C3×C4C2×Q1612yes—
266C2×C3×C4C2×Q1612yes—
267C2×C3×C4#13712yes—
268C2×C3×C4C22×D411yescharacteristic
269C2×C3×C4C22×D412yes—
270C2×C3×C4C2×#4111yescharacteristic
271C2×C3×C4C2×#4112yes—
272C2×Dic3—216no—
273C2×Dic3—24no—
274C2×Dic3—24no—
275C2×Dic3—24no—
276C2×Dic3—24no—
277C2×Dic3—48no—
278C2×Dic3—42no—
279C2×Dic3—42no—
280C2×Dic3—42no—
281C2×Dic3—42no—
282C2×Dic3—42no—
283C2×Dic3—42no—
284C2×Dic3—42no—
285C2×Dic3—42no—
286Dic6—216no—
287Dic6—24no—
288Dic6—24no—
289Dic6—24no—
290Dic6—44no—
291Dic6—44no—
292Dic6#12118yes—
293C3×C8—44no—
294C24C2×D1211yescharacteristic
295C2×Q8—34no—
296C2×Q8—34no—
297C2×Q8—34no—
298C2×Q8—616no—
299C2×Q8—64no—
300C2×Q8—64no—
301C2×Q8—616no—
302C2×Q8—64no—
303C2×Q8—62no—
304C2×Q8—62no—
305C2×Q8—62no—
306C2×Q8—64no—
307C2×Q8—62no—
308C2×Q8—62no—
309C2×Q8—64no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—41no—
317C22×C4—64no—
318C22×C4—64no—
319C22×C4—64no—
320C22×C4—62no—
321C22×C4—64no—
322C22×C4—62no—
323C22×C4—62no—
324C22×C4—62no—
325C22×C4—122no—
326C22×C4—122no—
327C22×C4Dic1214yescomplemented
328C22×C4C2×D1211yescharacteristic
329C22×C4C2×D1211yescharacteristic
330C22×C4C2×D1211yescharacteristic
331C22×C4C2×D1211yescharacteristic
332C22×C4C2×D1211yescharacteristic
333C22×C4D3×D412yes—
334C22×C4#23912yes—
335C22×C4C23×D311yescharacteristic
336Q16—1216no—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—24no—
340C4⋊C4—34no—
341C4⋊C4—68no—
342C4⋊C4—64no—
343C4⋊C4—68no—
344C4⋊C4—62no—
345C4⋊C4—62no—
346C4⋊C4—62no—
347C4⋊C4—122no—
348C4⋊C4—122no—
349C4⋊C4—122no—
350C4⋊C4—124no—
351C4⋊C4—122no—
352C22⋊C4—124no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—68no—
357C42—68no—
358C42D3×D414yes—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—22no—
362C22×C3—22no—
363C22×C3—22no—
364C22×C3—22no—
365C22×C3#46214yescomplemented
366C22×C3C2×#12011yescharacteristic
367C22×C3C2×#12111yescharacteristic
368C22×C3#52511yescharacteristic
369C22×C3C22×Q1611yescharacteristic
370C22×C3C2×#13711yescharacteristic
371C22×C3C2×#13711yescharacteristic
372C22×C3#57511yescharacteristic
373C3×C4—22no—
374C3×C4—22no—
375C3×C4—22no—
376C3×C4—41no—
377C3×C4#45114yescomplemented
378C3×C4C2×#12112yes—
379C3×C4#52512yes—
380Dic3—28no—
381Dic3—44no—
382Dic3—82no—
383Dic3—82no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23#89512yes—
389C23C2×Dic1212yes—
390C23#90912yes—
391C23C22×D1211yescharacteristic
392Q8—38no—
393Q8—616no—
394Q8—64no—
395Q8—64no—
396Q8—64no—
397Q8—124no—
398Q8—124no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—28no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—44no—
412C2×C4—616no—
413C2×C4—64no—
414C2×C4—64no—
415C2×C4—64no—
416C2×C4—64no—
417C2×C4—128no—
418C2×C4—122no—
419C2×C4—122no—
420C2×C4—122no—
421C2×C4—122no—
422C2×C4—122no—
423C2×C4—122no—
424C2×C4—122no—
425C2×C4—122no—
426C2×C4#88212yes—
427C2×C4#88212yes—
428C2×C4#89512yes—
429C2×C4C2×Dic1212yes—
430C2×C4C2×Dic1212yes—
431C2×C4#90912yes—
432C2×C4C2×D3×D411yescharacteristic
433C2×C4C2×D3×D412yes—
434C2×C4C2×#23911yescharacteristic
435C2×C4D3×#4112yes—
436C8—44no—
437C2×C3—24no—
438C2×C3#284711yescharacteristic
439C2×C3C2×#45111yescharacteristic
440C2×C3#300711yescharacteristic
441C2×C3C2×#46211yescharacteristic
442C2×C3#303711yescharacteristic
443C2×C3#310411yescharacteristic
444C2×C3#311311yescharacteristic
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22#484914yescomplemented
452C22C2×#88211yescharacteristic
453C22C2×#89511yescharacteristic
454C22#552411yescharacteristic
455C22C22×Dic1211yescharacteristic
456C22C2×#90911yescharacteristic
457C22C2×#90911yescharacteristic
458C22#574811yescharacteristic
459C4—22no—
460C4—22no—
461C4—22no—
462C4—41no—
463C4—68no—
464C4—124no—
465C4—242no—
466C4—242no—
467C4#473914yescomplemented
468C4D3×#12112yes—
469C4#562712yes—
470C3#3290611yescharacteristic, complemented, Sylow
471C2—24no—
472C2#7404811yescharacteristic
473C2C2×#473911yescharacteristic
474C2#7864011yescharacteristic
475C2C2×#484911yescharacteristic
476C2#7906711yescharacteristic
477C2#7931011yescharacteristic
478C2#7935611yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1797C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1797D3
hypocenterC3#32906
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
3. C2×#4849C2semidirect
6. #70485C2semidirect
8. #68685C2semidirect
132. C22×C3×C4Q16semidirect
327. C22×C4Dic12semidirect
365. C22×C3#462semidirect
377. C3×C4#451semidirect
451. C22#4849semidirect
467. C4#4739semidirect
470. C3#32906semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic62
#820762
C2×#48492
C2×#47392
C3×#17972
#704852
#699302
#686852
#329063