Finite Group Databasebeta

← #10695401

Subgroups — #10695401

Order 768.

872 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#448C211yesmaximal, characteristic
2C2×#5230C211yesmaximal, characteristic, complemented
3#82675C211yesmaximal, characteristic
4#72132C211yesmaximal, characteristic, complemented
5C2×#4537C211yesmaximal, characteristic
6C2×#4469C211yesmaximal, characteristic, complemented
7#67675C211yesmaximal, characteristic, complemented
8#9917—31nomaximal, Sylow
9C23×C3×Q8C2211yescharacteristic
10C22×C3×SD16—21no—
11C2×C3×#121C2211yescharacteristic
12C2×#935—21no—
13C22×Dic3⋊C4—21no—
14C22×Dic3⋊C4C2211yescharacteristic
15C3×#448—22no—
16C3×#448—22no—
17C2×C3×#103—21no—
18C2×C3×C22⋊C8C2211yescharacteristic
19#5230—22no—
20#5230—22no—
21C2×#835—22no—
22#5222—21no—
23#5220—21no—
24#5212C2211yescharacteristic
25C2×C3⋊#95—21no—
26C2×C3⋊#95C2211yescharacteristic
27C2×#830C2211yescharacteristic
28#4983—21no—
29#4948—22no—
30#4537—22no—
31#4469—22no—
32C2×#448D311yescharacteristic
33#2618—31no—
34C2×#395—31no—
35#1787—31no—
36C2×#327—31no—
37C2×#324—31no—
38#1294—31no—
39C22×C3×Q8—21no—
40C22×C3×Q8—21no—
41C22×C3×Q8—21no—
42C22×C3×Q8—21no—
43C22×C3×Q8—21no—
44C22×C3×Q8—44no—
45C22×C3×Q8D412yes—
46C22×C3×D4—21no—
47C22×C3×D4D411yescharacteristic
48C23×C3×C4—21no—
49C23×C3×C4C2311yescharacteristic
50C23×Dic3—21no—
51C23×Dic3D411yescharacteristic
52C23×Dic3D411yescharacteristic
53C2×C3×SD16—24no—
54C2×C3×SD16—44no—
55C22×C3×C8—21no—
56C3×#121—21no—
57C3×#121—21no—
58C3×#121—21no—
59C3×#121—21no—
60C2×C3×C4⋊C4D411yescharacteristic
61C2×C3×C22⋊C4—21no—
62C2×#217—22no—
63C2×#217—21no—
64C2×#217—22no—
65#935—42no—
66#935—42no—
67C2×C4⋊Dic3D412yes—
68C2×Dic3⋊C4—22no—
69C2×Dic3⋊C4—24no—
70C2×Dic3⋊C4—21no—
71C2×Dic3⋊C4—21no—
72C2×Dic3⋊C4—21no—
73C2×Dic3⋊C4—21no—
74C2×Dic3⋊C4—21no—
75C2×Dic3⋊C4—21no—
76C2×Dic3⋊C4—44no—
77C22×C3⋊C8—21no—
78C3×#103—24no—
79C3×C22⋊C8—21no—
80C3×C22⋊C8D412yes—
81#835—44no—
82C3⋊#95—22no—
83C3⋊#95—24no—
84C3⋊#95—21no—
85C3⋊#95—21no—
86C3⋊#95D412yes—
87#830—21no—
88#830—21no—
89C23×Q8C2×D311yescharacteristic
90C22×SD16—21no—
91C2×#123—61no—
92C2×#121C2×D311yescharacteristic
93C22×C4⋊C4—31no—
94C22×C4⋊C4—61no—
95#448—22no—
96#448—22no—
97C2×#103—21no—
98C2×#103—62no—
99C2×C22⋊C8—31no—
100C2×C22⋊C8C2×D311yescharacteristic
101#398—61no—
102#396—62no—
103#395—62no—
104#395—62no—
105#394—61no—
106#386—61no—
107#380—31no—
108C2×#95—31no—
109C2×#95—61no—
110#327—62no—
111#324—62no—
112C24×C3—21no—
113C24×C3C2×D411yescharacteristic
114C2×C3×Q8—24no—
115C2×C3×Q8—24no—
116C2×C3×Q8—22no—
117C2×C3×Q8—22no—
118C2×C3×Q8—44no—
119C2×C3×Q8—42no—
120C2×C3×Q8—42no—
121C2×C3×Q8—44no—
122C2×C3×Q8—44no—
123C2×C3×Q8—44no—
124C2×C3×Q8—41no—
125C2×C3×Q8—41no—
126C2×C3×D4—21no—
127C2×C3×D4—21no—
128C2×C3×D4—42no—
129C2×C3×D4—41no—
130C2×C3×D4—41no—
131C2×C3×D4—41no—
132C2×C3×D4—41no—
133C2×C3×D4—41no—
134C2×C3×D4—41no—
135C22×C3×C4—24no—
136C22×C3×C4—22no—
137C22×C3×C4—21no—
138C22×C3×C4—42no—
139C22×C3×C4—42no—
140C22×C3×C4—41no—
141C22×C3×C4—41no—
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4#4111yescharacteristic
148C22×Dic3—24no—
149C22×Dic3—21no—
150C22×Dic3—22no—
151C22×Dic3—21no—
152C22×Dic3—21no—
153C22×Dic3—21no—
154C22×Dic3—21no—
155C22×Dic3—21no—
156C22×Dic3—21no—
157C22×Dic3—22no—
158C22×Dic3—21no—
159C22×Dic3—21no—
160C22×Dic3—42no—
161C22×Dic3—41no—
162C22×Dic3—41no—
163C22×Dic3—42no—
164C22×Dic3SD1612yescomplemented
165C3×SD16—48no—
166C2×C3×C8—22no—
167C2×C3×C8—42no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—21no—
170C3×C22⋊C4—41no—
171C3×C22⋊C4—41no—
172#217—44no—
173#217—41no—
174#217—41no—
175#217—82no—
176C4⋊Dic3—42no—
177Dic3⋊C4—44no—
178Dic3⋊C4—48no—
179Dic3⋊C4—41no—
180Dic3⋊C4—41no—
181Dic3⋊C4—41no—
182Dic3⋊C4—41no—
183C2×C3⋊C8—42no—
184C2×C3⋊C8—41no—
185C22×Q8—21no—
186C22×Q8—21no—
187C22×Q8—21no—
188C22×Q8—21no—
189C22×Q8—21no—
190C22×Q8—44no—
191C22×Q8#6712yes—
192C22×D4—21no—
193C22×D4#6711yescharacteristic
194C23×C4—21no—
195C23×C4—31no—
196C23×C4—31no—
197C23×C4—61no—
198C23×C4C22×D311yescharacteristic
199C2×SD16—24no—
200C2×SD16—44no—
201C22×C8—21no—
202C22×C8—61no—
203#123—122no—
204#123—122no—
205#121—21no—
206#121—21no—
207#121—21no—
208#121—21no—
209C2×C4⋊C4—32no—
210C2×C4⋊C4—64no—
211C2×C4⋊C4—62no—
212C2×C4⋊C4—61no—
213C2×C4⋊C4—61no—
214C2×C4⋊C4—61no—
215C2×C4⋊C4—61no—
216C2×C4⋊C4—61no—
217C2×C4⋊C4—61no—
218C2×C4⋊C4—124no—
219C2×C4⋊C4#6711yescharacteristic
220C2×C22⋊C4—21no—
221C2×C22⋊C4—62no—
222C2×C22⋊C4—61no—
223C2×C22⋊C4—62no—
224#103—24no—
225#103—124no—
226C22⋊C8—21no—
227C22⋊C8—61no—
228C22⋊C8—61no—
229C22⋊C8#6712yes—
230#95—32no—
231#95—64no—
232#95—62no—
233#95—61no—
234#95—61no—
235C23×C3—22no—
236C23×C3—22no—
237C23×C3—41no—
238C23×C3—41no—
239C23×C3—41no—
240C23×C3—42no—
241C23×C3—42no—
242C23×C3#12011yescharacteristic
243C23×C3#12011yescharacteristic
244C23×C3#12111yescharacteristic
245C23×C3C2×SD1611yescharacteristic
246C23×C3#13511yescharacteristic
247C23×C3AGL(1,ℤ/8)11yescharacteristic
248C23×C3#13711yescharacteristic
249C3×Q8—24no—
250C3×Q8—24no—
251C3×Q8—44no—
252C3×Q8—42no—
253C3×Q8—44no—
254C3×Q8—42no—
255C3×D4—42no—
256C3×D4—81no—
257C3×D4—81no—
258C3×D4—81no—
259C3×D4—81no—
260C3×D4—81no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—24no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—42no—
269C2×C3×C4—44no—
270C2×C3×C4—42no—
271C2×C3×C4—44no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4#12011yescharacteristic
276C2×C3×C4#12111yescharacteristic
277C2×C3×C4#12111yescharacteristic
278C2×C3×C4#12711yescharacteristic
279C2×Dic3—24no—
280C2×Dic3—22no—
281C2×Dic3—22no—
282C2×Dic3—22no—
283C2×Dic3—42no—
284C2×Dic3—44no—
285C2×Dic3—44no—
286C2×Dic3—41no—
287C2×Dic3—41no—
288C2×Dic3—42no—
289C2×Dic3—41no—
290C2×Dic3—41no—
291C2×Dic3—42no—
292C2×Dic3—41no—
293C2×Dic3—42no—
294C2×Dic3—42no—
295C2×Dic3—42no—
296C2×Dic3—42no—
297C2×Dic3—81no—
298C3×C8—42no—
299C3⋊C8—81no—
300C24—21no—
301C24D3×D411yescharacteristic
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—44no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—44no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—42no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—32no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—64no—
332C22×C4—61no—
333C22×C4—61no—
334C22×C4—61no—
335C22×C4—62no—
336C22×C4—61no—
337C22×C4—61no—
338C22×C4—61no—
339C22×C4—61no—
340C22×C4—62no—
341C22×C4—61no—
342C22×C4—61no—
343C22×C4—122no—
344C22×C4—121no—
345C22×C4—121no—
346C22×C4—122no—
347C22×C4D3×D411yescharacteristic
348C22×C4D3×D411yescharacteristic
349C22×C4#23711yescharacteristic
350C22×C4C2×#6711yescharacteristic
351C22×C4C2×#6711yescharacteristic
352C22×C4C2×#6711yescharacteristic
353SD16—48no—
354C2×C8—22no—
355C2×C8—42no—
356C2×C8—122no—
357C2×C8—121no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—122no—
361C4⋊C4—128no—
362C4⋊C4—124no—
363C4⋊C4—121no—
364C4⋊C4—121no—
365C4⋊C4—121no—
366C4⋊C4—121no—
367C22⋊C4—41no—
368C22⋊C4—41no—
369C22⋊C4—124no—
370C22⋊C4—121no—
371C22⋊C4—121no—
372C22⋊C4—242no—
373C22×C3—22no—
374C22×C3—22no—
375C22×C3—22no—
376C22×C3—22no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—82no—
382C22×C3—82no—
383C22×C3—81no—
384C22×C3—81no—
385C22×C3—81no—
386C22×C3#39211yescharacteristic
387C22×C3#44811yescharacteristic
388C22×C3#44911yescharacteristic
389C22×C3#45011yescharacteristic
390C22×C3#45211yescharacteristic
391C22×C3#45612yes—
392C22×C3B(4,𝔽2)12yes—
393C22×C3#46511yescharacteristic
394C22×C3#46811yescharacteristic
395C3×C4—21no—
396C3×C4—21no—
397C3×C4—21no—
398C3×C4—21no—
399C3×C4—44no—
400C3×C4—81no—
401Dic3—44no—
402Dic3—42no—
403Dic3—82no—
404Dic3—81no—
405C23—22no—
406C23—22no—
407C23—41no—
408C23—41no—
409C23—42no—
410C23—41no—
411C23—42no—
412C23D3×SD1611yescharacteristic
413C23#91411yescharacteristic
414C23#91511yescharacteristic
415C23#91611yescharacteristic
416C23#93711yescharacteristic
417C23#93711yescharacteristic
418C23#93811yescharacteristic
419Q8—24no—
420Q8—24no—
421Q8—42no—
422Q8—44no—
423Q8—44no—
424Q8—42no—
425D4—42no—
426D4—81no—
427D4—81no—
428D4—81no—
429D4—81no—
430D4—81no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—64no—
446C2×C4—62no—
447C2×C4—62no—
448C2×C4—62no—
449C2×C4—122no—
450C2×C4—124no—
451C2×C4—124no—
452C2×C4—121no—
453C2×C4—122no—
454C2×C4—121no—
455C2×C4—121no—
456C2×C4—121no—
457C2×C4—122no—
458C2×C4—122no—
459C2×C4—122no—
460C2×C4—121no—
461C2×C4—122no—
462C2×C4—122no—
463C2×C4—241no—
464C2×C4#93911yescharacteristic
465C2×C4#93911yescharacteristic
466C2×C4#94011yescharacteristic
467C2×C4#95311yescharacteristic
468C8—42no—
469C8—241no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—161no—
475C2×C3#160312yes—
476C2×C3#160412yes—
477C2×C3#200011yescharacteristic
478C2×C3#200511yescharacteristic
479C2×C3#202111yescharacteristic
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—82no—
489C22—82no—
490C22—81no—
491C22—81no—
492C22—81no—
493C22#503212yes—
494C22#516911yescharacteristic
495C22#517011yescharacteristic
496C22#517111yescharacteristic
497C22#517211yescharacteristic
498C22#517311yescharacteristic
499C22#517411yescharacteristic
500C22#520512yes—
501C22#522311yescharacteristic
502C4—21no—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—44no—
507C4—81no—
508C4—124no—
509C4—122no—
510C4—242no—
511C4—241no—
512C3#991711yescharacteristic, complemented, Sylow
513C2—21no—
514C2—21no—
515C2—21no—
516C2—21no—
517C2—161no—
518C2#7097412yes—
519C2#7097512yes—
520C2#7167111yescharacteristic
521C2#7205411yescharacteristic
522C2#7210811yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#448C2
socleC23×C3#120
prime radicalC2×C3×#121C22
hypercenterC2×#448D3
hypocenterC3#9917
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#448

Complements

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

NH ≅ G/Ntype
2. C2×#5230C2semidirect
4. #72132C2semidirect
6. C2×#4469C2semidirect
7. #67675C2semidirect
164. C22×Dic3SD16semidirect
512. C3#9917semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4482
C2×#52302
#826752
#721322
C2×#45372
C2×#44692
#676752
#99173