Finite Group Databasebeta

← #10695453

Subgroups — #10695453

Order 768.

890 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#393C211yesmaximal, characteristic
2C2×#5228C211yesmaximal, characteristic, complemented
3#82117C211yesmaximal, characteristic
4#72132C211yesmaximal, characteristic, complemented
5C2×#4525C211yesmaximal, characteristic
6C2×#4452C211yesmaximal, characteristic, complemented
7#67612C211yesmaximal, characteristic, complemented
8#9924—31nomaximal, Sylow
9C23×C3×Q8C2211yescharacteristic
10C22×C3×C22⋊C4—21no—
11C22×C3×C22⋊C4C2211yescharacteristic
12C22×#216—21no—
13C2×#924C2211yescharacteristic
14C3×#393—22no—
15C3×#393—22no—
16C2×C3×#95—21no—
17C2×C3×#95C2211yescharacteristic
18#5247—21no—
19#5228—22no—
20#5228—22no—
21C2×#835—21no—
22C2×#835—22no—
23#5212C2211yescharacteristic
24#5210—21no—
25C2×#830C2211yescharacteristic
26C2×#830C2211yescharacteristic
27#4949—21no—
28#4948—22no—
29#4947—21no—
30#4525—22no—
31#4452—22no—
32C2×#451—31no—
33#2626—31no—
34C2×#393D311yescharacteristic
35#1787—31no—
36C2×#328—31no—
37C2×#323—31no—
38#1294—31no—
39C25×C3D411yescharacteristic
40C22×C3×Q8—21no—
41C22×C3×Q8—21no—
42C22×C3×Q8—21no—
43C22×C3×Q8—21no—
44C22×C3×Q8—21no—
45C22×C3×Q8—44no—
46C22×C3×Q8D412yes—
47C23×C3×C4—21no—
48C23×C3×C4—21no—
49C23×C3×C4D411yescharacteristic
50C23×C3×C4C2311yescharacteristic
51C22×Dic6D411yescharacteristic
52C2×C3×C22⋊C4—24no—
53C2×C3×C22⋊C4—21no—
54C2×C3×C22⋊C4—21no—
55C2×C3×C22⋊C4—21no—
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—44no—
60C2×#217—21no—
61C2×#217—21no—
62C2×#217—22no—
63C2×#217—22no—
64C2×#216—24no—
65C2×#216—44no—
66C2×C4⋊Dic3D411yescharacteristic
67C2×C4⋊Dic3D412yes—
68#924—21no—
69#924—21no—
70#924—21no—
71#924—21no—
72C2×Dic3⋊C4—21no—
73C2×Dic3⋊C4—22no—
74C22×C3⋊C8—21no—
75C22×C3⋊C8—21no—
76C3×#95—24no—
77C3×#95—21no—
78C3×#95D412yes—
79#835—24no—
80#835—44no—
81C3⋊#95—22no—
82C3⋊#95—21no—
83#830—21no—
84#830—21no—
85#830—21no—
86#830D412yes—
87C23×Q8C2×D311yescharacteristic
88C22×Q16—61no—
89C2×C22⋊Q8—31no—
90C22×C22⋊C4—21no—
91C22×C22⋊C4C2×D311yescharacteristic
92#451—62no—
93#451—62no—
94C2×#103—61no—
95C2×#103—62no—
96C2×C22⋊C8—31no—
97C2×C22⋊C8—31no—
98#399—61no—
99#396—62no—
100#393—22no—
101#393—22no—
102#393—61no—
103#385—61no—
104#380—31no—
105#379—61no—
106C2×#95—21no—
107C2×#95C2×D311yescharacteristic
108#328—62no—
109#323—62no—
110C24×C3—21no—
111C24×C3—21no—
112C24×C3—21no—
113C24×C3—21no—
114C24×C3—21no—
115C24×C3—21no—
116C24×C3—42no—
117C24×C3—42no—
118C24×C3Q1612yescomplemented
119C24×C3C2×D411yescharacteristic
120C2×C3×Q8—24no—
121C2×C3×Q8—24no—
122C2×C3×Q8—22no—
123C2×C3×Q8—22no—
124C2×C3×Q8—44no—
125C2×C3×Q8—42no—
126C2×C3×Q8—42no—
127C2×C3×Q8—44no—
128C2×C3×Q8—44no—
129C2×C3×Q8—44no—
130C2×C3×Q8—41no—
131C2×C3×Q8—41no—
132C22×C3×C4—24no—
133C22×C3×C4—22no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
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×C4—42no—
143C22×C3×C4—42no—
144C22×C3×C4—42no—
145C22×C3×C4—41no—
146C22×C3×C4—41no—
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4#4111yescharacteristic
153C22×Dic3—22no—
154C22×Dic3—21no—
155C22×Dic3—21no—
156C2×Dic6—21no—
157C2×Dic6—21no—
158C2×Dic6—42no—
159C3×C22⋊C4—48no—
160C3×C22⋊C4—41no—
161C3×C22⋊C4—41no—
162C3×C22⋊C4—41no—
163C3×C22⋊C4—41no—
164#217—41no—
165#217—41no—
166#217—41no—
167#217—41no—
168#217—82no—
169#217—82no—
170#216—48no—
171C4⋊Dic3—21no—
172C4⋊Dic3—21no—
173C4⋊Dic3—42no—
174Dic3⋊C4—41no—
175Dic3⋊C4—41no—
176Dic3⋊C4—82no—
177C2×C3⋊C8—22no—
178C2×C3⋊C8—42no—
179C2×C3⋊C8—42no—
180C2×C3⋊C8—41no—
181C25#6711yescharacteristic
182C22×Q8—21no—
183C22×Q8—21no—
184C22×Q8—21no—
185C22×Q8—21no—
186C22×Q8—21no—
187C22×Q8—31no—
188C22×Q8—44no—
189C22×Q8#6712yes—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4#6711yescharacteristic
193C23×C4C22×D311yescharacteristic
194C2×Q16—64no—
195C2×Q16—124no—
196C22×C8—61no—
197C22×C8—61no—
198C22⋊Q8—61no—
199C22⋊Q8—61no—
200C22⋊Q8—61no—
201C22⋊Q8—61no—
202C2×C4⋊C4—31no—
203C2×C4⋊C4—32no—
204C2×C4⋊C4—61no—
205C2×C4⋊C4—62no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—61no—
215C2×C22⋊C4—61no—
216C2×C22⋊C4—62no—
217C2×C22⋊C4—62no—
218#103—64no—
219#103—124no—
220C22⋊C8—32no—
221C22⋊C8—61no—
222C22⋊C8—61no—
223C22⋊C8—61no—
224#95—24no—
225#95—21no—
226#95—62no—
227#95—61no—
228#95#6712yes—
229C23×C3—22no—
230C23×C3—22no—
231C23×C3—22no—
232C23×C3—22no—
233C23×C3—22no—
234C23×C3—44no—
235C23×C3—42no—
236C23×C3—44no—
237C23×C3—42no—
238C23×C3—41no—
239C23×C3—41no—
240C23×C3—41no—
241C23×C3—42no—
242C23×C3—42no—
243C23×C3—41no—
244C23×C3—82no—
245C23×C3—82no—
246C23×C3—81no—
247C23×C3—81no—
248C23×C3#12011yescharacteristic
249C23×C3#12011yescharacteristic
250C23×C3#12111yescharacteristic
251C23×C3C2×Q1611yescharacteristic
252C23×C3#13511yescharacteristic
253C23×C3#13711yescharacteristic
254C23×C3#13711yescharacteristic
255C3×Q8—24no—
256C3×Q8—24no—
257C3×Q8—42no—
258C3×Q8—44no—
259C3×Q8—44no—
260C3×Q8—42no—
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—42no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—41no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—82no—
279C2×C3×C4—82no—
280C2×C3×C4—81no—
281C2×C3×C4#12011yescharacteristic
282C2×C3×C4#12111yescharacteristic
283C2×C3×C4#12111yescharacteristic
284C2×C3×C4#12711yescharacteristic
285C2×Dic3—41no—
286C2×Dic3—41no—
287C2×Dic3—42no—
288C2×Dic3—42no—
289C2×Dic3—42no—
290C2×Dic3—42no—
291C2×Dic3—42no—
292Dic6—42no—
293Dic6—81no—
294C3⋊C8—42no—
295C3⋊C8—81no—
296C24—21no—
297C24—21no—
298C24—21no—
299C24—21no—
300C24—21no—
301C24—21no—
302C24—42no—
303C24—42no—
304C24#21612yescomplemented
305C24C2×#6711yescharacteristic
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—44no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—44no—
314C2×Q8—44no—
315C2×Q8—44no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—61no—
319C2×Q8—61no—
320C2×Q8—122no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—62no—
337C22×C4—61no—
338C22×C4—61no—
339C22×C4D3×D411yescharacteristic
340C22×C4D3×D411yescharacteristic
341C22×C4D3×D411yescharacteristic
342C22×C4#23711yescharacteristic
343C22×C4C2×#6711yescharacteristic
344C22×C4C2×#6711yescharacteristic
345Q16—128no—
346C2×C8—62no—
347C2×C8—122no—
348C2×C8—122no—
349C2×C8—121no—
350C4⋊C4—61no—
351C4⋊C4—61no—
352C4⋊C4—122no—
353C4⋊C4—121no—
354C4⋊C4—121no—
355C4⋊C4—242no—
356C22⋊C4—48no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—121no—
362C22⋊C4—121no—
363C22⋊C4—121no—
364C22⋊C4—121no—
365C22⋊C4—242no—
366C22⋊C4—242no—
367C22×C3—22no—
368C22×C3—22no—
369C22×C3—22no—
370C22×C3—22no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—44no—
376C22×C3—44no—
377C22×C3—42no—
378C22×C3—82no—
379C22×C3—82no—
380C22×C3—82no—
381C22×C3—82no—
382C22×C3—81no—
383C22×C3—81no—
384C22×C3#39211yescharacteristic
385C22×C3#45111yescharacteristic
386C22×C3#45111yescharacteristic
387C22×C3#45211yescharacteristic
388C22×C3#45211yescharacteristic
389C22×C3#45512yes—
390C22×C3B(4,𝔽2)12yes—
391C22×C3#46711yescharacteristic
392C22×C3#47011yescharacteristic
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—21no—
396C3×C4—21no—
397C3×C4—44no—
398C3×C4—82no—
399C3×C4—81no—
400Dic3—82no—
401Dic3—81no—
402Dic3—81no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—44no—
409C23—42no—
410C23—42no—
411C23—44no—
412C23—41no—
413C23—41no—
414C23—42no—
415C23—41no—
416C23—41no—
417C23—42no—
418C23—82no—
419C23—81no—
420C23—82no—
421C23—81no—
422C23#93711yescharacteristic
423C23#93911yescharacteristic
424C23#94211yescharacteristic
425C23C2×#21611yescharacteristic
426C23#95011yescharacteristic
427C23#95111yescharacteristic
428C23#95311yescharacteristic
429Q8—24no—
430Q8—24no—
431Q8—42no—
432Q8—44no—
433Q8—44no—
434Q8—42no—
435Q8—122no—
436Q8—241no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—81no—
457C2×C4—121no—
458C2×C4—121no—
459C2×C4—122no—
460C2×C4—122no—
461C2×C4—122no—
462C2×C4—122no—
463C2×C4—122no—
464C2×C4#93711yescharacteristic
465C2×C4#93811yescharacteristic
466C2×C4#93911yescharacteristic
467C2×C4#94011yescharacteristic
468C8—122no—
469C8—241no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—82no—
475C2×C3#160212yes—
476C2×C3#160612yes—
477C2×C3#200111yescharacteristic
478C2×C3#200411yescharacteristic
479C2×C3#202111yescharacteristic
480C22—21no—
481C22—21no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—21no—
487C22—21no—
488C22—44no—
489C22—42no—
490C22—44no—
491C22—82no—
492C22—82no—
493C22—82no—
494C22—81no—
495C22—82no—
496C22—81no—
497C22#503212yes—
498C22#504711yescharacteristic
499C22#505011yescharacteristic
500C22#505111yescharacteristic
501C22#505211yescharacteristic
502C22#506112yes—
503C22#522311yescharacteristic
504C22#522811yescharacteristic
505C22#523911yescharacteristic
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—44no—
511C4—82no—
512C4—81no—
513C4—241no—
514C4—242no—
515C4—241no—
516C3#992411yescharacteristic, complemented, Sylow
517C2—21no—
518C2—21no—
519C2—21no—
520C2—21no—
521C2—82no—
522C2#6970012yes—
523C2#6970112yes—
524C2#7167111yescharacteristic
525C2#7170011yescharacteristic
526C2#7173111yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#393C2
socleC23×C3#120
prime radicalC25×C3D4
hypercenterC2×#393D3
hypocenterC3#9924
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
2. C2×#5228C2semidirect
4. #72132C2semidirect
6. C2×#4452C2semidirect
7. #67612C2semidirect
118. C24×C3Q16semidirect
304. C24#216semidirect
516. C3#9924semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3932
C2×#52282
#821172
#721322
C2×#45252
C2×#44522
#676122
#99243