Finite Group Databasebeta

← #10690319

Subgroups — #10690319

Order 768.

750 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#451C211yesmaximal, characteristic
2#82745C211yesmaximal, characteristic
3#72134C211yesmaximal, characteristic
4#72132C211yesmaximal, characteristic
5C2×#4538C211yesmaximal, characteristic
6#67675C211yesmaximal, characteristic
7#67674C211yesmaximal, characteristic
8#9040—31nomaximal, Sylow
9C23×C3×Q8C2211yescharacteristic
10C22×C3×Q16—21no—
11C2×C3×C22⋊Q8C2211yescharacteristic
12C3×#451—22no—
13C3×#451C414yescomplemented
14C2×C3×#103—21no—
15C2×C3×C22⋊C8C2211yescharacteristic
16#5229—21no—
17C2×#835—21no—
18C2×#835—21no—
19#5212C2211yescharacteristic
20C2×C3⋊#95—21no—
21C2×C3⋊#95—21no—
22C2×C3⋊#95C2211yescharacteristic
23C2×C3⋊#95C2211yescharacteristic
24C2×#830C2211yescharacteristic
25#4984—21no—
26#4980—21no—
27#4980—21no—
28#4950—21no—
29Dic3×C22⋊C4—21no—
30#4538—22no—
31C2×#451D311yescharacteristic
32#2329—31no—
33#1787—31no—
34C2×#328—31no—
35#1446—31no—
36#1294—31no—
37#1280—31no—
38C22×C3×Q8—21no—
39C22×C3×Q8—21no—
40C22×C3×Q8—21no—
41C22×C3×Q8—21no—
42C22×C3×Q8—21no—
43C22×C3×Q8—42no—
44C22×C3×Q8—42no—
45C22×C3×Q8C2×C412yes—
46C22×C3×Q8D411yescharacteristic
47C23×C3×C4—21no—
48C23×C3×C4C2311yescharacteristic
49C23×Dic3—21no—
50C23×Dic3D411yescharacteristic
51C23×Dic3D411yescharacteristic
52C2×C3×Q16—24no—
53C2×C3×Q16—42no—
54C2×C3×Q16—42no—
55C22×C3×C8—21no—
56C3×C22⋊Q8—21no—
57C3×C22⋊Q8—21no—
58C3×C22⋊Q8C2×C412yes—
59C3×C22⋊Q8D412yes—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4D411yescharacteristic
62C2×C3×C22⋊C4—21no—
63C2×#217—21no—
64C2×C4⋊Dic3—21no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3D411yescharacteristic
67C2×C4⋊Dic3D411yescharacteristic
68C2×C4×Dic3—21no—
69C2×C4×Dic3—21no—
70C22×C3⋊C8—21no—
71C3×#103—24no—
72C3×C22⋊C8—21no—
73C3×C22⋊C8C2×C412yes—
74#835—42no—
75#835—42no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82C3⋊#95—21no—
83C3⋊#95—21no—
84C3⋊#95—21no—
85C3⋊#95—42no—
86C3⋊#95—42no—
87#830—21no—
88#830—21no—
89C23×Q8C2×D311yescharacteristic
90C22×Q16—21no—
91C2×C22⋊Q8C2×D311yescharacteristic
92#451—22no—
93#451Dic314yescomplemented
94C2×#103—21no—
95C2×#103—61no—
96C2×#103—61no—
97C2×C22⋊C8—31no—
98C2×C22⋊C8C2×D311yescharacteristic
99#391—61no—
100#389—61no—
101#387—61no—
102#387—61no—
103#385—61no—
104#380—31no—
105C4×C22⋊C4—61no—
106C2×#95—31no—
107C2×#95—31no—
108C2×#95—61no—
109C2×#95—61no—
110#328—62no—
111C24×C3C2×D411yescharacteristic
112C2×C3×Q8—21no—
113C2×C3×Q8—22no—
114C2×C3×Q8—22no—
115C2×C3×Q8—22no—
116C2×C3×Q8—22no—
117C2×C3×Q8—22no—
118C2×C3×Q8—22no—
119C2×C3×Q8—44no—
120C2×C3×Q8—41no—
121C2×C3×Q8—41no—
122C2×C3×Q8—41no—
123C2×C3×Q8—41no—
124C2×C3×Q8—44no—
125C2×C3×Q8—41no—
126C2×C3×Q8—41no—
127C2×C3×Q8—44no—
128C2×C3×Q8—44no—
129C2×C3×Q8—41no—
130C2×C3×Q8—41no—
131C2×C3×Q8C22⋊C412yes—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4—41no—
138C22×C3×C4—41no—
139C22×C3×C4—41no—
140C22×C3×C4—41no—
141C22×C3×C4—41no—
142C22×C3×C4—41no—
143C22×C3×C4C22×C411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4#4111yescharacteristic
148C22×C3×C4#4111yescharacteristic
149C22×Dic3—21no—
150C22×Dic3—21no—
151C22×Dic3—22no—
152C22×Dic3—21no—
153C22×Dic3—21no—
154C22×Dic3—21no—
155C22×Dic3—21no—
156C22×Dic3—21no—
157C22×Dic3—41no—
158C22×Dic3—41no—
159C22×Dic3—41no—
160C22×Dic3—41no—
161C22×Dic3—41no—
162C22×Dic3—41no—
163C22×Dic3—41no—
164C22×Dic3—41no—
165C22×Dic3—41no—
166C22×Dic3—41no—
167C22×Dic3Q1612yescomplemented
168C3×Q16—48no—
169C2×C3×C8—22no—
170C2×C3×C8—41no—
171C2×C3×C8—41no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4C22⋊C412yes—
176C3×C22⋊C4—22no—
177C3×C22⋊C4—22no—
178#217—41no—
179#217—41no—
180C4⋊Dic3—41no—
181C4⋊Dic3—41no—
182C4⋊Dic3—81no—
183C4⋊Dic3—81no—
184C4×Dic3—81no—
185C4×Dic3—81no—
186C2×C3⋊C8—41no—
187C2×C3⋊C8—41no—
188C2×C3⋊C8—41no—
189C22×Q8—21no—
190C22×Q8—21no—
191C22×Q8—21no—
192C22×Q8—21no—
193C22×Q8—21no—
194C22×Q8—42no—
195C22×Q8—42no—
196C22×Q8C2×Dic312yes—
197C22×Q8#6711yescharacteristic
198C23×C4—21no—
199C23×C4—31no—
200C23×C4—31no—
201C23×C4—61no—
202C23×C4C22×D311yescharacteristic
203C2×Q16—24no—
204C2×Q16—42no—
205C2×Q16—42no—
206C22×C8—21no—
207C22×C8—61no—
208C22⋊Q8—21no—
209C22⋊Q8—21no—
210C22⋊Q8C2×Dic312yes—
211C22⋊Q8#6712yes—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—31no—
214C2×C4⋊C4—31no—
215C2×C4⋊C4—61no—
216C2×C4⋊C4—61no—
217C2×C4⋊C4#6711yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—61no—
220C2×C42—61no—
221C2×C42—61no—
222#103—24no—
223#103—122no—
224#103—122no—
225C22⋊C8—21no—
226C22⋊C8—61no—
227C22⋊C8—61no—
228C22⋊C8C2×Dic312yes—
229#95—61no—
230#95—61no—
231#95—61no—
232#95—61no—
233#95—61no—
234#95—61no—
235#95—61no—
236#95—61no—
237#95—61no—
238#95—122no—
239#95—122no—
240C23×C3—21no—
241C23×C3—21no—
242C23×C3—21no—
243C23×C3—21no—
244C23×C3C4×D411yescharacteristic
245C23×C3#12011yescharacteristic
246C23×C3#12111yescharacteristic
247C23×C3C2×Q1611yescharacteristic
248C23×C3#13511yescharacteristic
249C23×C3#13711yescharacteristic
250C23×C3#13711yescharacteristic
251C3×Q8—24no—
252C3×Q8—24no—
253C3×Q8—42no—
254C3×Q8—42no—
255C3×Q8—42no—
256C3×Q8—42no—
257C3×Q8—42no—
258C3×Q8—42no—
259C3×Q8—42no—
260C3×Q8—81no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—24no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4C2×C22⋊C411yescharacteristic
283C2×C3×C4C4×D411yescharacteristic
284C2×C3×C4#12111yescharacteristic
285C2×C3×C4#12311yescharacteristic
286C2×Dic3—22no—
287C2×Dic3—41no—
288C2×Dic3—41no—
289C2×Dic3—42no—
290C2×Dic3—42no—
291C2×Dic3—42no—
292C2×Dic3—42no—
293C2×Dic3—42no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296C2×Dic3—41no—
297C2×Dic3—41no—
298C2×Dic3—41no—
299C2×Dic3—41no—
300C2×Dic3—41no—
301C2×Dic3—41no—
302C2×Dic3—41no—
303C2×Dic3—81no—
304C2×Dic3—81no—
305C2×Dic3—81no—
306C2×Dic3—81no—
307C2×Dic3—81no—
308C2×Dic3—81no—
309C3×C8—42no—
310C3⋊C8—81no—
311C24D3×D411yescharacteristic
312C2×Q8—21no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—44no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—41no—
324C2×Q8—44no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—44no—
328C2×Q8—44no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8#21712yes—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—32no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—61no—
345C22×C4—61no—
346C22×C4—62no—
347C22×C4—61no—
348C22×C4—61no—
349C22×C4—61no—
350C22×C4—61no—
351C22×C4—61no—
352C22×C4—121no—
353C22×C4—121no—
354C22×C4—121no—
355C22×C4—121no—
356C22×C4—121no—
357C22×C4—121no—
358C22×C4—121no—
359C22×C4—121no—
360C22×C4—121no—
361C22×C4—121no—
362C22×C4D3×D411yescharacteristic
363C22×C4#23711yescharacteristic
364C22×C4#23711yescharacteristic
365C22×C4C22×Dic311yescharacteristic
366C22×C4C2×#6711yescharacteristic
367C22×C4C2×#6711yescharacteristic
368Q16—48no—
369C2×C8—22no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—121no—
373C2×C8—121no—
374C2×C8—121no—
375C4⋊C4—22no—
376C4⋊C4—21no—
377C4⋊C4—22no—
378C4⋊C4—121no—
379C4⋊C4—121no—
380C4⋊C4—241no—
381C4⋊C4—241no—
382C4⋊C4#21712yes—
383C22⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—121no—
386C22⋊C4—121no—
387C42—241no—
388C42—241no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3#38611yescharacteristic
402C22×C3C4×Q1611yescharacteristic
403C22×C3#44111yescharacteristic
404C22×C3#45111yescharacteristic
405C22×C3#45211yescharacteristic
406C22×C3#45511yescharacteristic
407C22×C3#45611yescharacteristic
408C22×C3B(4,𝔽2)11yescharacteristic
409C22×C3B(4,𝔽2)11yescharacteristic
410C22×C3#46211yescharacteristic
411C22×C3#46411yescharacteristic
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—21no—
416C3×C4—42no—
417C3×C4—44no—
418C3×C4—42no—
419Dic3—42no—
420Dic3—82no—
421Dic3—81no—
422Dic3—81no—
423Dic3—81no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23D3×Q1611yescharacteristic
429C23#91811yescharacteristic
430C23#91811yescharacteristic
431C23#91911yescharacteristic
432C23D4×Dic311yescharacteristic
433C23#93711yescharacteristic
434C23#93911yescharacteristic
435Q8—24no—
436Q8—24no—
437Q8—42no—
438Q8—42no—
439Q8—42no—
440Q8—42no—
441Q8—42no—
442Q8—42no—
443Q8—42no—
444Q8—81no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—62no—
467C2×C4—121no—
468C2×C4—121no—
469C2×C4—122no—
470C2×C4—122no—
471C2×C4—122no—
472C2×C4—122no—
473C2×C4—122no—
474C2×C4—121no—
475C2×C4—121no—
476C2×C4—121no—
477C2×C4—121no—
478C2×C4—121no—
479C2×C4—121no—
480C2×C4—121no—
481C2×C4—121no—
482C2×C4—121no—
483C2×C4—241no—
484C2×C4—241no—
485C2×C4—241no—
486C2×C4—241no—
487C2×C4—241no—
488C2×C4—241no—
489C2×C4D4×Dic311yescharacteristic
490C2×C4#93511yescharacteristic
491C2×C4#93911yescharacteristic
492C2×C4C2×#21711yescharacteristic
493C8—42no—
494C8—241no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3#160211yescharacteristic
500C2×C3#160311yescharacteristic
501C2×C3#160511yescharacteristic
502C2×C3#160611yescharacteristic
503C2×C3#187811yescharacteristic
504C2×C3#188211yescharacteristic
505C2×C3#188911yescharacteristic
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22#503211yescharacteristic
519C22#503211yescharacteristic
520C22Dic3×Q1611yescharacteristic
521C22#518211yescharacteristic
522C22#518411yescharacteristic
523C22#518511yescharacteristic
524C22#518611yescharacteristic
525C22#518711yescharacteristic
526C22#520211yescharacteristic
527C22#520511yescharacteristic
528C22#522011yescharacteristic
529C4—21no—
530C4—21no—
531C4—21no—
532C4—21no—
533C4—42no—
534C4—44no—
535C4—42no—
536C4—122no—
537C4—242no—
538C4—241no—
539C4—241no—
540C4—241no—
541C3#904011yescharacteristic, complemented, Sylow
542C2—21no—
543C2—21no—
544C2—21no—
545C2—21no—
546C2#7098011yescharacteristic
547C2#7098111yescharacteristic
548C2#7098211yescharacteristic
549C2#7098311yescharacteristic
550C2#7166911yescharacteristic
551C2#7206511yescharacteristic
552C2#7210011yescharacteristic
subgroupG/H
centerC23#937
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#451C2
socleC23×C3#120
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenterC2×#451D3
hypocenterC3#9040
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#451

Complements

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

NH ≅ G/Ntype
13. C3×#451C4semidirect
93. #451Dic3semidirect
167. C22×Dic3Q16semidirect
541. C3#9040semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4512
#827452
#721342
#721322
C2×#45382
#676752
#676742
#90403