Finite Group Databasebeta

← #10698505

Subgroups — #10698505

Order 768.

570 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82183C211yesmaximal, characteristic
2C2×C3×#324C211yesmaximal, characteristic
3#71565C211yesmaximal, characteristic
4C2×#4467C211yesmaximal, characteristic
5#69930C211yesmaximal, characteristic
6#69928C211yesmaximal, characteristic
7#67675C211yesmaximal, characteristic
8#10401—31nomaximal, Sylow
9C22×C3×C4⋊C4C2211yescharacteristic
10C22×C4⋊Dic3C2211yescharacteristic
11C2×#924C2211yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13C2×C3×C22⋊C8C2211yescharacteristic
14#5212C2211yescharacteristic
15#5211—21no—
16#5210—21no—
17C2×C3⋊#95—21no—
18C2×C3⋊#95—21no—
19C2×C3⋊#95C2211yescharacteristic
20C2×#817—21no—
21C2×#816—21no—
22#4986—21no—
23#4982—21no—
24#4982—21no—
25#4979—21no—
26C3×#324—22no—
27#4551—21no—
28#4551—21no—
29#4467—22no—
30#2627—31no—
31#1872—31no—
32#1853—31no—
33C2×#328—31no—
34C2×#324D311yescharacteristic
35#1441—31no—
36#1294—31no—
37C23×C3×C4D411yescharacteristic
38C23×C3×C4D411yescharacteristic
39C23×C3×C4C2311yescharacteristic
40C23×Dic3—21no—
41C23×Dic3—21no—
42C22×Dic6D411yescharacteristic
43C22×C3×C8—21no—
44C22×C3×C8—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×#217—21no—
52C2×#217—21no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3—42no—
56C2×C4⋊Dic3D411yescharacteristic
57C2×C4⋊Dic3D411yescharacteristic
58C2×C4⋊Dic3D411yescharacteristic
59#924—21no—
60#924—21no—
61#924—21no—
62#924—21no—
63C2×Dic3⋊C4—21no—
64C2×Dic3⋊C4—21no—
65C2×Dic3⋊C4—21no—
66C3×C22⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×C22⋊C8D412yes—
70C3⋊#95—21no—
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—42no—
80C3⋊#95—42no—
81#817—42no—
82#816—42no—
83C2×C22⋊Q8—31no—
84C22×C4⋊C4—31no—
85C22×C4⋊C4C2×D311yescharacteristic
86C2×#106—61no—
87C2×#103—61no—
88C2×C22⋊C8C2×D311yescharacteristic
89C2×C22⋊C8C2×D311yescharacteristic
90#400—61no—
91#398—61no—
92#398—61no—
93#395—61no—
94#385—61no—
95#381—61no—
96#380—31no—
97C2×#95—31no—
98C2×#95—61no—
99C2×#95—61no—
100#340—61no—
101#340—61no—
102#328—62no—
103#324—22no—
104C24×C3C2×D411yescharacteristic
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4—21no—
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—41no—
115C22×C3×C4—41no—
116C22×C3×C4—41no—
117C22×C3×C4—41no—
118C22×C3×C4SD1612yes—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4#4111yescharacteristic
123C22×C3×C4#4111yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C22×Dic3—22no—
126C22×Dic3—22no—
127C22×Dic3—21no—
128C22×Dic3—21no—
129C22×Dic3—21no—
130C22×Dic3—21no—
131C22×Dic3—41no—
132C22×Dic3—41no—
133C22×Dic3—41no—
134C22×Dic3—41no—
135C22×Dic3—41no—
136C22×Dic3—41no—
137C22×Dic3—41no—
138C22×Dic3—41no—
139C22×Dic3—41no—
140C22×Dic3—41no—
141C22×Dic3—41no—
142C22×Dic3—41no—
143C2×Dic6—21no—
144C2×Dic6—21no—
145C2×Dic6—41no—
146C2×Dic6—41no—
147C2×C3×C8—22no—
148C2×C3×C8—41no—
149C2×C3×C8—41no—
150C2×C3×C8—41no—
151C2×C3×C8—41no—
152C2×C3×C8—41no—
153C3×C4⋊C4—41no—
154C3×C4⋊C4—41no—
155C3×C4⋊C4—41no—
156C3×C4⋊C4—41no—
157#217—41no—
158#217—41no—
159#217—41no—
160#217—41no—
161C4⋊Dic3—22no—
162C4⋊Dic3—22no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165C4⋊Dic3—41no—
166C4⋊Dic3—41no—
167C4⋊Dic3—41no—
168C4⋊Dic3—41no—
169Dic3⋊C4—41no—
170Dic3⋊C4—41no—
171Dic3⋊C4—81no—
172Dic3⋊C4—81no—
173C22×Q8—31no—
174C23×C4—61no—
175C23×C4—61no—
176C23×C4D1211yescharacteristic
177C23×C4D1211yescharacteristic
178C23×C4C22×D311yescharacteristic
179C22×C8—21no—
180C22×C8—21no—
181C22⋊Q8—61no—
182C22⋊Q8—61no—
183C22⋊Q8—61no—
184C22⋊Q8—61no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—31no—
192C2×C4⋊C4—31no—
193C2×C4⋊C4—31no—
194C2×C4⋊C4—61no—
195C2×C4⋊C4—61no—
196C2×C4⋊C4—61no—
197C2×C4⋊C4—61no—
198C2×C4⋊C4—61no—
199C2×C4⋊C4—122no—
200C2×C22⋊C4—61no—
201C2×C22⋊C4—61no—
202#106—122no—
203#103—122no—
204C22⋊C8—21no—
205C22⋊C8—21no—
206C22⋊C8—21no—
207C22⋊C8#6712yes—
208#95—61no—
209#95—61no—
210#95—61no—
211#95—61no—
212#95—61no—
213#95—61no—
214#95—61no—
215#95—61no—
216#95—61no—
217#95—122no—
218#95—122no—
219C23×C3—21no—
220C23×C3—21no—
221C23×C3—21no—
222C23×C3—21no—
223C23×C3#12011yescharacteristic
224C23×C3#12111yescharacteristic
225C23×C3#12311yescharacteristic
226C23×C3C2×SD1611yescharacteristic
227C23×C3#13511yescharacteristic
228C23×C3#13711yescharacteristic
229C23×C3#13711yescharacteristic
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—42no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—81no—
250C2×C3×C4#12111yescharacteristic
251C2×C3×C4#12111yescharacteristic
252C2×C3×C4#12311yescharacteristic
253C2×C3×C4#12411yescharacteristic
254C2×Dic3—42no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3—42no—
258C2×Dic3—42no—
259C2×Dic3—41no—
260C2×Dic3—41no—
261C2×Dic3—41no—
262C2×Dic3—41no—
263C2×Dic3—41no—
264C2×Dic3—41no—
265C2×Dic3—41no—
266C2×Dic3—41no—
267C2×Dic3—41no—
268C2×Dic3—41no—
269C2×Dic3—42no—
270C2×Dic3—41no—
271C2×Dic3—41no—
272C2×Dic3—81no—
273C2×Dic3—81no—
274C2×Dic3—81no—
275C2×Dic3—81no—
276C2×Dic3—81no—
277C2×Dic3—81no—
278C2×Dic3—81no—
279C2×Dic3—81no—
280Dic6—42no—
281Dic6—81no—
282C3×C8—42no—
283C3×C8—81no—
284C24C2×D1211yescharacteristic
285C2×Q8—61no—
286C2×Q8—61no—
287C2×Q8—121no—
288C2×Q8—121no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—62no—
303C22×C4—62no—
304C22×C4—61no—
305C22×C4—61no—
306C22×C4—61no—
307C22×C4—61no—
308C22×C4—121no—
309C22×C4—121no—
310C22×C4—121no—
311C22×C4—121no—
312C22×C4—121no—
313C22×C4—121no—
314C22×C4—121no—
315C22×C4—121no—
316C22×C4—121no—
317C22×C4—121no—
318C22×C4—121no—
319C22×C4—121no—
320C22×C4#20412yes—
321C22×C4#23511yescharacteristic
322C22×C4D3×D411yescharacteristic
323C22×C4D3×D411yescharacteristic
324C22×C4#23711yescharacteristic
325C22×C4#23711yescharacteristic
326C22×C4C2×#6711yescharacteristic
327C2×C8—22no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—62no—
338C4⋊C4—62no—
339C4⋊C4—61no—
340C4⋊C4—61no—
341C4⋊C4—121no—
342C4⋊C4—121no—
343C4⋊C4—121no—
344C4⋊C4—121no—
345C4⋊C4—121no—
346C4⋊C4—121no—
347C4⋊C4—241no—
348C4⋊C4—241no—
349C22⋊C4—121no—
350C22⋊C4—121no—
351C22⋊C4—121no—
352C22⋊C4—121no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—21no—
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3#39411yescharacteristic
366C22×C3#44811yescharacteristic
367C22×C3#45211yescharacteristic
368C22×C3#45411yescharacteristic
369C22×C3#45611yescharacteristic
370C22×C3B(4,𝔽2)11yescharacteristic
371C22×C3#45811yescharacteristic
372C22×C3#46511yescharacteristic
373C22×C3#47011yescharacteristic
374C22×C3#48311yescharacteristic
375C22×C3#48511yescharacteristic
376C3×C4—21no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—42no—
381C3×C4—81no—
382Dic3—82no—
383Dic3—82no—
384Dic3—81no—
385Dic3—81no—
386Dic3—81no—
387Dic3—81no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23#88211yescharacteristic
393C23#88611yescharacteristic
394C23C2×#20411yescharacteristic
395C23#90411yescharacteristic
396C23#90911yescharacteristic
397C23#90911yescharacteristic
398C23#93011yescharacteristic
399Q8—122no—
400Q8—241no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—81no—
421C2×C4—122no—
422C2×C4—122no—
423C2×C4—122no—
424C2×C4—122no—
425C2×C4—122no—
426C2×C4—121no—
427C2×C4—121no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4—122no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4—121no—
434C2×C4—121no—
435C2×C4—121no—
436C2×C4—121no—
437C2×C4—121no—
438C2×C4—121no—
439C2×C4—241no—
440C2×C4—241no—
441C2×C4—241no—
442C2×C4—241no—
443C2×C4—241no—
444C2×C4—241no—
445C2×C4—241no—
446C2×C4—241no—
447C2×C4#88311yescharacteristic
448C2×C4#88511yescharacteristic
449C2×C4#93811yescharacteristic
450C2×C4#93911yescharacteristic
451C8—42no—
452C8—81no—
453C2×C3—21no—
454C2×C3—21no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3#159911yescharacteristic
458C2×C3#160311yescharacteristic
459C2×C3#161111yescharacteristic
460C2×C3#161511yescharacteristic
461C2×C3#203511yescharacteristic
462C2×C3#203811yescharacteristic
463C2×C3#204411yescharacteristic
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22#472111yescharacteristic
477C22#472311yescharacteristic
478C22#473311yescharacteristic
479C22#473811yescharacteristic
480C22#474111yescharacteristic
481C22#474211yescharacteristic
482C22#475211yescharacteristic
483C22#475311yescharacteristic
484C22#496111yescharacteristic
485C22#511411yescharacteristic
486C22#513711yescharacteristic
487C4—21no—
488C4—21no—
489C4—21no—
490C4—21no—
491C4—42no—
492C4—81no—
493C4—242no—
494C4—241no—
495C4—241no—
496C4—242no—
497C4—241no—
498C4—241no—
499C3#1040111yescharacteristic, complemented, Sylow
500C2—21no—
501C2—21no—
502C2—21no—
503C2—21no—
504C2#6813211yescharacteristic
505C2#6813611yescharacteristic
506C2#6813811yescharacteristic
507C2#6814011yescharacteristic
508C2#6993611yescharacteristic
509C2#7003211yescharacteristic
510C2#7010011yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#324C2
socleC23×C3#120
prime radicalC24×C3C2×D4
hypercenterC2×#324D3
hypocenterC3#10401
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
499. C3#10401semidirect

Maximal subgroups

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

subgroupindex
#821832
C2×C3×#3242
#715652
C2×#44672
#699302
#699282
#676752
#104013