Finite Group Databasebeta

← #10698539

Subgroups — #10698539

Order 768.

576 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82789C211yesmaximal, characteristic
2C2×C3×#328C211yesmaximal, characteristic
3#71692C211yesmaximal, characteristic
4#69930C211yesmaximal, characteristic
5#69412C211yesmaximal, characteristic
6C2×#4452C211yesmaximal, characteristic
7#67674C211yesmaximal, characteristic
8#10407—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×C4⋊Dic3C2211yescharacteristic
11C2×#924C2211yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13C2×C3×#95C2211yescharacteristic
14#5230—21no—
15C2×#835—21no—
16C2×C3⋊#95—21no—
17C2×C3⋊#95—21no—
18C2×C3⋊#95C2211yescharacteristic
19C2×#830C2211yescharacteristic
20C2×#816—21no—
21#4983—21no—
22#4981—21no—
23#4979—21no—
24#4942—21no—
25#4942—21no—
26C3×#328—22no—
27#4551—21no—
28#4529—21no—
29#4452—22no—
30#2659—31no—
31#1872—31no—
32#1872—31no—
33C2×#328—31no—
34C2×#328D311yescharacteristic
35#1442—31no—
36#1280—31no—
37C22×C3×Q8D411yescharacteristic
38C23×C3×C4—21no—
39C23×C3×C4C2311yescharacteristic
40C23×Dic3—21no—
41C23×Dic3D411yescharacteristic
42C23×Dic3D411yescharacteristic
43C22×Dic6D411yescharacteristic
44C22×C3×C8—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C22⋊C4—21no—
52C2×#217—21no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3—42no—
56C2×C4⋊Dic3D411yescharacteristic
57#924—21no—
58#924—21no—
59#924—21no—
60#924—21no—
61C2×Dic3⋊C4—21no—
62C2×Dic3⋊C4—21no—
63C2×Dic3⋊C4—21no—
64C2×Dic3⋊C4—21no—
65C2×Dic3⋊C4—21no—
66C22×C3⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×#95—21no—
70C3×#95D412yes—
71#835—42no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—42no—
79C3⋊#95—42no—
80#830—21no—
81#830—21no—
82#816—42no—
83C2×C22⋊Q8—31no—
84C2×C22⋊Q8C2×D311yescharacteristic
85C22×C4⋊C4—31no—
86C2×#103—61no—
87C2×#103—61no—
88C2×C22⋊C8—31no—
89C2×C22⋊C8C2×D311yescharacteristic
90#398—61no—
91#398—61no—
92#395—61no—
93#395—61no—
94#380—61no—
95#380—61no—
96C2×#95—31no—
97C2×#95—61no—
98C2×#95—61no—
99C2×#95C2×D311yescharacteristic
100#340—61no—
101#340—61no—
102#328—22no—
103#328—62no—
104C24×C3C2×D411yescharacteristic
105C2×C3×Q8—21no—
106C2×C3×Q8—21no—
107C2×C3×Q8—41no—
108C2×C3×Q8—41no—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4—22no—
112C22×C3×C4—41no—
113C22×C3×C4—41no—
114C22×C3×C4—41no—
115C22×C3×C4—41no—
116C22×C3×C4—41no—
117C22×C3×C4—41no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4#4111yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C22×Dic3—21no—
127C22×Dic3—21no—
128C22×Dic3—21no—
129C22×Dic3—21no—
130C22×Dic3—22no—
131C22×Dic3—21no—
132C22×Dic3—21no—
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—
143C22×Dic3SD1612yes—
144C22×Dic3Q1612yes—
145C2×Dic6—21no—
146C2×Dic6—21no—
147C2×Dic6—41no—
148C2×Dic6—41no—
149C2×C3×C8—41no—
150C2×C3×C8—41no—
151C2×C3×C8—41no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—41no—
155C3×C4⋊C4—41no—
156C3×C22⋊C4—41no—
157C3×C22⋊C4—41no—
158#217—41no—
159#217—41no—
160C4⋊Dic3—22no—
161C4⋊Dic3—22no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164C4⋊Dic3—41no—
165C4⋊Dic3—41no—
166Dic3⋊C4—41no—
167Dic3⋊C4—41no—
168Dic3⋊C4—81no—
169Dic3⋊C4—81no—
170Dic3⋊C4—81no—
171Dic3⋊C4—81no—
172C2×C3⋊C8—41no—
173C2×C3⋊C8—41no—
174C2×C3⋊C8—41no—
175C22×Q8—31no—
176C22×Q8D1211yescharacteristic
177C23×C4—21no—
178C23×C4—31no—
179C23×C4—31no—
180C23×C4—61no—
181C23×C4C22×D311yescharacteristic
182C22×C8—21no—
183C22×C8—61no—
184C22⋊Q8—21no—
185C22⋊Q8—21no—
186C22⋊Q8—21no—
187C22⋊Q8—21no—
188C22⋊Q8—61no—
189C22⋊Q8—61no—
190C22⋊Q8—61no—
191C22⋊Q8—61no—
192C2×C4⋊C4—21no—
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—61no—
200C2×C4⋊C4—61no—
201C2×C4⋊C4—122no—
202C2×C4⋊C4D1211yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—61no—
205#103—122no—
206#103—122no—
207C22⋊C8—21no—
208C22⋊C8—21no—
209C22⋊C8—61no—
210C22⋊C8—61no—
211#95—21no—
212#95—61no—
213#95—61no—
214#95—61no—
215#95—61no—
216#95—61no—
217#95—61no—
218#95—122no—
219#95—122no—
220#95#6712yes—
221C23×C3—21no—
222C23×C3—21no—
223C23×C3—21no—
224C23×C3—21no—
225C23×C3#12011yescharacteristic
226C23×C3#12111yescharacteristic
227C23×C3#12411yescharacteristic
228C23×C3C2×SD1611yescharacteristic
229C23×C3C2×Q1611yescharacteristic
230C23×C3#13711yescharacteristic
231C23×C3#13711yescharacteristic
232C3×Q8—42no—
233C3×Q8—81no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—42no—
251C2×C3×C4—81no—
252C2×C3×C4—81no—
253C2×C3×C4—81no—
254C2×C3×C4—81no—
255C2×C3×C4#12111yescharacteristic
256C2×C3×C4#12111yescharacteristic
257C2×C3×C4#12311yescharacteristic
258C2×C3×C4#12311yescharacteristic
259C2×Dic3—22no—
260C2×Dic3—22no—
261C2×Dic3—41no—
262C2×Dic3—41no—
263C2×Dic3—41no—
264C2×Dic3—41no—
265C2×Dic3—42no—
266C2×Dic3—42no—
267C2×Dic3—42no—
268C2×Dic3—42no—
269C2×Dic3—41no—
270C2×Dic3—41no—
271C2×Dic3—41no—
272C2×Dic3—41no—
273C2×Dic3—41no—
274C2×Dic3—42no—
275C2×Dic3—41no—
276C2×Dic3—81no—
277C2×Dic3—81no—
278C2×Dic3—81no—
279C2×Dic3—81no—
280C2×Dic3—81no—
281C2×Dic3—81no—
282Dic6—42no—
283Dic6—81no—
284C3×C8—81no—
285C3⋊C8—81no—
286C24D3×D411yescharacteristic
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—61no—
292C2×Q8—61no—
293C2×Q8—121no—
294C2×Q8—121no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—32no—
299C22×C4—32no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—61no—
309C22×C4—61no—
310C22×C4—61no—
311C22×C4—61no—
312C22×C4—62no—
313C22×C4—61no—
314C22×C4—61no—
315C22×C4—121no—
316C22×C4—121no—
317C22×C4—121no—
318C22×C4—121no—
319C22×C4—121no—
320C22×C4—121no—
321C22×C4—121no—
322C22×C4—121no—
323C22×C4—121no—
324C22×C4—121no—
325C22×C4C2×D1211yescharacteristic
326C22×C4#23511yescharacteristic
327C22×C4D3×D411yescharacteristic
328C22×C4#23711yescharacteristic
329C22×C4#23711yescharacteristic
330C22×C4C2×#6711yescharacteristic
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—121no—
335C2×C8—121no—
336C2×C8—121no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—62no—
342C4⋊C4—62no—
343C4⋊C4—61no—
344C4⋊C4—61no—
345C4⋊C4—121no—
346C4⋊C4—121no—
347C4⋊C4—121no—
348C4⋊C4—121no—
349C4⋊C4—241no—
350C4⋊C4—241no—
351C4⋊C4—241no—
352C4⋊C4—241no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—121no—
356C22⋊C4—121no—
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—21no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3#39411yescharacteristic
370C22×C3#44811yescharacteristic
371C22×C3#45111yescharacteristic
372C22×C3#45411yescharacteristic
373C22×C3#45511yescharacteristic
374C22×C3B(4,𝔽2)11yescharacteristic
375C22×C3#45811yescharacteristic
376C22×C3#46111yescharacteristic
377C22×C3#46211yescharacteristic
378C22×C3#48711yescharacteristic
379C22×C3#49111yescharacteristic
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—82no—
385C3×C4—81no—
386C3×C4—81no—
387Dic3—42no—
388Dic3—42no—
389Dic3—82no—
390Dic3—81no—
391Dic3—81no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23#88211yescharacteristic
397C23#88511yescharacteristic
398C23D3×SD1611yescharacteristic
399C23#91511yescharacteristic
400C23D3×Q1611yescharacteristic
401C23#91811yescharacteristic
402C23#93911yescharacteristic
403Q8—42no—
404Q8—81no—
405Q8—122no—
406Q8—241no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—62no—
425C2×C4—62no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—121no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4—121no—
434C2×C4—122no—
435C2×C4—122no—
436C2×C4—122no—
437C2×C4—122no—
438C2×C4—121no—
439C2×C4—121no—
440C2×C4—122no—
441C2×C4—121no—
442C2×C4—121no—
443C2×C4—121no—
444C2×C4—121no—
445C2×C4—241no—
446C2×C4—241no—
447C2×C4—241no—
448C2×C4—241no—
449C2×C4—241no—
450C2×C4—241no—
451C2×C4#88311yescharacteristic
452C2×C4#88611yescharacteristic
453C2×C4#93011yescharacteristic
454C2×C4#93811yescharacteristic
455C8—81no—
456C8—241no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3#159911yescharacteristic
462C2×C3#160211yescharacteristic
463C2×C3#160811yescharacteristic
464C2×C3#161711yescharacteristic
465C2×C3#204011yescharacteristic
466C2×C3#204311yescharacteristic
467C2×C3#204411yescharacteristic
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22#474111yescharacteristic
481C22#474211yescharacteristic
482C22#479211yescharacteristic
483C22#479711yescharacteristic
484C22#480611yescharacteristic
485C22#480911yescharacteristic
486C22#482511yescharacteristic
487C22#482711yescharacteristic
488C22#496111yescharacteristic
489C22#516211yescharacteristic
490C22#518211yescharacteristic
491C4—21no—
492C4—21no—
493C4—21no—
494C4—21no—
495C4—82no—
496C4—81no—
497C4—81no—
498C4—122no—
499C4—122no—
500C4—241no—
501C4—241no—
502C4—242no—
503C3#1040711yescharacteristic, complemented, Sylow
504C2—21no—
505C2—21no—
506C2—21no—
507C2—21no—
508C2#6831811yescharacteristic
509C2#6832011yescharacteristic
510C2#6832511yescharacteristic
511C2#6833111yescharacteristic
512C2#7003211yescharacteristic
513C2#7024611yescharacteristic
514C2#7036211yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#328C2
socleC23×C3#120
prime radicalC24×C3C2×D4
hypercenterC2×#328D3
hypocenterC3#10407
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
503. C3#10407semidirect

Maximal subgroups

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

subgroupindex
#827892
C2×C3×#3282
#716922
#699302
#694122
C2×#44522
#676742
#104073