Finite Group Databasebeta

← #10690766

Subgroups — #10690766

Order 768.

612 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4719C211yesmaximal, characteristic
2#74000C211yesmaximal, characteristic
3C2×#4467C211yesmaximal, characteristic
4#69402C211yesmaximal, characteristic
5C2×#4452C211yesmaximal, characteristic
6C3×#1294C211yesmaximal, characteristic
7#67624C211yesmaximal, characteristic
8#9116—31nomaximal, Sylow
9C2×#924C2211yescharacteristic
10C22×C4×Dic3C2211yescharacteristic
11C2×C3×C22⋊C8C2211yescharacteristic
12C3×#380C2211yescharacteristic
13C2×C3×#95C2211yescharacteristic
14C2×C3⋊#95—21no—
15C2×C3⋊#95—21no—
16C2×C3⋊#95C2211yescharacteristic
17C2×#830C2211yescharacteristic
18C2×#815—21no—
19C2×Dic3⋊C8—21no—
20#4984C411yescharacteristic, complemented
21#4984C411yescharacteristic, complemented
22#4973C411yescharacteristic, complemented
23#4973C411yescharacteristic, complemented
24#4950—21no—
25#4943—21no—
26#4719—22no—
27#4719—22no—
28#4653—21no—
29#4653—21no—
30#4467—22no—
31#4452—22no—
32C2×#432—31no—
33#2855—31no—
34C2×#328—31no—
35C2×#328—31no—
36#1434—31no—
37#1294—31no—
38#1294D311yescharacteristic
39C23×C3×C4—21no—
40C23×C3×C4C2311yescharacteristic
41C23×Dic3—21no—
42C23×Dic3D411yescharacteristic
43C23×Dic3D411yescharacteristic
44C22×Dic6C2×C411yescharacteristic
45C22×C3×C8—21no—
46C2×C3×C4⋊C4C2×C411yescharacteristic
47C2×C3×C4⋊C4C2×C411yescharacteristic
48C2×C3×C22⋊C4—21no—
49C2×#217—21no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3C2×C411yescharacteristic
53#924—21no—
54#924—21no—
55#924—21no—
56#924—21no—
57C2×Dic3⋊C4—21no—
58C2×C4×Dic3—21no—
59C2×C4×Dic3—21no—
60C2×C4×Dic3—21no—
61C2×C4×Dic3—21no—
62C2×C4×Dic3C2×C411yescharacteristic
63C2×C4×Dic3C2×C411yescharacteristic
64C2×C4×Dic3D411yescharacteristic
65C2×C4×Dic3D411yescharacteristic
66C22×C3⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×C22⋊C8D412yes—
69C3×#95—21no—
70C3×#95—21no—
71C3×#95—21no—
72C3⋊#95—24no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81C3⋊#95—42no—
82#830—21no—
83#830D412yes—
84#815—24no—
85Dic3⋊C8—24no—
86C2×C22⋊Q8—31no—
87C22×C42—31no—
88#432—62no—
89#432—62no—
90C2×C4⋊C8—61no—
91C2×C22⋊C8—31no—
92C2×C22⋊C8C2×D311yescharacteristic
93C2×#97—61no—
94#391—31no—
95#391—31no—
96#389—31no—
97#389—31no—
98#387—61no—
99#387—61no—
100#385—61no—
101#381—61no—
102#380C2×D311yescharacteristic
103C2×#95—31no—
104C2×#95—61no—
105C2×#95—61no—
106C2×#95C2×D311yescharacteristic
107#328—62no—
108#328—62no—
109C24×C3C2×D411yescharacteristic
110C22×C3×C4—22no—
111C22×C3×C4—21no—
112C22×C3×C4—21no—
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×C4—41no—
121C22×C3×C4C22×C411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C22×C3×C4#4111yescharacteristic
127C22×Dic3—21no—
128C22×Dic3—21no—
129C22×Dic3—21no—
130C22×Dic3—21no—
131C22×Dic3—21no—
132C22×Dic3—21no—
133C22×Dic3—21no—
134C22×Dic3—21no—
135C22×Dic3—22no—
136C22×Dic3—22no—
137C22×Dic3—21no—
138C22×Dic3—21no—
139C22×Dic3—41no—
140C22×Dic3—41no—
141C22×Dic3—41no—
142C22×Dic3—41no—
143C22×Dic3—41no—
144C22×Dic3—41no—
145C22×Dic3—41no—
146C22×Dic3C22⋊C411yescharacteristic
147C22×Dic3C22⋊C411yescharacteristic
148C22×Dic3C22⋊C411yescharacteristic
149C22×Dic3C22⋊C411yescharacteristic
150C2×Dic6—21no—
151C2×Dic6—21no—
152C2×Dic6—41no—
153C2×Dic6—41no—
154C2×C3×C8—22no—
155C2×C3×C8—41no—
156C2×C3×C8—41no—
157C3×C4⋊C4—41no—
158C3×C4⋊C4—41no—
159C3×C22⋊C4—41no—
160C3×C22⋊C4—41no—
161#217—41no—
162#217—41no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165C4⋊Dic3—42no—
166C4⋊Dic3—42no—
167Dic3⋊C4—41no—
168Dic3⋊C4—41no—
169C4×Dic3—21no—
170C4×Dic3—21no—
171C4×Dic3—21no—
172C4×Dic3—21no—
173C4×Dic3—21no—
174C4×Dic3—21no—
175C4×Dic3—21no—
176C4×Dic3—21no—
177C2×C3⋊C8—22no—
178C2×C3⋊C8—41no—
179C2×C3⋊C8—41no—
180C22×Q8—31no—
181C23×C4—21no—
182C23×C4—31no—
183C23×C4—31no—
184C23×C4—61no—
185C23×C4C22×D311yescharacteristic
186C22×C8—21no—
187C22×C8—61no—
188C22⋊Q8—61no—
189C22⋊Q8—61no—
190C22⋊Q8—61no—
191C22⋊Q8—61no—
192C2×C4⋊C4—31no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4—61no—
195C2×C4⋊C4—61no—
196C2×C4⋊C4C4×D311yescharacteristic
197C2×C4⋊C4C4×D311yescharacteristic
198C2×C22⋊C4—21no—
199C2×C22⋊C4—61no—
200C2×C42—31no—
201C2×C42—31no—
202C2×C42—31no—
203C2×C42—31no—
204C2×C42—61no—
205C2×C42—61no—
206C2×C42—61no—
207C2×C42—61no—
208C4⋊C8—64no—
209C22⋊C8—21no—
210C22⋊C8—32no—
211C22⋊C8—61no—
212C22⋊C8D1212yes—
213#97—64no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—61no—
218#95—61no—
219#95—61no—
220#95—61no—
221#95—61no—
222#95—61no—
223#95—64no—
224#95—61no—
225#95—61no—
226#95—122no—
227C23×C3—21no—
228C23×C3—21no—
229C23×C3—21no—
230C23×C3—21no—
231C23×C3C2×C22⋊C411yescharacteristic
232C23×C3#12111yescharacteristic
233C23×C3#12111yescharacteristic
234C23×C3AGL(1,ℤ/8)11yescharacteristic
235C23×C3#13711yescharacteristic
236C23×C3#13711yescharacteristic
237C23×C3#13711yescharacteristic
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—81no—
256C2×C3×C4—81no—
257C2×C3×C4—81no—
258C2×C3×C4—81no—
259C2×C3×C4C4×D411yescharacteristic
260C2×C3×C4C4×D411yescharacteristic
261C2×C3×C4#12011yescharacteristic
262C2×C3×C4#12311yescharacteristic
263C2×Dic3—21no—
264C2×Dic3—21no—
265C2×Dic3—21no—
266C2×Dic3—21no—
267C2×Dic3—21no—
268C2×Dic3—21no—
269C2×Dic3—21no—
270C2×Dic3—21no—
271C2×Dic3—21no—
272C2×Dic3—21no—
273C2×Dic3—21no—
274C2×Dic3—42no—
275C2×Dic3—42no—
276C2×Dic3—41no—
277C2×Dic3—41no—
278C2×Dic3—41no—
279C2×Dic3—41no—
280C2×Dic3—42no—
281C2×Dic3—42no—
282C2×Dic3—42no—
283C2×Dic3—41no—
284C2×Dic3—41no—
285C2×Dic3—41no—
286C2×Dic3—41no—
287C2×Dic3—41no—
288C2×Dic3—41no—
289C2×Dic3—81no—
290C2×Dic3—81no—
291C2×Dic3—81no—
292C2×Dic3—81no—
293C2×Dic3#10412yes—
294Dic6—81no—
295Dic6—81no—
296C3×C8—42no—
297C3⋊C8—42no—
298C24D3×D411yescharacteristic
299C2×Q8—61no—
300C2×Q8—61no—
301C2×Q8—121no—
302C2×Q8—121no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—31no—
307C22×C4—31no—
308C22×C4—31no—
309C22×C4—31no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—62no—
319C22×C4—61no—
320C22×C4—61no—
321C22×C4—61no—
322C22×C4—61no—
323C22×C4—61no—
324C22×C4—61no—
325C22×C4—61no—
326C22×C4—61no—
327C22×C4—62no—
328C22×C4—61no—
329C22×C4—61no—
330C22×C4—121no—
331C22×C4—121no—
332C22×C4—121no—
333C22×C4—121no—
334C22×C4—121no—
335C22×C4—121no—
336C22×C4—121no—
337C22×C4C2×C4×D311yescharacteristic
338C22×C4C2×D1211yescharacteristic
339C22×C4#23511yescharacteristic
340C22×C4D3×D411yescharacteristic
341C22×C4D3×D411yescharacteristic
342C22×C4#23911yescharacteristic
343C2×C8—22no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—62no—
347C2×C8—121no—
348C2×C8—121no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—61no—
352C4⋊C4—61no—
353C4⋊C4—122no—
354C4⋊C4—122no—
355C4⋊C4—121no—
356C4⋊C4—121no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—121no—
360C22⋊C4—121no—
361C42—61no—
362C42—61no—
363C42—61no—
364C42—61no—
365C42—61no—
366C42—61no—
367C42—61no—
368C42—61no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3#38611yescharacteristic
382C22×C3#41011yescharacteristic
383C22×C3#41011yescharacteristic
384C22×C3#41711yescharacteristic
385C22×C3#41911yescharacteristic
386C22×C3C2×#10411yescharacteristic
387C22×C3#42111yescharacteristic
388C22×C3#46811yescharacteristic
389C22×C3#46811yescharacteristic
390C22×C3#47011yescharacteristic
391C22×C3#47011yescharacteristic
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—21no—
396C3×C4—82no—
397C3×C4—81no—
398C3×C4—81no—
399Dic3—22no—
400Dic3—41no—
401Dic3—41no—
402Dic3—41no—
403Dic3—82no—
404Dic3—81no—
405Dic3—81no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23D3×C22⋊C411yescharacteristic
411C23#88411yescharacteristic
412C23#89511yescharacteristic
413C23#91111yescharacteristic
414C23#91511yescharacteristic
415C23#91511yescharacteristic
416C23#91811yescharacteristic
417Q8—241no—
418Q8—241no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—32no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—61no—
438C2×C4—61no—
439C2×C4—61no—
440C2×C4—61no—
441C2×C4—61no—
442C2×C4—61no—
443C2×C4—61no—
444C2×C4—61no—
445C2×C4—61no—
446C2×C4—61no—
447C2×C4—61no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—122no—
453C2×C4—122no—
454C2×C4—121no—
455C2×C4—121no—
456C2×C4—121no—
457C2×C4—121no—
458C2×C4—122no—
459C2×C4—122no—
460C2×C4—122no—
461C2×C4—121no—
462C2×C4—121no—
463C2×C4—121no—
464C2×C4—121no—
465C2×C4—121no—
466C2×C4—121no—
467C2×C4—241no—
468C2×C4—241no—
469C2×C4—241no—
470C2×C4—241no—
471C2×C4C4×D1211yescharacteristic
472C2×C4#88211yescharacteristic
473C2×C4#89311yescharacteristic
474C2×C4#89411yescharacteristic
475C8—42no—
476C8—122no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3#150611yescharacteristic
482C2×C3#150911yescharacteristic
483C2×C3#151211yescharacteristic
484C2×C3#151511yescharacteristic
485C2×C3#187111yescharacteristic
486C2×C3#189411yescharacteristic
487C2×C3#189911yescharacteristic
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22#466511yescharacteristic
501C22#474011yescharacteristic
502C22#474011yescharacteristic
503C22#477111yescharacteristic
504C22#478011yescharacteristic
505C22#480211yescharacteristic
506C22#481611yescharacteristic
507C22D3×#10411yescharacteristic
508C22#482111yescharacteristic
509C22#486711yescharacteristic
510C22#488611yescharacteristic
511C4—21no—
512C4—21no—
513C4—21no—
514C4—21no—
515C4—62no—
516C4—82no—
517C4—81no—
518C4—81no—
519C4—121no—
520C4—121no—
521C4—121no—
522C4—242no—
523C4—241no—
524C4—241no—
525C3#911611yescharacteristic, complemented, Sylow
526C2—21no—
527C2—21no—
528C2—21no—
529C2—21no—
530C2#6827911yescharacteristic
531C2#6828011yescharacteristic
532C2#6831011yescharacteristic
533C2#6831311yescharacteristic
534C2#6875511yescharacteristic
535C2#6888411yescharacteristic
536C2#6895411yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1294C2
socleC23×C3C2×C22⋊C4
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#1294D3
hypocenterC3#9116
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
20. #4984C4semidirect
21. #4984C4semidirect
22. #4973C4semidirect
23. #4973C4semidirect
525. C3#9116semidirect

Maximal subgroups

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

subgroupindex
C2×#47192
#740002
C2×#44672
#694022
C2×#44522
C3×#12942
#676242
#91163