Finite Group Databasebeta

← #10695742

Subgroups — #10695742

Order 768.

1020 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 1020 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82789C211yesmaximal, characteristic, complemented
2C2×#5046C211yesmaximal, characteristic
3C2×#4731C211yesmaximal, characteristic
4C2×C3×#328C211yesmaximal, characteristic, complemented
5#69426C211yesmaximal, characteristic
6C2×#4452C211yesmaximal, characteristic, complemented
7#67674C211yesmaximal, characteristic, complemented
8#9966—31nomaximal, Sylow
9C23×D12C2211yescharacteristic
10C2×C3×C22⋊Q8C2211yescharacteristic
11C22×#215—21no—
12C22×(C2×D3)⋊C4—22no—
13C2×#924C2211yescharacteristic
14C22×#204—21no—
15C2×C3×C22⋊C8C2211yescharacteristic
16C2×C3×#95C2211yescharacteristic
17#5230—21no—
18C2×C3⋊#95C2211yescharacteristic
19C2×#830C2211yescharacteristic
20C2×#821—21no—
21#5046—22no—
22#5046—22no—
23#4983—21no—
24#4981—21no—
25#4979—21no—
26C2×#809—21no—
27#4942—22no—
28#4731—22no—
29#4731—22no—
30C3×#328—22no—
31#4452—22no—
32C2×#450—31no—
33C2×#450—31no—
34#2659—31no—
35C2×#328—31no—
36C2×#328D311yescharacteristic, complemented
37#1438—31no—
38#1280—31no—
39C24×D3—21no—
40C22×C3×Q8D411yescharacteristic
41C23×C3×C4—21no—
42C23×C3×C4C2311yescharacteristic
43C23×Dic3D412yes—
44C22×D12—21no—
45C22×D12—21no—
46C22×D12—21no—
47C22×D12—21no—
48C22×D12—21no—
49C22×D12—44no—
50C22×D12D412yes—
51C22×Dic6D411yescharacteristic
52C22×C3×C8—21no—
53C3×C22⋊Q8—21no—
54C3×C22⋊Q8—21no—
55C3×C22⋊Q8—21no—
56C3×C22⋊Q8—21no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C22⋊C4—21no—
60C2×#217—21no—
61C2×#215—24no—
62C2×#215—44no—
63C2×(C2×D3)⋊C4—44no—
64C2×(C2×D3)⋊C4—44no—
65C2×(C2×D3)⋊C4—44no—
66C2×C4⋊Dic3D411yescharacteristic
67#924—21no—
68#924—21no—
69#924—21no—
70#924—21no—
71C2×Dic3⋊C4—21no—
72C2×Dic3⋊C4—22no—
73C2×Dic3⋊C4—22no—
74C22×C3⋊C8—21no—
75C2×#204—24no—
76C2×#204—44no—
77C3×C22⋊C8—21no—
78C3×C22⋊C8D412yes—
79C3×#95—21no—
80C3×#95—21no—
81C3⋊#95—22no—
82C3⋊#95—22no—
83C3⋊#95—21no—
84C3⋊#95—21no—
85#830—21no—
86#830D412yes—
87#821—24no—
88#809—24no—
89C23×D4—31no—
90C22×SD16—61no—
91C22×SD16—61no—
92C2×C22⋊Q8—31no—
93C2×C22⋊Q8C2×D311yescharacteristic
94C22×C22⋊C4—62no—
95#450—62no—
96#450—62no—
97#450—62no—
98#450—62no—
99C2×#102—61no—
100C2×#102—61no—
101C2×C22⋊C8—31no—
102C2×C22⋊C8C2×D311yescharacteristic
103#398—61no—
104#398—61no—
105#395—61no—
106#395—61no—
107#380—62no—
108C2×#95—31no—
109C2×#95C2×D311yescharacteristic
110#328—22no—
111#328—62no—
112C24×C3C2×D411yescharacteristic
113C23×D3—24no—
114C23×D3—22no—
115C23×D3—42no—
116C23×D3—42no—
117C23×D3—42no—
118C23×D3—41no—
119C23×D3—41no—
120C23×D3—42no—
121C23×D3—42no—
122C2×C3×Q8—21no—
123C2×C3×Q8—21no—
124C2×C3×Q8—42no—
125C22×C3×C4—22no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—42no—
129C22×C3×C4—42no—
130C22×C3×C4—41no—
131C22×C3×C4—41no—
132C22×C3×C4—42no—
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4#4111yescharacteristic
139C22×Dic3—22no—
140C22×Dic3—22no—
141C22×Dic3—21no—
142C22×Dic3—21no—
143C22×Dic3—42no—
144C22×Dic3—42no—
145C22×Dic3SD1614yescomplemented
146C2×D12—24no—
147C2×D12—24no—
148C2×D12—22no—
149C2×D12—22no—
150C2×D12—44no—
151C2×D12—44no—
152C2×D12—44no—
153C2×D12—42no—
154C2×D12—42no—
155C2×D12—44no—
156C2×D12—41no—
157C2×D12—41no—
158C2×Dic6—21no—
159C2×Dic6—21no—
160C2×Dic6—42no—
161C2×C3×C8—22no—
162C2×C3×C8—42no—
163C3×C4⋊C4—21no—
164C3×C4⋊C4—21no—
165C3×C4⋊C4—41no—
166C3×C4⋊C4—41no—
167C3×C22⋊C4—41no—
168C3×C22⋊C4—41no—
169#217—41no—
170#217—41no—
171#215—48no—
172(C2×D3)⋊C4—416no—
173C4⋊Dic3—21no—
174C4⋊Dic3—21no—
175Dic3⋊C4—41no—
176Dic3⋊C4—41no—
177Dic3⋊C4—82no—
178Dic3⋊C4—82no—
179C2×C3⋊C8—22no—
180C2×C3⋊C8—42no—
181#204—48no—
182C25—61no—
183C22×Q8—31no—
184C22×Q8D1211yescharacteristic
185C22×D4—32no—
186C22×D4—61no—
187C22×D4—61no—
188C22×D4—61no—
189C22×D4—61no—
190C22×D4—61no—
191C22×D4—124no—
192C23×C4—21no—
193C23×C4—32no—
194C23×C4C22×D311yescharacteristic
195C2×SD16—64no—
196C2×SD16—64no—
197C2×SD16—124no—
198C2×SD16—124no—
199C22×C8—21no—
200C22×C8—61no—
201C22⋊Q8—21no—
202C22⋊Q8—21no—
203C22⋊Q8—21no—
204C22⋊Q8—21no—
205C22⋊Q8—61no—
206C22⋊Q8—61no—
207C22⋊Q8—61no—
208C22⋊Q8—61no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—31no—
211C2×C4⋊C4—61no—
212C2×C4⋊C4—62no—
213C2×C4⋊C4—62no—
214C2×C4⋊C4D1211yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—61no—
217C2×C22⋊C4—124no—
218C2×C22⋊C4—124no—
219C2×C22⋊C4—124no—
220#102—64no—
221#102—64no—
222C22⋊C8—21no—
223C22⋊C8—32no—
224C22⋊C8—61no—
225C22⋊C8#6712yes—
226#95—21no—
227#95—21no—
228#95—62no—
229#95—62no—
230#95—61no—
231#95—61no—
232C23×C3—21no—
233C23×C3—21no—
234C23×C3—22no—
235C23×C3#12011yescharacteristic
236C23×C3#12111yescharacteristic
237C23×C3#12711yescharacteristic
238C23×C3C2×SD1612yes—
239C23×C3#13712yes—
240C22×D3—24no—
241C22×D3—44no—
242C22×D3—44no—
243C22×D3—42no—
244C22×D3—44no—
245C22×D3—44no—
246C22×D3—44no—
247C22×D3—44no—
248C22×D3—42no—
249C22×D3—44no—
250C22×D3—44no—
251C22×D3—82no—
252C22×D3—82no—
253C22×D3—82no—
254C22×D3—82no—
255C22×D3—82no—
256C22×D3—82no—
257C22×D3—81no—
258C22×D3—81no—
259C22×D3—81no—
260C22×D3—81no—
261C3×Q8—42no—
262C3×Q8—81no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—44no—
270C2×C3×C4—42no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—42no—
276C2×C3×C4—82no—
277C2×C3×C4—81no—
278C2×C3×C4—81no—
279C2×C3×C4#12011yescharacteristic
280C2×C3×C4#12011yescharacteristic
281C2×C3×C4#12111yescharacteristic
282C2×C3×C4#12111yescharacteristic
283C2×Dic3—24no—
284C2×Dic3—42no—
285C2×Dic3—42no—
286C2×Dic3—44no—
287C2×Dic3—41no—
288C2×Dic3—41no—
289C2×Dic3—41no—
290C2×Dic3—41no—
291C2×Dic3—42no—
292C2×Dic3—82no—
293D12—24no—
294D12—24no—
295D12—42no—
296D12—42no—
297D12—44no—
298D12—44no—
299Dic6—42no—
300Dic6—81no—
301C3×C8—42no—
302C3⋊C8—42no—
303C24—64no—
304C24—62no—
305C24—122no—
306C24—122no—
307C24—122no—
308C24—121no—
309C24—122no—
310C24—122no—
311C24—121no—
312C24D3×D411yescharacteristic
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—42no—
316C2×Q8—61no—
317C2×Q8—61no—
318C2×Q8—122no—
319C2×D4—64no—
320C2×D4—64no—
321C2×D4—62no—
322C2×D4—62no—
323C2×D4—122no—
324C2×D4—122no—
325C2×D4—124no—
326C2×D4—124no—
327C2×D4—124no—
328C2×D4—124no—
329C2×D4—121no—
330C2×D4—121no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—34no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—62no—
341C22×C4—62no—
342C22×C4—61no—
343C22×C4—61no—
344C22×C4—122no—
345C22×C4—122no—
346C22×C4C2×D1211yescharacteristic
347C22×C4#23511yescharacteristic
348C22×C4D3×D411yescharacteristic
349C22×C4D3×D411yescharacteristic
350C22×C4D3×D411yescharacteristic
351C22×C4C2×#6711yescharacteristic
352SD16—128no—
353SD16—128no—
354C2×C8—22no—
355C2×C8—42no—
356C2×C8—62no—
357C2×C8—122no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—61no—
363C4⋊C4—61no—
364C4⋊C4—121no—
365C4⋊C4—121no—
366C4⋊C4—242no—
367C4⋊C4—242no—
368C22⋊C4—41no—
369C22⋊C4—41no—
370C22⋊C4—1216no—
371C22⋊C4—121no—
372C22⋊C4—121no—
373C22×C3—22no—
374C22×C3—22no—
375C22×C3—22no—
376C22×C3—22no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3#39211yescharacteristic
382C22×C3#44812yes—
383C22×C3#45311yescharacteristic
384C22×C3#45411yescharacteristic
385C22×C3B(4,𝔽2)12yes—
386C22×C3#46112yes—
387C22×C3#49211yescharacteristic
388C22×C3#49711yescharacteristic
389C2×D3—48no—
390C2×D3—44no—
391C2×D3—84no—
392C2×D3—84no—
393C2×D3—84no—
394C2×D3—84no—
395C2×D3—84no—
396C2×D3—82no—
397C2×D3—82no—
398C3×C4—21no—
399C3×C4—21no—
400C3×C4—21no—
401C3×C4—21no—
402C3×C4—82no—
403C3×C4—81no—
404C3×C4—81no—
405Dic3—44no—
406Dic3—81no—
407Dic3—81no—
408C23—22no—
409C23—21no—
410C23—21no—
411C23—64no—
412C23—124no—
413C23—124no—
414C23—122no—
415C23—124no—
416C23—124no—
417C23—124no—
418C23—124no—
419C23—122no—
420C23—124no—
421C23—124no—
422C23—242no—
423C23—242no—
424C23—242no—
425C23—242no—
426C23—242no—
427C23—241no—
428C23—241no—
429C23—242no—
430C23—241no—
431C23—241no—
432C23#88211yescharacteristic
433C23#88411yescharacteristic
434C23D3×SD1612yes—
435C23#91812yes—
436C23#94011yescharacteristic
437Q8—42no—
438Q8—81no—
439Q8—122no—
440Q8—241no—
441D4—64no—
442D4—64no—
443D4—122no—
444D4—122no—
445D4—124no—
446D4—124no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—64no—
461C2×C4—81no—
462C2×C4—82no—
463C2×C4—81no—
464C2×C4—122no—
465C2×C4—122no—
466C2×C4—124no—
467C2×C4—121no—
468C2×C4—121no—
469C2×C4—122no—
470C2×C4—121no—
471C2×C4—121no—
472C2×C4—242no—
473C2×C4#88211yescharacteristic
474C2×C4#88411yescharacteristic
475C2×C4#93011yescharacteristic
476C2×C4#93711yescharacteristic
477C8—42no—
478C8—122no—
479C2×C3—21no—
480C2×C3—22no—
481C2×C3—21no—
482C2×C3#159612yes—
483C2×C3#159912yes—
484C2×C3#200611yescharacteristic
485C2×C3#201311yescharacteristic
486C2×C3#202111yescharacteristic
487D3—84no—
488C22—21no—
489C22—21no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—21no—
495C22—21no—
496C22—128no—
497C22—124no—
498C22—244no—
499C22—244no—
500C22—244no—
501C22—242no—
502C22—244no—
503C22—244no—
504C22—242no—
505C22#474112yes—
506C22#480612yes—
507C22#481812yes—
508C22#482211yescharacteristic
509C22#482711yescharacteristic
510C22#496011yescharacteristic
511C22#517511yescharacteristic
512C22#519011yescharacteristic
513C4—21no—
514C4—21no—
515C4—21no—
516C4—21no—
517C4—82no—
518C4—81no—
519C4—81no—
520C4—124no—
521C4—241no—
522C4—241no—
523C3#996611yescharacteristic, complemented, Sylow
524C2—21no—
525C2—22no—
526C2—21no—
527C2—244no—
528C2#6831612yes—
529C2#6832512yes—
530C2#7003711yescharacteristic
531C2#7028211yescharacteristic
532C2#7039211yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#328C2
socleC23×C3#120
prime radicalC23×D12C22
hypercenterC2×#328D3
hypocenterC3#9966
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
1. #82789C2semidirect
4. C2×C3×#328C2semidirect
6. C2×#4452C2semidirect
7. #67674C2semidirect
36. C2×#328D3semidirect
145. C22×Dic3SD16semidirect
523. C3#9966semidirect

Maximal subgroups

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

subgroupindex
#827892
C2×#50462
C2×#47312
C2×C3×#3282
#694262
C2×#44522
#676742
#99663