Finite Group Databasebeta

← #10695740

Subgroups — #10695740

Order 768.

1020 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 1020 conjugacy classes of subgroups — too many to draw. Only its 117 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
1C2×C3×#450C211yesmaximal, characteristic
2C2×#5218C211yesmaximal, characteristic
3#82789C211yesmaximal, characteristic, complemented
4#72131C211yesmaximal, characteristic
5C2×#4538C211yesmaximal, characteristic, complemented
6C2×#4467C211yesmaximal, characteristic, complemented
7#67674C211yesmaximal, characteristic, complemented
8#9966—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C22×C3×SD16—21no—
11C2×C3×C22⋊Q8C2211yescharacteristic
12C22×#217—22no—
13C22×#214—21no—
14C2×#924C2211yescharacteristic
15C3×#450—22no—
16C3×#450—22no—
17C2×C3×#102—21no—
18C2×C3×C22⋊C8C2211yescharacteristic
19#5230—21no—
20#5218—22no—
21#5218—22no—
22C2×#832—21no—
23C2×C3⋊#95C2211yescharacteristic
24C2×C3⋊#95C2211yescharacteristic
25C2×#830C2211yescharacteristic
26#4983—21no—
27#4981—21no—
28#4979—21no—
29#4942—22no—
30#4538—22no—
31#4467—22no—
32C2×#450—31no—
33C2×#450D311yescharacteristic
34#2659—31no—
35C2×#328—31no—
36C2×#328—31no—
37#1438—31no—
38#1280—31no—
39C25×C3—21no—
40C22×C3×Q8D411yescharacteristic
41C22×C3×D4—21no—
42C22×C3×D4—21no—
43C22×C3×D4—21no—
44C22×C3×D4—21no—
45C22×C3×D4—21no—
46C22×C3×D4—44no—
47C22×C3×D4D412yes—
48C23×C3×C4C2311yescharacteristic
49C23×Dic3—21no—
50C23×Dic3D412yes—
51C22×Dic6D411yescharacteristic
52C2×C3×SD16—24no—
53C2×C3×SD16—44no—
54C22×C3×C8—21no—
55C3×C22⋊Q8—21no—
56C3×C22⋊Q8—21no—
57C3×C22⋊Q8—21no—
58C3×C22⋊Q8—21no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4D411yescharacteristic
61C2×C3×C22⋊C4—21no—
62C2×#217—21no—
63C2×#217—44no—
64C2×#217—44no—
65C2×#217—44no—
66C2×#214—24no—
67C2×#214—44no—
68C2×C4⋊Dic3D411yescharacteristic
69#924—21no—
70#924—21no—
71#924—21no—
72#924—21no—
73C2×Dic3⋊C4—21no—
74C2×Dic3⋊C4—22no—
75C2×Dic3⋊C4—22no—
76C22×C3⋊C8—21no—
77C3×#102—24no—
78C3×C22⋊C8—21no—
79C3×C22⋊C8D412yes—
80#832—24no—
81C3⋊#95—22no—
82C3⋊#95—22no—
83C3⋊#95—21no—
84C3⋊#95—21no—
85C3⋊#95—21no—
86C3⋊#95—21no—
87#830—21no—
88#830D412yes—
89C23×D4C2×D311yescharacteristic
90C22×SD16—21no—
91C22×SD16—61no—
92C2×C22⋊Q8—31no—
93C2×C22⋊Q8C2×D311yescharacteristic
94C22×C22⋊C4—62no—
95#450—22no—
96#450—22no—
97#450—62no—
98#450—62no—
99C2×#102—21no—
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×#95—31no—
110#328—62no—
111#328—62no—
112C24×C3—24no—
113C24×C3—22no—
114C24×C3—42no—
115C24×C3—42no—
116C24×C3—42no—
117C24×C3—41no—
118C24×C3—41no—
119C24×C3—42no—
120C24×C3—42no—
121C24×C3C2×D411yescharacteristic
122C2×C3×Q8—21no—
123C2×C3×Q8—21no—
124C2×C3×Q8—42no—
125C2×C3×D4—24no—
126C2×C3×D4—24no—
127C2×C3×D4—22no—
128C2×C3×D4—22no—
129C2×C3×D4—42no—
130C2×C3×D4—42no—
131C2×C3×D4—44no—
132C2×C3×D4—44no—
133C2×C3×D4—44no—
134C2×C3×D4—44no—
135C2×C3×D4—41no—
136C2×C3×D4—41no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—42no—
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4#4111yescharacteristic
146C22×Dic3—22no—
147C22×Dic3—22no—
148C22×Dic3—22no—
149C22×Dic3—21no—
150C22×Dic3—21no—
151C22×Dic3—42no—
152C22×Dic3—42no—
153C22×Dic3—42no—
154C22×Dic3—41no—
155C22×Dic3—41no—
156C22×Dic3—42no—
157C22×Dic3SD1614yescomplemented
158C2×Dic6—21no—
159C2×Dic6—21no—
160C2×Dic6—42no—
161C3×SD16—48no—
162C2×C3×C8—22no—
163C2×C3×C8—42no—
164C3×C4⋊C4—21no—
165C3×C4⋊C4—21no—
166C3×C4⋊C4—41no—
167C3×C4⋊C4—41no—
168C3×C22⋊C4—41no—
169C3×C22⋊C4—41no—
170#217—416no—
171#217—41no—
172#217—41no—
173#214—48no—
174C4⋊Dic3—21no—
175C4⋊Dic3—21no—
176Dic3⋊C4—41no—
177Dic3⋊C4—41no—
178Dic3⋊C4—82no—
179Dic3⋊C4—82no—
180C2×C3⋊C8—22no—
181C2×C3⋊C8—42no—
182C25—21no—
183C22×Q8—31no—
184C22×Q8#6711yescharacteristic
185C22×D4—21no—
186C22×D4—21no—
187C22×D4—21no—
188C22×D4—21no—
189C22×D4—21no—
190C22×D4—44no—
191C22×D4#6712yes—
192C23×C4—32no—
193C23×C4—61no—
194C23×C4C22×D311yescharacteristic
195C2×SD16—24no—
196C2×SD16—44no—
197C2×SD16—64no—
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⋊C4#6711yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—61no—
217C2×C22⋊C4—124no—
218C2×C22⋊C4—124no—
219C2×C22⋊C4—124no—
220#102—24no—
221#102—64no—
222C22⋊C8—21no—
223C22⋊C8—32no—
224C22⋊C8—61no—
225C22⋊C8#6712yes—
226#95—62no—
227#95—62no—
228#95—61no—
229#95—61no—
230#95—61no—
231#95—61no—
232C23×C3—21no—
233C23×C3—21no—
234C23×C3—22no—
235C23×C3—24no—
236C23×C3—44no—
237C23×C3—44no—
238C23×C3—42no—
239C23×C3—44no—
240C23×C3—44no—
241C23×C3—44no—
242C23×C3—44no—
243C23×C3—42no—
244C23×C3—44no—
245C23×C3—44no—
246C23×C3—82no—
247C23×C3—82no—
248C23×C3—82no—
249C23×C3—82no—
250C23×C3—82no—
251C23×C3—82no—
252C23×C3—81no—
253C23×C3—81no—
254C23×C3—81no—
255C23×C3—81no—
256C23×C3#12011yescharacteristic
257C23×C3#12111yescharacteristic
258C23×C3#12711yescharacteristic
259C23×C3C2×SD1612yes—
260C23×C3#13712yes—
261C3×Q8—42no—
262C3×Q8—81no—
263C3×D4—24no—
264C3×D4—24no—
265C3×D4—42no—
266C3×D4—42no—
267C3×D4—44no—
268C3×D4—44no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—42no—
280C2×C3×C4#12011yescharacteristic
281C2×C3×C4#12011yescharacteristic
282C2×C3×C4#12111yescharacteristic
283C2×C3×C4#12111yescharacteristic
284C2×Dic3—24no—
285C2×Dic3—42no—
286C2×Dic3—42no—
287C2×Dic3—44no—
288C2×Dic3—44no—
289C2×Dic3—42no—
290C2×Dic3—41no—
291C2×Dic3—41no—
292C2×Dic3—41no—
293C2×Dic3—41no—
294C2×Dic3—42no—
295C2×Dic3—82no—
296C2×Dic3—82no—
297C2×Dic3—81no—
298C2×Dic3—81no—
299Dic6—42no—
300Dic6—81no—
301C3×C8—42no—
302C3⋊C8—42no—
303C24—24no—
304C24—22no—
305C24—42no—
306C24—42no—
307C24—42no—
308C24—42no—
309C24—42no—
310C24—41no—
311C24—41no—
312C24D3×D411yescharacteristic
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—42no—
316C2×Q8—61no—
317C2×Q8—61no—
318C2×Q8—122no—
319C2×D4—24no—
320C2×D4—24no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4—44no—
329C2×D4—41no—
330C2×D4—41no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—34no—
334C22×C4—42no—
335C22×C4—62no—
336C22×C4—62no—
337C22×C4—62no—
338C22×C4—61no—
339C22×C4—61no—
340C22×C4—122no—
341C22×C4—122no—
342C22×C4—122no—
343C22×C4—121no—
344C22×C4—122no—
345C22×C4—121no—
346C22×C4D3×D411yescharacteristic
347C22×C4D3×D411yescharacteristic
348C22×C4#23711yescharacteristic
349C22×C4C2×#6711yescharacteristic
350C22×C4C2×#6711yescharacteristic
351C22×C4C2×#6711yescharacteristic
352SD16—48no—
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—48no—
382C22×C3—44no—
383C22×C3—84no—
384C22×C3—84no—
385C22×C3—84no—
386C22×C3—84no—
387C22×C3—84no—
388C22×C3—82no—
389C22×C3—82no—
390C22×C3#39211yescharacteristic
391C22×C3#44812yes—
392C22×C3#45311yescharacteristic
393C22×C3#45411yescharacteristic
394C22×C3B(4,𝔽2)12yes—
395C22×C3#46112yes—
396C22×C3#49211yescharacteristic
397C22×C3#49711yescharacteristic
398C3×C4—21no—
399C3×C4—21no—
400C3×C4—21no—
401C3×C4—21no—
402C3×C4—81no—
403C3×C4—81no—
404Dic3—44no—
405Dic3—82no—
406Dic3—81no—
407Dic3—81no—
408C23—21no—
409C23—21no—
410C23—22no—
411C23—24no—
412C23—44no—
413C23—44no—
414C23—42no—
415C23—44no—
416C23—44no—
417C23—44no—
418C23—44no—
419C23—44no—
420C23—42no—
421C23—44no—
422C23—82no—
423C23—82no—
424C23—82no—
425C23—82no—
426C23—82no—
427C23—81no—
428C23—82no—
429C23—81no—
430C23—81no—
431C23—81no—
432C23D3×SD1612yes—
433C23#91512yes—
434C23#93711yescharacteristic
435C23#93911yescharacteristic
436C23#94011yescharacteristic
437Q8—42no—
438Q8—81no—
439Q8—122no—
440Q8—241no—
441D4—24no—
442D4—24no—
443D4—42no—
444D4—42no—
445D4—44no—
446D4—44no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—64no—
459C2×C4—122no—
460C2×C4—122no—
461C2×C4—124no—
462C2×C4—124no—
463C2×C4—122no—
464C2×C4—121no—
465C2×C4—121no—
466C2×C4—121no—
467C2×C4—121no—
468C2×C4—122no—
469C2×C4—242no—
470C2×C4—242no—
471C2×C4—241no—
472C2×C4—241no—
473C2×C4#93711yescharacteristic
474C2×C4#93811yescharacteristic
475C2×C4#93911yescharacteristic
476C2×C4#95311yescharacteristic
477C8—42no—
478C8—122no—
479C2×C3—21no—
480C2×C3—22no—
481C2×C3—21no—
482C2×C3—84no—
483C2×C3#159612yes—
484C2×C3#159912yes—
485C2×C3#200611yescharacteristic
486C2×C3#201311yescharacteristic
487C2×C3#202111yescharacteristic
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—48no—
497C22—44no—
498C22—84no—
499C22—84no—
500C22—84no—
501C22—84no—
502C22—84no—
503C22—82no—
504C22—82no—
505C22#503212yes—
506C22#516212yes—
507C22#516711yescharacteristic
508C22#517012yes—
509C22#517511yescharacteristic
510C22#519811yescharacteristic
511C22#520111yescharacteristic
512C22#522311yescharacteristic
513C4—21no—
514C4—21no—
515C4—21no—
516C4—21no—
517C4—81no—
518C4—81no—
519C4—124no—
520C4—242no—
521C4—241no—
522C4—241no—
523C3#996611yescharacteristic, complemented, Sylow
524C2—21no—
525C2—22no—
526C2—21no—
527C2—84no—
528C2#7097612yes—
529C2#7097912yes—
530C2#7167111yescharacteristic
531C2#7205711yescharacteristic
532C2#7209811yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#450C2
socleC23×C3#120
prime radicalC23×C3×D4C22
hypercenterC2×#450D3
hypocenterC3#9966
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#450

Complements

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

NH ≅ G/Ntype
3. #82789C2semidirect
5. C2×#4538C2semidirect
6. C2×#4467C2semidirect
7. #67674C2semidirect
157. 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
C2×C3×#4502
C2×#52182
#827892
#721312
C2×#45382
C2×#44672
#676742
#99663