Finite Group Databasebeta

← #10712699

Subgroups — #10712699

Order 768.

820 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 820 conjugacy classes of subgroups — too many to draw. Only its 103 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#84096C211yesmaximal, characteristic
2C2×#5240C211yesmaximal, characteristic, complemented
3C3×#1785C211yesmaximal, characteristic, complemented
4C2×#4475C212yesmaximal, complemented
5#67698C212yesmaximal, complemented
6#13087—31nomaximal, Sylow
7C22×C3×#41C2211yescharacteristic
8C2×#937C411yescharacteristic, complemented
9C2×#937C2211yescharacteristic
10C22×(C2×D3)⋊C4C411yescharacteristic, complemented
11C2×#926—21no—
12C2×C3×#101—21no—
13C2×C3×#100C2212yes—
14C3×#407—21no—
15#5240—24no—
16#5231—21no—
17#5223—21no—
18#5220C411yescharacteristic, complemented
19#5220C2211yescharacteristic
20C2×#834—21no—
21C2×#834C2212yes—
22#5213—21no—
23#5213—21no—
24#5211C411yescharacteristic, complemented
25#4475—24no—
26C2×#410—31no—
27#2404—31no—
28C2×AΣL(1,𝔽16)—32no—
29#1785D311yescharacteristic, complemented
30#1390—32no—
31C24×D3C2×C411yescharacteristic
32C2×C3×#41—22no—
33C2×C3×#41—22no—
34C2×C3×#41—41no—
35C2×C3×#41—41no—
36C2×C3×#41—42no—
37C22×C3×Q8D411yescharacteristic
38C22×C3×D4C2×C411yescharacteristic
39C22×C3×D4D411yescharacteristic
40C22×C3×D4C2311yescharacteristic
41C23×C3×C4C2×C411yescharacteristic
42C23×C3×C4D411yescharacteristic
43C23×C3×C4D411yescharacteristic
44C22×#67—21no—
45C22×#67—21no—
46C23×Dic3C2×C411yescharacteristic
47C2×C3×M16—22no—
48C2×#217—22no—
49C2×#217—21no—
50C2×#217—21no—
51C2×#217C2×C411yescharacteristic
52C2×#217C2×C411yescharacteristic
53#937—22no—
54#937—22no—
55#937—42no—
56C2×(C2×D3)⋊C4—21no—
57C2×(C2×D3)⋊C4—21no—
58C2×(C2×D3)⋊C4—21no—
59C2×(C2×D3)⋊C4—21no—
60C2×(C2×D3)⋊C4—41no—
61C2×(C2×D3)⋊C4—41no—
62#926—24no—
63#926—42no—
64C2×C4⋊Dic3—41no—
65C2×C4×Dic3—41no—
66C3×#101—42no—
67C3×#100—22no—
68C3×#100—22no—
69C3×C22⋊C8—41no—
70C3×C22⋊C8—41no—
71#834—24no—
72#834—22no—
73#834D414yescomplemented
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C22×#41C2×D311yescharacteristic
79C2×#120—31no—
80C2×#120—31no—
81C2×#117—61no—
82C22×C22⋊C4—31no—
83C2×#101—21no—
84C2×#100C2×D312yes—
85#410—64no—
86C2×#99—32no—
87C2×#99—61no—
88#407—21no—
89#393—61no—
90#392—61no—
91#386—31no—
92#386—31no—
93#386—61no—
94#386—61no—
95#381—31no—
96AΣL(1,𝔽16)—64no—
97C24×C3C22⋊C411yescharacteristic
98C24×C3C22×C411yescharacteristic
99C24×C3C2×D411yescharacteristic
100C23×D3—21no—
101C23×D3—22no—
102C23×D3—42no—
103C23×D3—41no—
104C23×D3—41no—
105C23×D3—81no—
106C3×#41—44no—
107C3×#41—44no—
108C3×#41—44no—
109C3×#41—44no—
110C2×C3×Q8—22no—
111C2×C3×Q8—22no—
112C2×C3×Q8—41no—
113C2×C3×D4—22no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—22no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4C2×D412yes—
123C2×C3×D4#4112yes—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—22no—
135C22×C3×C4—41no—
136C22×C3×C4—41no—
137C22×C3×C4C22⋊C411yescharacteristic
138C22×C3×C4C22⋊C411yescharacteristic
139C22×C3×C4C22⋊C411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C2×#67—42no—
142C2×#67—42no—
143C2×#67—42no—
144C2×#67—41no—
145C2×#67—41no—
146C2×#67—81no—
147C2×#67—81no—
148C22×Dic3—21no—
149C22×Dic3—21no—
150C22×Dic3—21no—
151C22×Dic3—42no—
152C22×Dic3—41no—
153C22×Dic3—81no—
154C3×M16—42no—
155C3×M16—42no—
156C2×C3×C8—42no—
157#217—22no—
158#217—24no—
159#217—42no—
160#217—41no—
161#217—81no—
162#217—81no—
163(C2×D3)⋊C4—42no—
164(C2×D3)⋊C4—41no—
165(C2×D3)⋊C4—41no—
166(C2×D3)⋊C4—42no—
167(C2×D3)⋊C4—42no—
168C4⋊Dic3—44no—
169C4×Dic3—44no—
170C25—31no—
171C2×#41—22no—
172C2×#41—22no—
173C2×#41—41no—
174C2×#41—41no—
175C2×#41—42no—
176C22×Q8#6711yescharacteristic
177C22×D4—61no—
178C22×D4—61no—
179C22×D4C4×D311yescharacteristic
180C22×D4D1211yescharacteristic
181C22×D4C22×D311yescharacteristic
182C23×C4—31no—
183C23×C4C4×D311yescharacteristic
184C23×C4D1211yescharacteristic
185C23×C4#6711yescharacteristic
186C2×M16—22no—
187#120—62no—
188#120—62no—
189#120—122no—
190#117—64no—
191#117—122no—
192C2×C4⋊C4—121no—
193C2×C22⋊C4—31no—
194C2×C22⋊C4—31no—
195C2×C22⋊C4—62no—
196C2×C22⋊C4—61no—
197C2×C22⋊C4—61no—
198C2×C22⋊C4—61no—
199C2×C22⋊C4—61no—
200C2×C22⋊C4—61no—
201C2×C22⋊C4—61no—
202C2×C22⋊C4—121no—
203C2×C22⋊C4—121no—
204C2×C42—121no—
205#101—42no—
206#100—22no—
207#100—22no—
208#99—34no—
209#99—64no—
210#99—62no—
211C22⋊C8—41no—
212C22⋊C8—41no—
213#95—61no—
214#95—61no—
215#95—61no—
216#95—61no—
217C23×C3—21no—
218C23×C3—21no—
219C23×C3—21no—
220C23×C3—21no—
221C23×C3—21no—
222C23×C3—21no—
223C23×C3—21no—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3—41no—
227C23×C3—41no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3C2×C22⋊C411yescharacteristic
231C23×C3C4×D412yes—
232C23×C3#12011yescharacteristic
233C23×C3#12311yescharacteristic
234C22×D3—22no—
235C22×D3—41no—
236C22×D3—41no—
237C22×D3—41no—
238C22×D3—42no—
239C22×D3—41no—
240C22×D3—41no—
241C22×D3—41no—
242C22×D3—42no—
243C22×D3—84no—
244C22×D3—84no—
245C22×D3—82no—
246C22×D3—82no—
247C3×Q8—44no—
248C3×D4—42no—
249C3×D4—42no—
250C3×D4—44no—
251C3×D4—42no—
252C3×D4—42no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4#9911yescharacteristic
271C2×C3×C4#9911yescharacteristic
272C2×C3×C4#9912yes—
273C2×C3×C4#12112yes—
274#67—82no—
275#67—82no—
276#67—84no—
277C2×Dic3—44no—
278C2×Dic3—41no—
279C2×Dic3—41no—
280C2×Dic3—41no—
281C2×Dic3—41no—
282C2×Dic3—41no—
283C2×Dic3—41no—
284C2×Dic3—41no—
285C2×Dic3—41no—
286C2×Dic3—41no—
287C2×Dic3—82no—
288C2×Dic3—82no—
289C2×Dic3—82no—
290C3×C8—82no—
291C24—61no—
292C24—62no—
293C24—121no—
294C24—121no—
295C24—122no—
296C24—241no—
297C24(C2×D3)⋊C411yescharacteristic
298C24C2×C4×D311yescharacteristic
299C24C2×D1211yescharacteristic
300#41—44no—
301#41—44no—
302#41—44no—
303#41—44no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—41no—
307C2×D4—22no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—22no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—122no—
317C2×D4—122no—
318C2×D4—122no—
319C2×D4—121no—
320C2×D4—121no—
321C2×D4—241no—
322C2×D4—241no—
323C2×D4D3×D412yes—
324C2×D4#23712yes—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—61no—
339C22×C4—61no—
340C22×C4—61no—
341C22×C4—122no—
342C22×C4—121no—
343C22×C4—241no—
344C22×C4(C2×D3)⋊C411yescharacteristic
345C22×C4(C2×D3)⋊C411yescharacteristic
346C22×C4(C2×D3)⋊C411yescharacteristic
347C22×C4C2×#6711yescharacteristic
348M16—42no—
349M16—42no—
350C2×C8—42no—
351C4⋊C4—124no—
352C22⋊C4—62no—
353C22⋊C4—64no—
354C22⋊C4—122no—
355C22⋊C4—122no—
356C22⋊C4—121no—
357C22⋊C4—121no—
358C22⋊C4—122no—
359C22⋊C4—122no—
360C22⋊C4—121no—
361C22⋊C4—241no—
362C22⋊C4—241no—
363C42—124no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—22no—
369C22×C3—41no—
370C22×C3—41no—
371C22×C3—42no—
372C22×C3—42no—
373C22×C3—42no—
374C22×C3—41no—
375C22×C3—41no—
376C22×C3—41no—
377C22×C3—41no—
378C22×C3—41no—
379C22×C3—41no—
380C22×C3—41no—
381C22×C3—81no—
382C22×C3—81no—
383C22×C3#38611yescharacteristic
384C22×C3C2×#9911yescharacteristic
385C22×C3C2×#9911yescharacteristic
386C2×D3—42no—
387C2×D3—42no—
388C2×D3—42no—
389C2×D3—81no—
390C2×D3—81no—
391C2×D3—84no—
392C2×D3—81no—
393C2×D3—81no—
394C2×D3—84no—
395C3×C4—22no—
396C3×C4—21no—
397C3×C4—21no—
398C3×C4—22no—
399C3×C4—41no—
400Dic3—81no—
401Dic3—81no—
402Dic3—84no—
403Dic3—81no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—62no—
418C23—121no—
419C23—121no—
420C23—121no—
421C23—122no—
422C23—122no—
423C23—121no—
424C23—121no—
425C23—121no—
426C23—242no—
427C23—244no—
428C23—244no—
429C23—242no—
430C23#67⋊C412yes—
431C23#88211yescharacteristic
432C23#88611yescharacteristic
433C23C2×(C2×D3)⋊C411yescharacteristic
434Q8—44no—
435D4—42no—
436D4—42no—
437D4—44no—
438D4—42no—
439D4—42no—
440D4—242no—
441D4—242no—
442D4—244no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—124no—
461C2×C4—121no—
462C2×C4—121no—
463C2×C4—121no—
464C2×C4—121no—
465C2×C4—121no—
466C2×C4—121no—
467C2×C4—121no—
468C2×C4—121no—
469C2×C4—121no—
470C2×C4—242no—
471C2×C4—242no—
472C2×C4—242no—
473C2×C4#80612yes—
474C2×C4#80611yescharacteristic
475C2×C4#80611yescharacteristic
476C2×C4#93912yes—
477C8—82no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—41no—
481C2×C3—42no—
482C2×C3—41no—
483C2×C3—81no—
484C2×C3#189811yescharacteristic
485C2×C3#212011yescharacteristic
486C2×C3#212211yescharacteristic
487D3—82no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—22no—
493C22—41no—
494C22—41no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—81no—
506C22—81no—
507C22—122no—
508C22—122no—
509C22—122no—
510C22—241no—
511C22—241no—
512C22—244no—
513C22—244no—
514C22—241no—
515C22—241no—
516C22C2×#80611yescharacteristic
517C22C2×#80611yescharacteristic
518C22#495811yescharacteristic
519C4—22no—
520C4—21no—
521C4—21no—
522C4—22no—
523C4—41no—
524C4—241no—
525C4—241no—
526C4—244no—
527C4—241no—
528C3#1308711yescharacteristic, complemented, Sylow
529C2—21no—
530C2—21no—
531C2—41no—
532C2—42no—
533C2—41no—
534C2—81no—
535C2—242no—
536C2#6949611yescharacteristic
537C2#7014511yescharacteristic
538C2#7014611yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC24×C3C22×C4
FrattiniC22×D4C22×D3
FittingC3×#1785C2
socleC22×C3C2×#99
prime radical#84096C2
cosocleC22×C3×D4C23
hypercenter#1785D3
hypocenterC3#13087
LCSG ▷ C24×C3 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #1785

Complements

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

NH ≅ G/Ntype
2. C2×#5240C2semidirect
3. C3×#1785C2semidirect
4. C2×#4475C2semidirect
5. #67698C2semidirect
8. C2×#937C4semidirect
10. C22×(C2×D3)⋊C4C4semidirect
18. #5220C4semidirect
24. #5211C4semidirect
29. #1785D3semidirect
73. #834D4semidirect
528. C3#13087semidirect

Maximal subgroups

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

subgroupindex
#840962
C2×#52402
C3×#17852
C2×#44752
#676982
#130873