Finite Group Databasebeta

← #10712681

Subgroups — #10712681

Order 768.

818 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 818 conjugacy classes of subgroups — too many to draw. Only its 107 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×#413C211yesmaximal, characteristic, complemented
2#84096C211yesmaximal, characteristic
3#72139C211yesmaximal, characteristic, complemented
4C2×#4475C212yesmaximal, complemented
5#67698C212yesmaximal, complemented
6#13083—31nomaximal, Sylow
7C22×C3×#41C2211yescharacteristic
8C22×C3×M16—21no—
9C2×#937C411yescharacteristic, complemented
10C2×#937C2211yescharacteristic
11C22×(C2×D3)⋊C4C411yescharacteristic, complemented
12C3×#413—24no—
13C2×C3×#101—21no—
14C2×C3×#100C2212yes—
15#5231—21no—
16#5223—21no—
17#5220C411yescharacteristic, complemented
18#5220C2211yescharacteristic
19C2×#834—21no—
20C2×#834C2212yes—
21#5213—21no—
22#5213—21no—
23#5211—21no—
24#5211C411yescharacteristic, complemented
25#4475—24no—
26C2×#413D311yescharacteristic, complemented
27#2404—31no—
28C2×AΣL(1,𝔽16)—32no—
29#1782—31no—
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—24no—
48C2×C3×M16—22no—
49C2×C3×M16—42no—
50C2×C3×M16—41no—
51C22×C3×C8—41no—
52C2×#217—22no—
53C2×#217—21no—
54C2×#217—21no—
55C2×#217C2×C411yescharacteristic
56C2×#217C2×C411yescharacteristic
57#937—22no—
58#937—22no—
59#937—42no—
60C2×(C2×D3)⋊C4—21no—
61C2×(C2×D3)⋊C4—21no—
62C2×(C2×D3)⋊C4—21no—
63C2×(C2×D3)⋊C4—21no—
64C2×(C2×D3)⋊C4—41no—
65C2×(C2×D3)⋊C4—41no—
66C3×#101—24no—
67C3×#100—22no—
68C3×#100D414yescomplemented
69#834—22no—
70#834—22no—
71#834—42no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—41no—
77C3⋊#95—41no—
78C22×#41C2×D311yescharacteristic
79C22×M16—21no—
80C2×#120—31no—
81C2×#120—31no—
82C22×C22⋊C4—31no—
83#413—24no—
84C2×#101—21no—
85C2×#100C2×D312yes—
86C2×#99—32no—
87C2×#99—61no—
88#393—61no—
89#392—61no—
90#386—31no—
91#386—31no—
92#386—61no—
93#386—61no—
94#381—31no—
95#381—61no—
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—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—41no—
113C2×C3×Q8—41no—
114C2×C3×D4—22no—
115C2×C3×D4—22no—
116C2×C3×D4—22no—
117C2×C3×D4—22no—
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—22no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
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—42no—
145C2×#67—81no—
146C2×#67—81no—
147C22×Dic3—21no—
148C22×Dic3—21no—
149C22×Dic3—21no—
150C22×Dic3—41no—
151C22×Dic3—42no—
152C22×Dic3—81no—
153C3×M16—24no—
154C3×M16—44no—
155C3×M16—42no—
156C2×C3×C8—44no—
157C2×C3×C8—42no—
158#217—22no—
159#217—41no—
160#217—42no—
161#217—42no—
162#217—81no—
163#217—81no—
164(C2×D3)⋊C4—42no—
165(C2×D3)⋊C4—42no—
166(C2×D3)⋊C4—41no—
167(C2×D3)⋊C4—41no—
168(C2×D3)⋊C4—41no—
169(C2×D3)⋊C4—41no—
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—24no—
187C2×M16—22no—
188C2×M16—42no—
189C2×M16—41no—
190C22×C8—41no—
191#120—62no—
192#120—62no—
193#120—122no—
194C2×C22⋊C4—31no—
195C2×C22⋊C4—31no—
196C2×C22⋊C4—61no—
197C2×C22⋊C4—61no—
198C2×C22⋊C4—61no—
199C2×C22⋊C4—61no—
200C2×C22⋊C4—62no—
201C2×C22⋊C4—61no—
202C2×C22⋊C4—61no—
203C2×C22⋊C4—121no—
204C2×C22⋊C4—121no—
205#101—24no—
206#100—22no—
207#100#6714yescomplemented
208#99—62no—
209#99—62no—
210#99—122no—
211#95—61no—
212#95—61no—
213#95—61no—
214#95—61no—
215#95—121no—
216#95—121no—
217C23×C3—21no—
218C23×C3—21no—
219C23×C3—21no—
220C23×C3—21no—
221C23×C3—21no—
222C23×C3—21no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3—41no—
227C23×C3—41no—
228C23×C3—41no—
229C23×C3#9912yes—
230C23×C3C2×C22⋊C411yescharacteristic
231C23×C3C4×D412yes—
232C23×C3#12112yes—
233C22×D3—22no—
234C22×D3—41no—
235C22×D3—41no—
236C22×D3—41no—
237C22×D3—42no—
238C22×D3—41no—
239C22×D3—41no—
240C22×D3—41no—
241C22×D3—42no—
242C22×D3—84no—
243C22×D3—84no—
244C22×D3—82no—
245C22×D3—82no—
246C3×Q8—42no—
247C3×Q8—42no—
248C3×D4—42no—
249C3×D4—42no—
250C3×D4—44no—
251C3×D4—44no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—21no—
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#12011yescharacteristic
273C2×C3×C4#12311yescharacteristic
274#67—84no—
275#67—84no—
276C2×Dic3—41no—
277C2×Dic3—41no—
278C2×Dic3—41no—
279C2×Dic3—41no—
280C2×Dic3—41no—
281C2×Dic3—41no—
282C2×Dic3—41no—
283C2×Dic3—41no—
284C2×Dic3—41no—
285C2×Dic3—82no—
286C2×Dic3—82no—
287C2×Dic3—82no—
288C2×Dic3—82no—
289C3×C8—44no—
290C24—61no—
291C24—62no—
292C24—121no—
293C24—121no—
294C24—122no—
295C24—241no—
296C24(C2×D3)⋊C411yescharacteristic
297C24C2×C4×D311yescharacteristic
298C24C2×D1211yescharacteristic
299#41—44no—
300#41—44no—
301#41—44no—
302#41—44no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—122no—
316C2×D4—122no—
317C2×D4—122no—
318C2×D4—122no—
319C2×D4—241no—
320C2×D4—241no—
321C2×D4#23512yes—
322C2×D4C2×#6712yes—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—61no—
337C22×C4—61no—
338C22×C4—61no—
339C22×C4—121no—
340C22×C4—122no—
341C22×C4—241no—
342C22×C4(C2×D3)⋊C411yescharacteristic
343C22×C4(C2×D3)⋊C411yescharacteristic
344C22×C4(C2×D3)⋊C411yescharacteristic
345C22×C4C2×#6711yescharacteristic
346M16—24no—
347M16—44no—
348M16—42no—
349C2×C8—44no—
350C2×C8—42no—
351C22⋊C4—62no—
352C22⋊C4—122no—
353C22⋊C4—122no—
354C22⋊C4—121no—
355C22⋊C4—121no—
356C22⋊C4—121no—
357C22⋊C4—121no—
358C22⋊C4—121no—
359C22⋊C4—122no—
360C22⋊C4—122no—
361C22⋊C4—241no—
362C22⋊C4—241no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—22no—
366C22×C3—22no—
367C22×C3—22no—
368C22×C3—42no—
369C22×C3—42no—
370C22×C3—42no—
371C22×C3—42no—
372C22×C3—41no—
373C22×C3—41no—
374C22×C3—41no—
375C22×C3—41no—
376C22×C3—41no—
377C22×C3—41no—
378C22×C3—81no—
379C22×C3—81no—
380C22×C3#38611yescharacteristic
381C22×C3C2×#9911yescharacteristic
382C22×C3C2×#9911yescharacteristic
383C2×D3—42no—
384C2×D3—42no—
385C2×D3—42no—
386C2×D3—81no—
387C2×D3—81no—
388C2×D3—84no—
389C2×D3—81no—
390C2×D3—81no—
391C2×D3—84no—
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—21no—
396C3×C4—41no—
397C3×C4—41no—
398Dic3—81no—
399Dic3—81no—
400Dic3—81no—
401Dic3—162no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—62no—
415C23—121no—
416C23—121no—
417C23—121no—
418C23—122no—
419C23—122no—
420C23—121no—
421C23—121no—
422C23—121no—
423C23—242no—
424C23—244no—
425C23—244no—
426C23—242no—
427C23#80612yes—
428C23C2×(C2×D3)⋊C411yescharacteristic
429C23C4×#6712yes—
430C23#93012yes—
431Q8—42no—
432Q8—42no—
433D4—42no—
434D4—42no—
435D4—44no—
436D4—44no—
437D4—244no—
438D4—244no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—121no—
458C2×C4—121no—
459C2×C4—121no—
460C2×C4—121no—
461C2×C4—121no—
462C2×C4—121no—
463C2×C4—121no—
464C2×C4—121no—
465C2×C4—121no—
466C2×C4—242no—
467C2×C4—242no—
468C2×C4—242no—
469C2×C4—242no—
470C2×C4#80611yescharacteristic
471C2×C4#80611yescharacteristic
472C2×C4#92911yescharacteristic
473C2×C4#95311yescharacteristic
474C8—44no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—42no—
478C2×C3—42no—
479C2×C3—42no—
480C2×C3#189811yescharacteristic
481C2×C3#212011yescharacteristic
482C2×C3#212211yescharacteristic
483D3—82no—
484C22—21no—
485C22—21no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—81no—
500C22—81no—
501C22—122no—
502C22—122no—
503C22—122no—
504C22—241no—
505C22—241no—
506C22—244no—
507C22—244no—
508C22—241no—
509C22—241no—
510C22C2×#80611yescharacteristic
511C22C2×#80611yescharacteristic
512C22#521311yescharacteristic
513C4—21no—
514C4—21no—
515C4—21no—
516C4—21no—
517C4—41no—
518C4—41no—
519C4—241no—
520C4—241no—
521C4—241no—
522C4—482no—
523C3#1308311yescharacteristic, complemented, Sylow
524C2—21no—
525C2—21no—
526C2—42no—
527C2—42no—
528C2—42no—
529C2—242no—
530C2#7014511yescharacteristic
531C2#7014611yescharacteristic
532C2#7154311yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC24×C3C22×C4
FrattiniC22×D4C22×D3
FittingC2×C3×#413C2
socleC22×C3C2×#99
prime radical#84096C2
cosocleC22×C3×D4C23
hypercenterC2×#413D3
hypocenterC3#13083
LCSG ▷ C24×C3 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#413

Complements

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

NH ≅ G/Ntype
1. C2×C3×#413C2semidirect
3. #72139C2semidirect
4. C2×#4475C2semidirect
5. #67698C2semidirect
9. C2×#937C4semidirect
11. C22×(C2×D3)⋊C4C4semidirect
17. #5220C4semidirect
24. #5211C4semidirect
26. C2×#413D3semidirect
68. C3×#100D4semidirect
207. #100#67semidirect
523. C3#13083semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4132
#840962
#721392
C2×#44752
#676982
#130833