Finite Group Databasebeta

← #10693166

Subgroups — #10693166

Order 768.

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

normal subgroups only

This group has 812 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
1C3×#2879C211yesmaximal, characteristic, complemented
2#84096C211yesmaximal, characteristic
3#72137C211yesmaximal, characteristic, complemented
4#69967C212yesmaximal, complemented
5#67686C212yesmaximal, complemented
6#9511—31nomaximal, Sylow
7C22×C3×#41C2211yescharacteristic
8C22×C3×M16—21no—
9C2×#937C411yescharacteristic
10C2×#937C411yescharacteristic
11C22×(C2×D3)⋊C4C2211yescharacteristic
12C3×#408—24no—
13C3×#407—21no—
14C3×#407C2212yes—
15#5231—21no—
16C2×#836—21no—
17#5223—21no—
18#5220C411yescharacteristic
19#5220C411yescharacteristic
20C2×#833—21no—
21#5213—21no—
22#5213—21no—
23#5211C2211yescharacteristic
24#5207C2212yes—
25#4468—24no—
26#2879D311yescharacteristic, complemented
27#2404—31no—
28#1785—31no—
29#1462—32no—
30#1306—32no—
31C24×D3C2×C411yescharacteristic
32C2×C3×#41—22no—
33C2×C3×#41—22no—
34C2×C3×#41—42no—
35C2×C3×#41—42no—
36C22×C3×Q8D411yescharacteristic
37C22×C3×D4C2×C411yescharacteristic
38C22×C3×D4C2×C411yescharacteristic
39C22×C3×D4D411yescharacteristic
40C23×C3×C4D411yescharacteristic
41C23×C3×C4D411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C22×#67—21no—
44C22×#67—21no—
45C23×Dic3C2×C411yescharacteristic
46C2×C3×M16—22no—
47C2×C3×M16—22no—
48C2×C3×M16—42no—
49C2×C3×M16—42no—
50C22×C3×C8—22no—
51C2×#217—21no—
52C2×#217—21no—
53C2×#217C2×C411yescharacteristic
54C2×#217C2×C411yescharacteristic
55#937—42no—
56#937—42no—
57C2×(C2×D3)⋊C4—21no—
58C2×(C2×D3)⋊C4—21no—
59C2×(C2×D3)⋊C4—21no—
60C2×(C2×D3)⋊C4—21no—
61C2×(C2×D3)⋊C4—22no—
62C2×(C2×D3)⋊C4—22no—
63C2×#208—22no—
64C3×C22⋊C8—24no—
65C3×C22⋊C8—22no—
66C3×C22⋊C8D414yescomplemented
67#836—42no—
68#833—42no—
69C3⋊#95—21no—
70C3⋊#95—21no—
71C3⋊#95—21no—
72C3⋊#95—21no—
73#830—22no—
74#830—22no—
75C22×#41C2×D311yescharacteristic
76C22×M16—21no—
77C2×#120—31no—
78C2×#120—31no—
79C22×C22⋊C4—31no—
80C2×#101—61no—
81C2×#100—61no—
82#408—24no—
83#407—21no—
84#407—32no—
85#407C2×D312yes—
86#393—61no—
87#392—61no—
88#386—31no—
89#386—31no—
90#386—61no—
91#386—61no—
92#381—31no—
93#323—64no—
94C24×C3C22⋊C411yescharacteristic
95C24×C3C22⋊C411yescharacteristic
96C24×C3C22×C411yescharacteristic
97C23×D3—21no—
98C23×D3—22no—
99C23×D3—42no—
100C23×D3—41no—
101C23×D3—41no—
102C23×D3—81no—
103C3×#41—44no—
104C3×#41—44no—
105C3×#41—48no—
106C2×C3×Q8—22no—
107C2×C3×Q8—22no—
108C2×C3×Q8—41no—
109C2×C3×D4—22no—
110C2×C3×D4—22no—
111C2×C3×D4—22no—
112C2×C3×D4—22no—
113C2×C3×D4—22no—
114C2×C3×D4—22no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—41no—
125C22×C3×C4—41no—
126C22×C3×C4—41no—
127C22×C3×C4—41no—
128C22×C3×C4C22⋊C411yescharacteristic
129C22×C3×C4C22⋊C411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D412yes—
133C22×C3×C4#4112yes—
134C2×#67—42no—
135C2×#67—42no—
136C2×#67—42no—
137C2×#67—42no—
138C2×#67—81no—
139C2×#67—81no—
140C22×Dic3—21no—
141C22×Dic3—21no—
142C22×Dic3—21no—
143C22×Dic3—22no—
144C22×Dic3—81no—
145C3×M16—44no—
146C3×M16—42no—
147C3×M16—42no—
148C2×C3×C8—24no—
149C2×C3×C8—44no—
150#217—41no—
151#217—41no—
152#217—81no—
153#217—81no—
154(C2×D3)⋊C4—42no—
155(C2×D3)⋊C4—42no—
156(C2×D3)⋊C4—41no—
157(C2×D3)⋊C4—42no—
158(C2×D3)⋊C4—41no—
159#208—42no—
160#208—42no—
161C2×C3⋊C8—42no—
162C25—31no—
163C2×#41—22no—
164C2×#41—22no—
165C2×#41—42no—
166C2×#41—42no—
167C22×Q8D1211yescharacteristic
168C22×D4—61no—
169C22×D4—61no—
170C22×D4C4×D311yescharacteristic
171C22×D4C4×D311yescharacteristic
172C22×D4D1211yescharacteristic
173C23×C4—31no—
174C23×C4#6711yescharacteristic
175C23×C4#6711yescharacteristic
176C23×C4C22×D311yescharacteristic
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—42no—
180C2×M16—42no—
181C2×M16—62no—
182C22×C8—22no—
183#120—122no—
184#120—122no—
185C2×C22⋊C4—31no—
186C2×C22⋊C4—31no—
187C2×C22⋊C4—61no—
188C2×C22⋊C4—61no—
189C2×C22⋊C4—61no—
190C2×C22⋊C4—62no—
191C2×C22⋊C4—62no—
192C2×C22⋊C4—61no—
193C2×C22⋊C4—61no—
194C2×C22⋊C4—61no—
195#101—122no—
196#100—122no—
197C22⋊C8—24no—
198C22⋊C8—22no—
199C22⋊C8—62no—
200C22⋊C8—62no—
201C22⋊C8#6714yescomplemented
202#95—61no—
203#95—61no—
204#95—61no—
205#95—61no—
206C23×C3—21no—
207C23×C3—21no—
208C23×C3—21no—
209C23×C3—21no—
210C23×C3—21no—
211C23×C3—21no—
212C23×C3—21no—
213C23×C3—41no—
214C23×C3—41no—
215C23×C3—41no—
216C23×C3—41no—
217C23×C3—41no—
218C23×C3—41no—
219C23×C3#9912yes—
220C23×C3C2×C22⋊C411yescharacteristic
221C23×C3C4×D412yes—
222C22×D3—22no—
223C22×D3—24no—
224C22×D3—41no—
225C22×D3—41no—
226C22×D3—41no—
227C22×D3—42no—
228C22×D3—41no—
229C22×D3—41no—
230C22×D3—41no—
231C22×D3—84no—
232C22×D3—84no—
233C22×D3—82no—
234C22×D3—82no—
235C3×Q8—44no—
236C3×D4—44no—
237C3×D4—44no—
238C3×D4—44no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—24no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4#9912yes—
256C2×C3×C4#12011yescharacteristic
257C2×C3×C4#12112yes—
258C2×C3×C4#12311yescharacteristic
259#67—84no—
260#67—84no—
261C2×Dic3—41no—
262C2×Dic3—41no—
263C2×Dic3—41no—
264C2×Dic3—41no—
265C2×Dic3—41no—
266C2×Dic3—41no—
267C2×Dic3—41no—
268C2×Dic3—41no—
269C2×Dic3—41no—
270C2×Dic3—82no—
271C3×C8—44no—
272C3⋊C8—82no—
273C24—61no—
274C24—62no—
275C24—121no—
276C24—121no—
277C24—122no—
278C24—241no—
279C24(C2×D3)⋊C411yescharacteristic
280C24(C2×D3)⋊C411yescharacteristic
281C24C2×C4×D311yescharacteristic
282#41—44no—
283#41—44no—
284#41—48no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—41no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—122no—
298C2×D4—122no—
299C2×D4—122no—
300C2×D4—122no—
301C2×D4—241no—
302C2×D4—241no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—61no—
314C22×C4—61no—
315C22×C4—61no—
316C22×C4—62no—
317C22×C4—241no—
318C22×C4(C2×D3)⋊C411yescharacteristic
319C22×C4(C2×D3)⋊C411yescharacteristic
320C22×C4C2×D1211yescharacteristic
321C22×C4#23512yes—
322C22×C4C2×#6711yescharacteristic
323C22×C4C2×#6712yes—
324M16—44no—
325M16—42no—
326M16—42no—
327M16—122no—
328M16—122no—
329C2×C8—24no—
330C2×C8—44no—
331C2×C8—122no—
332C22⋊C4—122no—
333C22⋊C4—122no—
334C22⋊C4—121no—
335C22⋊C4—122no—
336C22⋊C4—121no—
337C22⋊C4—121no—
338C22⋊C4—121no—
339C22⋊C4—241no—
340C22⋊C4—241no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—22no—
344C22×C3—22no—
345C22×C3—22no—
346C22×C3—22no—
347C22×C3—42no—
348C22×C3—42no—
349C22×C3—42no—
350C22×C3—42no—
351C22×C3—41no—
352C22×C3—41no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3—41no—
356C22×C3—81no—
357C22×C3—81no—
358C22×C3#38611yescharacteristic
359C22×C3C2×#9911yescharacteristic
360C22×C3C2×#9911yescharacteristic
361C2×D3—42no—
362C2×D3—42no—
363C2×D3—42no—
364C2×D3—81no—
365C2×D3—81no—
366C2×D3—84no—
367C2×D3—81no—
368C2×D3—81no—
369C2×D3—84no—
370C3×C4—22no—
371C3×C4—21no—
372C3×C4—21no—
373C3×C4—22no—
374C3×C4—41no—
375Dic3—81no—
376Dic3—81no—
377Dic3—81no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—62no—
392C23—64no—
393C23—121no—
394C23—121no—
395C23—121no—
396C23—122no—
397C23—121no—
398C23—121no—
399C23—121no—
400C23—242no—
401C23—242no—
402C23—244no—
403C23—244no—
404C23#80612yes—
405C23C2×(C2×D3)⋊C411yescharacteristic
406C23C4×#6712yes—
407Q8—44no—
408D4—44no—
409D4—44no—
410D4—44no—
411D4—244no—
412D4—244no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—121no—
430C2×C4—121no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4—121no—
434C2×C4—121no—
435C2×C4—121no—
436C2×C4—121no—
437C2×C4—121no—
438C2×C4—242no—
439C2×C4#80612yes—
440C2×C4#92911yescharacteristic
441C2×C4#93012yes—
442C2×C4#95311yescharacteristic
443C8—44no—
444C8—242no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—42no—
448C2×C3—42no—
449C2×C3—81no—
450C2×C3#189111yescharacteristic
451C2×C3#189811yescharacteristic
452C2×C3#191411yescharacteristic
453D3—82no—
454C22—21no—
455C22—21no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—81no—
470C22—81no—
471C22—122no—
472C22—122no—
473C22—122no—
474C22—241no—
475C22—241no—
476C22—244no—
477C22—244no—
478C22—241no—
479C22—241no—
480C22C2×#80611yescharacteristic
481C22C2×#80611yescharacteristic
482C22#521311yescharacteristic
483C4—21no—
484C4—22no—
485C4—21no—
486C4—22no—
487C4—41no—
488C4—241no—
489C4—241no—
490C4—241no—
491C3#951111yescharacteristic, complemented, Sylow
492C2—21no—
493C2—21no—
494C2—42no—
495C2—42no—
496C2—81no—
497C2—242no—
498C2#7154311yescharacteristic
499C2#7202911yescharacteristic
500C2#7203011yescharacteristic
subgroupG/H
centerC22#5213
commutatorC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2879C2
socleC22×C3#386
prime radical#84096C2
cosocleC23×C3×C4C23
hypercenter#2879D3
hypocenterC3#9511
LCSG ▷ C24×C3 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2879

Complements

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

NH ≅ G/Ntype
1. C3×#2879C2semidirect
3. #72137C2semidirect
4. #69967C2semidirect
5. #67686C2semidirect
26. #2879D3semidirect
66. C3×C22⋊C8D4semidirect
201. C22⋊C8#67semidirect
491. C3#9511semidirect

Maximal subgroups

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

subgroupindex
C3×#28792
#840962
#721372
#699672
#676862
#95113