Finite Group Databasebeta

← #10692826

Subgroups — #10692826

Order 768.

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

normal subgroups only

This group has 712 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
2#84093C211yesmaximal, characteristic
3#72139C211yesmaximal, characteristic
4#69981C212yesmaximal, complemented
5#67682C212yesmaximal, complemented
6#9443—31nomaximal, Sylow
7C22×C3×#41C2211yescharacteristic
8C22×C3×M16—21no—
9C2×#944C411yescharacteristic, complemented
10C2×#936C411yescharacteristic, complemented
11C2×#924C2211yescharacteristic
12C3×#408—24no—
13C3×#407—21no—
14C3×#407C2212yes—
15#5229C411yescharacteristic, complemented
16#5222—21no—
17#5222—21no—
18#5221C411yescharacteristic, complemented
19C2×#834—21no—
20C2×#834—21no—
21#5211C2212yes—
22#5210—21no—
23#5210—21no—
24C4×#217C2211yescharacteristic
25#4467—24no—
26#2879D311yescharacteristic
27#2432—31no—
28#1782—31no—
29#1511—32no—
30#1284—32no—
31C2×C3×#41—22no—
32C2×C3×#41—22no—
33C2×C3×#41—42no—
34C2×C3×#41—42no—
35C22×C3×Q8C2×C411yescharacteristic
36C22×C3×D4C2×C411yescharacteristic
37C22×C3×D4D411yescharacteristic
38C22×C3×D4D411yescharacteristic
39C23×C3×C4D411yescharacteristic
40C23×C3×C4D411yescharacteristic
41C23×C3×C4C2311yescharacteristic
42C22×Dic6C2×C411yescharacteristic
43C2×C3×M16—22no—
44C2×C3×M16—22no—
45C2×C3×M16—42no—
46C2×C3×M16—42no—
47C22×C3×C8—22no—
48C2×#217—21no—
49C2×#217—21no—
50C2×#217—21no—
51C2×#217—21no—
52C2×#217—21no—
53C2×#217—21no—
54C2×#217—22no—
55#944—42no—
56#936—42no—
57C2×C4⋊Dic3C2×C411yescharacteristic
58#924—22no—
59#924—22no—
60C2×Dic3⋊C4—21no—
61C2×Dic3⋊C4—21no—
62C2×Dic3⋊C4—21no—
63C2×C4×Dic3C2×C411yescharacteristic
64C2×C4×Dic3C2×C411yescharacteristic
65C3×C22⋊C8—24no—
66C3×C22⋊C8—22no—
67C3×C22⋊C8D414yes—
68#834—42no—
69#834—42no—
70C3⋊#95—21no—
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95—22no—
74C3⋊#95—22no—
75C22×#41C2×D311yescharacteristic
76C22×M16—21no—
77C2×C4⋊Q8—31no—
78C2×#124—31no—
79C2×C22⋊Q8—31no—
80C2×#99—61no—
81C2×#99—61no—
82#408—24no—
83#407—21no—
84#407C2×D312yes—
85#394—61no—
86#394—61no—
87#391—31no—
88#390—31no—
89#385—61no—
90#385—61no—
91#381—32no—
92C4×C22⋊C4—31no—
93#328—64no—
94C24×C3C22⋊C411yescharacteristic
95C24×C3C22⋊C411yescharacteristic
96C24×C3C2×D411yescharacteristic
97C3×#41—44no—
98C3×#41—44no—
99C3×#41—48no—
100C2×C3×Q8—22no—
101C2×C3×Q8—22no—
102C2×C3×Q8—41no—
103C2×C3×D4—22no—
104C2×C3×D4—22no—
105C2×C3×D4—22no—
106C2×C3×D4—22no—
107C2×C3×D4—22no—
108C2×C3×D4—22no—
109C2×C3×D4—41no—
110C2×C3×D4—41no—
111C2×C3×D4—41no—
112C22×C3×C4—22no—
113C22×C3×C4—22no—
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4—41no—
121C22×C3×C4—41no—
122C22×C3×C4C22⋊C411yescharacteristic
123C22×C3×C4C22⋊C411yescharacteristic
124C22×C3×C4C22×C411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D412yes—
127C22×C3×C4#4112yes—
128C22×Dic3—21no—
129C22×Dic3—21no—
130C22×Dic3—21no—
131C22×Dic3—21no—
132C22×Dic3—42no—
133C2×Dic6—22no—
134C2×Dic6—41no—
135C2×Dic6—41no—
136C3×M16—44no—
137C3×M16—42no—
138C3×M16—42no—
139C2×C3×C8—24no—
140C2×C3×C8—44no—
141#217—42no—
142#217—42no—
143#217—42no—
144#217—42no—
145#217—42no—
146#217—42no—
147#217—42no—
148#217—42no—
149C4⋊Dic3—22no—
150Dic3⋊C4—42no—
151Dic3⋊C4—42no—
152Dic3⋊C4—41no—
153Dic3⋊C4—41no—
154C4×Dic3—41no—
155C4×Dic3—41no—
156C2×#41—22no—
157C2×#41—22no—
158C2×#41—42no—
159C2×#41—42no—
160C22×Q8—31no—
161C22×Q8C4×D311yescharacteristic
162C22×D4C4×D311yescharacteristic
163C22×D4D1211yescharacteristic
164C22×D4D1211yescharacteristic
165C23×C4#6711yescharacteristic
166C23×C4#6711yescharacteristic
167C23×C4C22×D311yescharacteristic
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—42no—
171C2×M16—42no—
172C22×C8—22no—
173C4⋊Q8—122no—
174#124—122no—
175C22⋊Q8—62no—
176C22⋊Q8—62no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—61no—
181C2×C22⋊C4—61no—
182C2×C22⋊C4—61no—
183C2×C22⋊C4—61no—
184C2×C22⋊C4—61no—
185C2×C22⋊C4—61no—
186C2×C22⋊C4—61no—
187C2×C22⋊C4—62no—
188C2×C42—31no—
189C2×C42—31no—
190#99—122no—
191#99—122no—
192C22⋊C8—24no—
193C22⋊C8—22no—
194C22⋊C8#6714yes—
195#95—61no—
196#95—61no—
197#95—61no—
198#95—62no—
199#95—62no—
200C23×C3—21no—
201C23×C3—21no—
202C23×C3—21no—
203C23×C3—21no—
204C23×C3—21no—
205C23×C3—21no—
206C23×C3—21no—
207C23×C3—41no—
208C23×C3—41no—
209C23×C3—41no—
210C23×C3—41no—
211C23×C3—41no—
212C23×C3—41no—
213C23×C3#9912yes—
214C23×C3C2×C22⋊C411yescharacteristic
215C23×C3#12112yes—
216C3×Q8—44no—
217C3×D4—44no—
218C3×D4—44no—
219C3×D4—44no—
220C2×C3×C4—22no—
221C2×C3×C4—22no—
222C2×C3×C4—24no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—42no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4—42no—
236C2×C3×C4#9912yes—
237C2×C3×C4C4×D412yes—
238C2×C3×C4#12011yescharacteristic
239C2×C3×C4#12311yescharacteristic
240C2×Dic3—22no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251C2×Dic3—41no—
252C2×Dic3—82no—
253C2×Dic3—82no—
254C2×Dic3—82no—
255Dic6—82no—
256C3×C8—44no—
257C24(C2×D3)⋊C411yescharacteristic
258C24(C2×D3)⋊C411yescharacteristic
259C24C2×D1211yescharacteristic
260#41—44no—
261#41—44no—
262#41—48no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—41no—
266C2×Q8—62no—
267C2×Q8—121no—
268C2×Q8—121no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—61no—
289C22×C4—61no—
290C22×C4—61no—
291C22×C4—61no—
292C22×C4—122no—
293C22×C4(C2×D3)⋊C411yescharacteristic
294C22×C4(C2×D3)⋊C411yescharacteristic
295C22×C4C2×C4×D311yescharacteristic
296C22×C4#23512yes—
297C22×C4C2×#6711yescharacteristic
298C22×C4C2×#6712yes—
299M16—44no—
300M16—42no—
301M16—42no—
302C2×C8—24no—
303C2×C8—44no—
304C4⋊C4—62no—
305C4⋊C4—122no—
306C4⋊C4—122no—
307C4⋊C4—121no—
308C4⋊C4—121no—
309C22⋊C4—122no—
310C22⋊C4—122no—
311C22⋊C4—122no—
312C22⋊C4—122no—
313C22⋊C4—122no—
314C22⋊C4—122no—
315C22⋊C4—122no—
316C22⋊C4—122no—
317C42—121no—
318C42—121no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—22no—
322C22×C3—22no—
323C22×C3—22no—
324C22×C3—22no—
325C22×C3—42no—
326C22×C3—42no—
327C22×C3—41no—
328C22×C3—41no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3—41no—
332C22×C3—81no—
333C22×C3—81no—
334C22×C3—81no—
335C22×C3—81no—
336C22×C3#38611yescharacteristic
337C22×C3C2×#9911yescharacteristic
338C22×C3C2×#9911yescharacteristic
339C3×C4—21no—
340C3×C4—22no—
341C3×C4—21no—
342C3×C4—22no—
343C3×C4—41no—
344Dic3—42no—
345Dic3—81no—
346Dic3—81no—
347Dic3—81no—
348Dic3—162no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23#80612yes—
363C23C2×(C2×D3)⋊C411yescharacteristic
364C23#93012yes—
365Q8—44no—
366Q8—242no—
367D4—44no—
368D4—44no—
369D4—44no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—62no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4—121no—
391C2×C4—121no—
392C2×C4—121no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4—121no—
396C2×C4—121no—
397C2×C4—121no—
398C2×C4—242no—
399C2×C4—242no—
400C2×C4—242no—
401C2×C4#80612yes—
402C2×C4C4×#6712yes—
403C2×C4#92911yescharacteristic
404C2×C4#95311yescharacteristic
405C8—44no—
406C2×C3—21no—
407C2×C3—21no—
408C2×C3—42no—
409C2×C3—42no—
410C2×C3—81no—
411C2×C3#188411yescharacteristic
412C2×C3#189811yescharacteristic
413C2×C3#190911yescharacteristic
414C22—21no—
415C22—21no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—42no—
421C22—42no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—81no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22C2×#80611yescharacteristic
432C22C2×#80611yescharacteristic
433C22#521311yescharacteristic
434C4—21no—
435C4—21no—
436C4—22no—
437C4—22no—
438C4—41no—
439C4—122no—
440C4—241no—
441C4—241no—
442C4—241no—
443C4—482no—
444C3#944311yescharacteristic, complemented, Sylow
445C2—21no—
446C2—21no—
447C2—42no—
448C2—42no—
449C2—81no—
450C2#7154311yescharacteristic
451C2#7202111yescharacteristic
452C2#7202411yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2879C2
socleC22×C3#386
prime radicalC22×C3×#41C22
cosocleC23×C3×C4C23
hypercenter#2879D3
hypocenterC3#9443
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2879

Complements

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

NH ≅ G/Ntype
4. #69981C2semidirect
5. #67682C2semidirect
9. C2×#944C4semidirect
10. C2×#936C4semidirect
15. #5229C4semidirect
18. #5221C4semidirect
444. C3#9443semidirect

Maximal subgroups

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

subgroupindex
C3×#28792
#840932
#721392
#699812
#676822
#94433