Finite Group Databasebeta

← #10690157

Subgroups — #10690157

Order 768.

784 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 111 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×#2388C211yesmaximal, characteristic
2#82302C211yesmaximal, characteristic, complemented
3#72138C212yesmaximal, complemented
4C4×#834C211yesmaximal, characteristic, complemented
5#67618C212yesmaximal, complemented
6#9009—31nomaximal, Sylow
7C22×C3×#41C2211yescharacteristic
8C2×C3×C4×D4—21no—
9C2×C3×C4×D4C2211yescharacteristic
10C3×#395C411yescharacteristic, complemented
11C3×#394—21no—
12C3×#394C411yescharacteristic, complemented
13C3×#393C411yescharacteristic, complemented
14C3×#392—21no—
15C3×#392C411yescharacteristic, complemented
16C3×C4×C22⋊C4—21no—
17C3×C4×C22⋊C4C2211yescharacteristic
18C2×#837—24no—
19C2×#834—21no—
20C4×#217C2212yes—
21#5207C2212yes—
22C2×#830—21no—
23#5011—24no—
24#4926—21no—
25#4526—21no—
26#2973—31no—
27#2388D311yescharacteristic
28#1789—32no—
29C4×#99—31no—
30#1297—32no—
31C2×C3×#41—22no—
32C2×C3×#41—22no—
33C2×C3×#41—44no—
34C22×C3×Q8C2×C411yescharacteristic
35C22×C3×D4—21no—
36C22×C3×D4—21no—
37C22×C3×D4C2×C411yescharacteristic
38C22×C3×D4D411yescharacteristic
39C23×C3×C4—21no—
40C23×C3×C4D411yescharacteristic
41C23×C3×C4C2311yescharacteristic
42C3×C4×D4—22no—
43C3×C4×D4—22no—
44C3×C4×D4—44no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4C2×C411yescharacteristic
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C22⋊C4C2×C411yescharacteristic
53C2×C3×C22⋊C4D411yescharacteristic
54C2×C3×C42—21no—
55C2×C3×C42D411yescharacteristic
56C2×#217—22no—
57C2×C4×Dic3—22no—
58C2×#208D414yescomplemented
59C22×C3⋊C8—22no—
60C3×#95—21no—
61C3×#95—21no—
62C3×#95C2×C411yescharacteristic
63C3×#95C2×C411yescharacteristic
64#837—48no—
65#834—24no—
66C3⋊#95—22no—
67#830—22no—
68#830—22no—
69#830—22no—
70C4⋊(C3⋊C8)—22no—
71#803—22no—
72C22×#41C2×D311yescharacteristic
73C2×C4×D4—21no—
74C2×C4×D4C2×D311yescharacteristic
75#435—64no—
76#432—61no—
77C2×#104—64no—
78C2×#99—61no—
79#407—32no—
80C2×C22⋊C8—61no—
81#395Dic311yescharacteristic, complemented
82#394—21no—
83#394Dic311yescharacteristic, complemented
84#393Dic311yescharacteristic, complemented
85#392—21no—
86#392Dic311yescharacteristic, complemented
87C4×C22⋊C4—21no—
88C4×C22⋊C4—32no—
89C4×C22⋊C4C2×D311yescharacteristic
90#342—61no—
91C24×C3—21no—
92C24×C3C22⋊C411yescharacteristic
93C24×C3C2×D411yescharacteristic
94C3×#41—28no—
95C3×#41—44no—
96C3×#41—44no—
97C3×#41—44no—
98C2×C3×Q8—22no—
99C2×C3×Q8—42no—
100C2×C3×D4—22no—
101C2×C3×D4—22no—
102C2×C3×D4—22no—
103C2×C3×D4—41no—
104C2×C3×D4—42no—
105C2×C3×D4—42no—
106C2×C3×D4—42no—
107C2×C3×D4—42no—
108C2×C3×D4—42no—
109C2×C3×D4—42no—
110C2×C3×D4—42no—
111C22×C3×C4—24no—
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4—42no—
118C22×C3×C4—42no—
119C22×C3×C4—42no—
120C22×C3×C4—41no—
121C22×C3×C4—41no—
122C22×C3×C4C22⋊C411yescharacteristic
123C22×C3×C4C22⋊C411yescharacteristic
124C22×C3×C4C22⋊C411yescharacteristic
125C22×C3×C4C22×C411yescharacteristic
126C22×C3×C4C2×D412yes—
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4#4112yes—
129C22×Dic3—42no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—42no—
132C3×C22⋊C4—22no—
133C3×C22⋊C4—22no—
134C3×C22⋊C4—42no—
135C3×C22⋊C4—42no—
136C3×C22⋊C4—42no—
137C3×C22⋊C4—42no—
138C3×C22⋊C4—42no—
139C3×C42—21no—
140C3×C42—21no—
141C3×C42—42no—
142#217—24no—
143#217—42no—
144C4×Dic3—44no—
145#208—44no—
146C2×C3⋊C8—24no—
147C2×C3⋊C8—82no—
148C2×#41—22no—
149C2×#41—22no—
150C2×#41—44no—
151C22×Q8C2×Dic311yescharacteristic
152C22×D4—21no—
153C22×D4—21no—
154C22×D4C2×Dic311yescharacteristic
155C22×D4#6711yescharacteristic
156C23×C4—21no—
157C23×C4#6711yescharacteristic
158C23×C4C22×D311yescharacteristic
159C2×M16—34no—
160C22×C8—62no—
161C4×D4—22no—
162C4×D4—22no—
163C4×D4—44no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4C2×Dic311yescharacteristic
166C2×C22⋊C4—21no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—62no—
172C2×C22⋊C4C2×Dic311yescharacteristic
173C2×C22⋊C4#6711yescharacteristic
174C2×C42—21no—
175C2×C42—62no—
176C2×C42#6711yescharacteristic
177C4⋊C8—62no—
178#104—128no—
179#99—64no—
180C22⋊C8—62no—
181C22⋊C8—62no—
182C22⋊C8—62no—
183#97—62no—
184#95—21no—
185#95—21no—
186#95—62no—
187#95C2×Dic311yescharacteristic
188#95C2×Dic311yescharacteristic
189C23×C3—21no—
190C23×C3—21no—
191C23×C3—21no—
192C23×C3—21no—
193C23×C3—22no—
194C23×C3—41no—
195C23×C3—41no—
196C23×C3—41no—
197C23×C3—41no—
198C23×C3—41no—
199C23×C3—41no—
200C23×C3—41no—
201C23×C3—41no—
202C23×C3—41no—
203C23×C3—41no—
204C23×C3#10412yescomplemented
205C23×C3C2×C22⋊C411yescharacteristic
206C23×C3#12112yes—
207C3×Q8—24no—
208C3×Q8—42no—
209C3×D4—24no—
210C3×D4—42no—
211C3×D4—44no—
212C3×D4—48no—
213C3×D4—48no—
214C2×C3×C4—24no—
215C2×C3×C4—22no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—41no—
224C2×C3×C4—42no—
225C2×C3×C4—44no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—42no—
231C2×C3×C4—44no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—81no—
238C2×C3×C4C4×D412yes—
239C2×C3×C4#12011yescharacteristic
240C2×C3×C4#12311yescharacteristic
241C2×Dic3—44no—
242C2×Dic3—82no—
243C2×Dic3—82no—
244C3⋊C8—44no—
245C24—21no—
246C24#21711yescharacteristic
247C24C2×#6711yescharacteristic
248#41—28no—
249#41—44no—
250#41—44no—
251#41—44no—
252C2×Q8—22no—
253C2×Q8—42no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—41no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C22×C4—24no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—122no—
277C22×C4#21711yescharacteristic
278C22×C4#21711yescharacteristic
279C22×C4#21711yescharacteristic
280C22×C4D3×D412yes—
281C22×C4#23712yes—
282C22×C4C22×Dic311yescharacteristic
283C22×C4C2×#6711yescharacteristic
284M16—124no—
285C2×C8—64no—
286C2×C8—242no—
287C4⋊C4—22no—
288C4⋊C4—42no—
289C22⋊C4—22no—
290C22⋊C4—22no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—64no—
297C22⋊C4—122no—
298C42—21no—
299C42—21no—
300C42—42no—
301C42—124no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—22no—
305C22×C3—22no—
306C22×C3—22no—
307C22×C3—22no—
308C22×C3—42no—
309C22×C3—42no—
310C22×C3—42no—
311C22×C3—41no—
312C22×C3—41no—
313C22×C3—42no—
314C22×C3—81no—
315C22×C3—81no—
316C22×C3—81no—
317C22×C3—81no—
318C22×C3—81no—
319C22×C3—81no—
320C22×C3—81no—
321C22×C3#38611yescharacteristic
322C22×C3C2×#10411yescharacteristic
323C22×C3#42111yescharacteristic
324C3×C4—21no—
325C3×C4—22no—
326C3×C4—42no—
327C3×C4—41no—
328C3×C4—41no—
329C3×C4—42no—
330C3×C4—41no—
331C3×C4—81no—
332C3×C4B(4,𝔽2)12yes—
333Dic3—84no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—22no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23#83712yescomplemented
350C23#93912yes—
351C23C2×#21711yescharacteristic
352Q8—24no—
353Q8—42no—
354D4—24no—
355D4—42no—
356D4—44no—
357D4—48no—
358D4—48no—
359C2×C4—22no—
360C2×C4—24no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—44no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—81no—
383C2×C4—124no—
384C2×C4—242no—
385C2×C4—242no—
386C2×C4D4×Dic312yes—
387C2×C4#93511yescharacteristic
388C2×C4#93711yescharacteristic
389C8—124no—
390C2×C3—21no—
391C2×C3—21no—
392C2×C3—42no—
393C2×C3—42no—
394C2×C3—82no—
395C2×C3#187111yescharacteristic
396C2×C3#188411yescharacteristic
397C2×C3#188911yescharacteristic
398C22—21no—
399C22—21no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—42no—
405C22—42no—
406C22—42no—
407C22—41no—
408C22—41no—
409C22—42no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22#522011yescharacteristic
418C22C2×#83711yescharacteristic
419C22#523611yescharacteristic
420C4—21no—
421C4—22no—
422C4—42no—
423C4—41no—
424C4—41no—
425C4—42no—
426C4—41no—
427C4—81no—
428C4—244no—
429C4#503212yes—
430C3#900911yescharacteristic, complemented, Sylow
431C2—21no—
432C2—21no—
433C2—42no—
434C2—42no—
435C2—82no—
436C2#7166911yescharacteristic
437C2#7171311yescharacteristic
438C2#7181711yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#2388C2
socleC22×C3#386
cosocleC23×C3×C4C23
hypercenter#2388D3
hypocenterC3#9009
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2388

Complements

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

NH ≅ G/Ntype
2. #82302C2semidirect
3. #72138C2semidirect
4. C4×#834C2semidirect
5. #67618C2semidirect
10. C3×#395C4semidirect
12. C3×#394C4semidirect
13. C3×#393C4semidirect
15. C3×#392C4semidirect
58. C2×#208D4semidirect
81. #395Dic3semidirect
83. #394Dic3semidirect
84. #393Dic3semidirect
86. #392Dic3semidirect
204. C23×C3#104semidirect
349. C23#837semidirect
430. C3#9009semidirect

Maximal subgroups

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

subgroupindex
C3×#23882
#823022
#721382
C4×#8342
#676182
#90093