Finite Group Databasebeta

← #10692518

Subgroups — #10692518

Order 768.

618 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 119 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×#2543C211yesmaximal, characteristic
2#82302C211yesmaximal, characteristic, complemented
3#71675C212yesmaximal, complemented
4#71619C211yesmaximal, characteristic, complemented
5#67612C212yesmaximal, complemented
6#9386—31nomaximal, Sylow
7C2×C3×#121—21no—
8C2×C3×#121C2211yescharacteristic
9C2×C3×C4×D4C2211yescharacteristic
10C3×#398C411yescharacteristic, complemented
11C3×#397C411yescharacteristic, complemented
12C3×#394C411yescharacteristic, complemented
13C3×#392C411yescharacteristic, complemented
14C3×#388—21no—
15C3×#386—21no—
16C3×#386—21no—
17C2×C3×#95C2211yescharacteristic
18C2×#834—21no—
19#5212C2212yes—
20#5207—21no—
21C2×#830C2212yes—
22#5011—24no—
23#4926—21no—
24#4537—24no—
25#4526—21no—
26#2973—31no—
27#2543D311yescharacteristic
28#1799—31no—
29#1519—32no—
30#1294—32no—
31C22×C3×D4—21no—
32C22×C3×D4—21no—
33C22×C3×D4C2×C411yescharacteristic
34C22×C3×D4D411yescharacteristic
35C23×C3×C4—21no—
36C23×C3×C4D411yescharacteristic
37C23×C3×C4C2311yescharacteristic
38C3×#121—22no—
39C3×#121—22no—
40C3×#121—44no—
41C3×C4×D4—22no—
42C3×C4×D4—22no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4C2×C411yescharacteristic
45C2×C3×C4⋊C4C2×C411yescharacteristic
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4C2×C411yescharacteristic
52C2×C3×C22⋊C4D411yescharacteristic
53C2×C3×C42D411yescharacteristic
54C2×#217—22no—
55C2×C4⋊Dic3—22no—
56C2×#208—22no—
57C22×C3⋊C8—22no—
58C3×#95—21no—
59C3×#95—21no—
60C3×#95—21no—
61C3×#95C2×C411yescharacteristic
62C3×#95C2×C411yescharacteristic
63#834—42no—
64C3⋊#95—22no—
65#830—22no—
66#830—22no—
67#830D414yescomplemented
68C4⋊(C3⋊C8)—22no—
69#803—22no—
70C2×#121—21no—
71C2×#121C2×D311yescharacteristic
72C2×C4×D4C2×D311yescharacteristic
73#435—64no—
74#432—61no—
75C2×#99—61no—
76#407—61no—
77C2×C22⋊C8—32no—
78#398Dic311yescharacteristic, complemented
79#397Dic311yescharacteristic, complemented
80#394Dic311yescharacteristic, complemented
81#392Dic311yescharacteristic, complemented
82#388—21no—
83#386—21no—
84#386—21no—
85#380—32no—
86C2×#95C2×D311yescharacteristic
87#342—61no—
88#327—64no—
89C24×C3—21no—
90C24×C3C22⋊C411yescharacteristic
91C24×C3C2×D411yescharacteristic
92C2×C3×D4—22no—
93C2×C3×D4—21no—
94C2×C3×D4—22no—
95C2×C3×D4—21no—
96C2×C3×D4—22no—
97C2×C3×D4—42no—
98C2×C3×D4—42no—
99C2×C3×D4—42no—
100C2×C3×D4—42no—
101C2×C3×D4—41no—
102C2×C3×D4—41no—
103C2×C3×D4—81no—
104C2×C3×D4—81no—
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4—21no—
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4—42no—
111C22×C3×C4—42no—
112C22×C3×C4—81no—
113C22×C3×C4—81no—
114C22×C3×C4—81no—
115C22×C3×C4C22⋊C411yescharacteristic
116C22×C3×C4C22⋊C411yescharacteristic
117C22×C3×C4C22⋊C411yescharacteristic
118C22×C3×C4C22×C411yescharacteristic
119C22×C3×C4C2×D412yes—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4#4112yes—
122C22×Dic3—42no—
123C3×C4⋊C4—22no—
124C3×C4⋊C4—22no—
125C3×C4⋊C4—42no—
126C3×C22⋊C4—22no—
127C3×C22⋊C4—22no—
128C3×C22⋊C4—42no—
129C3×C22⋊C4—42no—
130C3×C22⋊C4—42no—
131C3×C22⋊C4—41no—
132C3×C22⋊C4—41no—
133C3×C22⋊C4—81no—
134C3×C42—21no—
135C3×C42—21no—
136#217—42no—
137#217—42no—
138C4⋊Dic3—82no—
139#208—82no—
140C2×C3⋊C8—24no—
141C2×C3⋊C8—82no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4C2×Dic311yescharacteristic
145C22×D4#6711yescharacteristic
146C23×C4—21no—
147C23×C4#6711yescharacteristic
148C23×C4C22×D311yescharacteristic
149C2×M16—62no—
150C22×C8—62no—
151#121—22no—
152#121—22no—
153#121—44no—
154C4×D4—22no—
155C4×D4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—62no—
158C2×C4⋊C4C2×Dic311yescharacteristic
159C2×C4⋊C4C2×Dic311yescharacteristic
160C2×C22⋊C4—21no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—62no—
166C2×C22⋊C4C2×Dic311yescharacteristic
167C2×C22⋊C4#6711yescharacteristic
168C2×C42#6711yescharacteristic
169C4⋊C8—62no—
170#99—122no—
171C22⋊C8—34no—
172C22⋊C8—62no—
173C22⋊C8—62no—
174#97—62no—
175#95—21no—
176#95—21no—
177#95—21no—
178#95—62no—
179#95C2×Dic311yescharacteristic
180#95C2×Dic311yescharacteristic
181C23×C3—21no—
182C23×C3—21no—
183C23×C3—21no—
184C23×C3—41no—
185C23×C3—41no—
186C23×C3—41no—
187C23×C3—41no—
188C23×C3—41no—
189C23×C3—41no—
190C23×C3—41no—
191C23×C3—41no—
192C23×C3—41no—
193C23×C3—41no—
194C23×C3—41no—
195C23×C3C2×C22⋊C411yescharacteristic
196C23×C3#12112yes—
197C23×C3AGL(1,ℤ/8)12yes—
198C23×C3AGL(1,ℤ/8)11yescharacteristic
199C23×C3#13711yescharacteristic
200C3×D4—42no—
201C3×D4—42no—
202C3×D4—82no—
203C3×D4—84no—
204C3×D4—82no—
205C3×D4—82no—
206C2×C3×C4—22no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—42no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—81no—
228C2×C3×C4—84no—
229C2×C3×C4C4×D412yes—
230C2×C3×C4#12011yescharacteristic
231C2×C3×C4#12311yescharacteristic
232C2×Dic3—82no—
233C2×Dic3—82no—
234C2×Dic3—82no—
235C3⋊C8—44no—
236C24—21no—
237C24#21711yescharacteristic
238C24C2×#6711yescharacteristic
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—22no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—81no—
251C2×D4—81no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—122no—
263C22×C4#21711yescharacteristic
264C22×C4#21711yescharacteristic
265C22×C4#21711yescharacteristic
266C22×C4D3×D412yes—
267C22×C4#23712yes—
268C22×C4C22×Dic311yescharacteristic
269C22×C4C2×#6711yescharacteristic
270M16—242no—
271C2×C8—64no—
272C2×C8—242no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—42no—
276C4⋊C4—242no—
277C22⋊C4—22no—
278C22⋊C4—22no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—81no—
285C22⋊C4—122no—
286C22⋊C4—122no—
287C42—21no—
288C42—21no—
289C22×C3—22no—
290C22×C3—21no—
291C22×C3—22no—
292C22×C3—21no—
293C22×C3—42no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—42no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—81no—
301C22×C3—81no—
302C22×C3—81no—
303C22×C3—81no—
304C22×C3—81no—
305C22×C3—81no—
306C22×C3—81no—
307C22×C3—81no—
308C22×C3#38611yescharacteristic
309C22×C3#41711yescharacteristic
310C22×C3#41811yescharacteristic
311C22×C3#46812yes—
312C22×C3#46912yes—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—22no—
316C3×C4—42no—
317C3×C4—41no—
318C3×C4—41no—
319C3×C4—41no—
320C3×C4—81no—
321C3×C4—81no—
322Dic3—162no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23#93212yes—
338C23#93912yes—
339C23#94911yescharacteristic
340C23#95111yescharacteristic
341C23C2×#21711yescharacteristic
342D4—42no—
343D4—42no—
344D4—82no—
345D4—84no—
346D4—82no—
347D4—82no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—81no—
370C2×C4—84no—
371C2×C4—242no—
372C2×C4—242no—
373C2×C4—242no—
374C2×C4D4×Dic312yes—
375C2×C4#93511yescharacteristic
376C2×C4#93711yescharacteristic
377C8—124no—
378C2×C3—22no—
379C2×C3—21no—
380C2×C3—21no—
381C2×C3—22no—
382C2×C3—81no—
383C2×C3—161no—
384C2×C3#188411yescharacteristic
385C2×C3#189511yescharacteristic
386C2×C3#191311yescharacteristic
387C22—21no—
388C22—22no—
389C22—21no—
390C22—22no—
391C22—42no—
392C22—41no—
393C22—41no—
394C22—42no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22#503912yes—
407C22#504212yes—
408C22#521511yescharacteristic
409C22#522011yescharacteristic
410C22#523211yescharacteristic
411C4—21no—
412C4—21no—
413C4—22no—
414C4—42no—
415C4—41no—
416C4—41no—
417C4—41no—
418C4—81no—
419C4—81no—
420C4—482no—
421C3#938611yescharacteristic, complemented, Sylow
422C2—21no—
423C2—22no—
424C2—21no—
425C2—22no—
426C2—81no—
427C2—161no—
428C2#7168211yescharacteristic
429C2#7181711yescharacteristic
430C2#7184811yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#2543C2
socleC22×C3#386
cosocleC23×C3×C4C23
hypercenter#2543D3
hypocenterC3#9386
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2543

Complements

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

NH ≅ G/Ntype
2. #82302C2semidirect
3. #71675C2semidirect
4. #71619C2semidirect
5. #67612C2semidirect
10. C3×#398C4semidirect
11. C3×#397C4semidirect
12. C3×#394C4semidirect
13. C3×#392C4semidirect
67. #830D4semidirect
78. #398Dic3semidirect
79. #397Dic3semidirect
80. #394Dic3semidirect
81. #392Dic3semidirect
421. C3#9386semidirect

Maximal subgroups

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

subgroupindex
C3×#25432
#823022
#716752
#716192
#676122
#93863