Finite Group Databasebeta

← #10692972

Subgroups — #10692972

Order 768.

912 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 912 conjugacy classes of subgroups — too many to draw. Only its 123 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×#2408C211yesmaximal, characteristic
2#82302C211yesmaximal, characteristic, complemented
3#72128C212yesmaximal, complemented
4#71619C211yesmaximal, characteristic, complemented
5#67625C212yesmaximal, complemented
6#9472—31nomaximal, Sylow
7C23×C3×D4C2211yescharacteristic
8C2×C3×#121—21no—
9C2×C3×#120—21no—
10C2×C3×C4×D4C2211yescharacteristic
11C3×#394C412yescomplemented
12C3×#392C412yescomplemented
13C3×#390—21no—
14C3×#380C2211yescharacteristic
15C3×#379—21no—
16C2×#834—21no—
17C2×#832—24no—
18#5212C2212yes—
19#5207—21no—
20C2×#830C2212yes—
21#5011—24no—
22#4926—21no—
23#4526—21no—
24#2973—31no—
25#2408D311yescharacteristic
26#1799—31no—
27#1786—32no—
28#1289—32no—
29C25×C3—21no—
30C22×C3×D4—22no—
31C22×C3×D4—22no—
32C22×C3×D4—21no—
33C22×C3×D4—22no—
34C22×C3×D4—44no—
35C22×C3×D4C2×C412yes—
36C22×C3×D4D411yescharacteristic
37C23×C3×C4D411yescharacteristic
38C23×C3×C4C2311yescharacteristic
39C3×#121—44no—
40C3×#120—44no—
41C3×C4×D4—22no—
42C3×C4×D4—22no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4C2×C411yescharacteristic
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4—22no—
47C2×C3×C22⋊C4—22no—
48C2×C3×C22⋊C4—22no—
49C2×C3×C22⋊C4C2×C411yescharacteristic
50C2×C3×C22⋊C4D411yescharacteristic
51C2×C3×C42D411yescharacteristic
52C2×#217—22no—
53C2×C4⋊Dic3—22no—
54C2×#208—22no—
55C22×C3⋊C8D414yescomplemented
56C3×#95C2×C412yes—
57#834—42no—
58#832—48no—
59C3⋊#95—22no—
60#830—22no—
61#830—22no—
62#830—22no—
63C4⋊(C3⋊C8)—22no—
64#803—22no—
65C23×D4C2×D311yescharacteristic
66C2×#121—21no—
67C2×#120—21no—
68C2×C4×D4C2×D311yescharacteristic
69#435—64no—
70#432—61no—
71C2×#102—64no—
72C2×#99—61no—
73#407—61no—
74C2×C22⋊C8—32no—
75#394Dic312yescomplemented
76#392Dic312yescomplemented
77#390—21no—
78#380—32no—
79#380C2×D311yescharacteristic
80#379—21no—
81#342—61no—
82C24×C3—22no—
83C24×C3—44no—
84C24×C3—42no—
85C24×C3—41no—
86C24×C3—42no—
87C24×C3—41no—
88C24×C3—82no—
89C24×C3C22⋊C411yescharacteristic
90C24×C3C2×D411yescharacteristic
91C2×C3×D4—28no—
92C2×C3×D4—24no—
93C2×C3×D4—22no—
94C2×C3×D4—21no—
95C2×C3×D4—22no—
96C2×C3×D4—21no—
97C2×C3×D4—24no—
98C2×C3×D4—44no—
99C2×C3×D4—42no—
100C2×C3×D4—44no—
101C2×C3×D4—48no—
102C2×C3×D4—44no—
103C2×C3×D4—44no—
104C2×C3×D4—41no—
105C2×C3×D4—41no—
106C2×C3×D4—82no—
107C22×C3×C4—21no—
108C22×C3×C4—21no—
109C22×C3×C4—22no—
110C22×C3×C4—42no—
111C22×C3×C4—42no—
112C22×C3×C4C22⋊C411yescharacteristic
113C22×C3×C4C22⋊C411yescharacteristic
114C22×C3×C4C22⋊C411yescharacteristic
115C22×C3×C4C22×C411yescharacteristic
116C22×C3×C4C2×D412yes—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4#4112yes—
119C22×Dic3—42no—
120C3×C4⋊C4—22no—
121C3×C4⋊C4—42no—
122C3×C22⋊C4—22no—
123C3×C22⋊C4—22no—
124C3×C22⋊C4—44no—
125C3×C22⋊C4—44no—
126C3×C22⋊C4—42no—
127C3×C22⋊C4—82no—
128C3×C42—21no—
129C3×C42—21no—
130#217—42no—
131#217—42no—
132C4⋊Dic3—44no—
133#208—82no—
134C2×C3⋊C8—24no—
135C2×C3⋊C8—44no—
136C25—21no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—21no—
141C22×D4—44no—
142C22×D4C2×Dic312yes—
143C22×D4#6711yescharacteristic
144C23×C4#6711yescharacteristic
145C23×C4C22×D311yescharacteristic
146C2×M16—62no—
147C22×C8—34no—
148#121—44no—
149#120—44no—
150C4×D4—22no—
151C4×D4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—62no—
154C2×C4⋊C4C2×Dic311yescharacteristic
155C2×C22⋊C4—21no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—62no—
160C2×C22⋊C4C2×Dic311yescharacteristic
161C2×C22⋊C4#6711yescharacteristic
162C2×C42#6711yescharacteristic
163C4⋊C8—62no—
164#102—128no—
165#99—122no—
166C22⋊C8—62no—
167C22⋊C8—62no—
168C22⋊C8—62no—
169#97—62no—
170#95—62no—
171#95C2×Dic312yes—
172C23×C3—24no—
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—21no—
176C23×C3—41no—
177C23×C3—41no—
178C23×C3—41no—
179C23×C3—41no—
180C23×C3—42no—
181C23×C3—48no—
182C23×C3—42no—
183C23×C3—42no—
184C23×C3—42no—
185C23×C3—42no—
186C23×C3—42no—
187C23×C3—84no—
188C23×C3—88no—
189C23×C3—84no—
190C23×C3—84no—
191C23×C3—82no—
192C23×C3—82no—
193C23×C3C2×C22⋊C411yescharacteristic
194C23×C3#12112yes—
195C23×C3AGL(1,ℤ/8)12yes—
196C23×C3AGL(1,ℤ/8)11yescharacteristic
197C23×C3#13711yescharacteristic
198C3×D4—28no—
199C3×D4—44no—
200C3×D4—44no—
201C3×D4—42no—
202C3×D4—42no—
203C3×D4—44no—
204C3×D4—88no—
205C2×C3×C4—24no—
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—41no—
213C2×C3×C4—42no—
214C2×C3×C4—42no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4—42no—
220C2×C3×C4—81no—
221C2×C3×C4C4×D412yes—
222C2×C3×C4#12011yescharacteristic
223C2×C3×C4#12311yescharacteristic
224C2×Dic3—82no—
225C2×Dic3—82no—
226C2×Dic3—82no—
227C3⋊C8—44no—
228C24—22no—
229C24—44no—
230C24—42no—
231C24—42no—
232C24—41no—
233C24—41no—
234C24—82no—
235C24#21711yescharacteristic
236C24C2×#6711yescharacteristic
237C2×D4—28no—
238C2×D4—24no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—24no—
244C2×D4—42no—
245C2×D4—44no—
246C2×D4—44no—
247C2×D4—48no—
248C2×D4—44no—
249C2×D4—44no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—82no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—122no—
259C22×C4#21711yescharacteristic
260C22×C4#21711yescharacteristic
261C22×C4#21711yescharacteristic
262C22×C4D3×D412yes—
263C22×C4#23712yes—
264C22×C4C22×Dic311yescharacteristic
265C22×C4C2×#6711yescharacteristic
266M16—242no—
267C2×C8—64no—
268C2×C8—124no—
269C4⋊C4—22no—
270C4⋊C4—42no—
271C4⋊C4—124no—
272C22⋊C4—22no—
273C22⋊C4—22no—
274C22⋊C4—44no—
275C22⋊C4—44no—
276C22⋊C4—42no—
277C22⋊C4—82no—
278C22⋊C4—122no—
279C22⋊C4—122no—
280C42—21no—
281C42—21no—
282C22×C3—22no—
283C22×C3—21no—
284C22×C3—22no—
285C22×C3—21no—
286C22×C3—42no—
287C22×C3—41no—
288C22×C3—41no—
289C22×C3—42no—
290C22×C3—44no—
291C22×C3—44no—
292C22×C3—41no—
293C22×C3—44no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—81no—
297C22×C3—82no—
298C22×C3—82no—
299C22×C3—88no—
300C22×C3—88no—
301C22×C3—82no—
302C22×C3—82no—
303C22×C3#38611yescharacteristic
304C22×C3#41711yescharacteristic
305C22×C3#41811yescharacteristic
306C22×C3#46812yes—
307C22×C3#46912yes—
308C3×C4—21no—
309C3×C4—22no—
310C3×C4—41no—
311C3×C4—41no—
312C3×C4—41no—
313C3×C4—82no—
314C3×C4B(4,𝔽2)12yes—
315Dic3—162no—
316C23—24no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—42no—
325C23—48no—
326C23—42no—
327C23—42no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23—84no—
332C23—88no—
333C23—84no—
334C23—84no—
335C23—82no—
336C23—82no—
337C23#93212yes—
338C23#93912yes—
339C23#94911yescharacteristic
340C23#95111yescharacteristic
341C23C2×#21711yescharacteristic
342D4—28no—
343D4—44no—
344D4—44no—
345D4—42no—
346D4—42no—
347D4—44no—
348D4—88no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—81no—
365C2×C4—242no—
366C2×C4—242no—
367C2×C4—242no—
368C2×C4D4×Dic312yes—
369C2×C4#93511yescharacteristic
370C2×C4#93711yescharacteristic
371C8—124no—
372C2×C3—22no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—22no—
376C2×C3—81no—
377C2×C3—84no—
378C2×C3#188911yescharacteristic
379C2×C3#189511yescharacteristic
380C2×C3#191011yescharacteristic
381C22—21no—
382C22—22no—
383C22—21no—
384C22—22no—
385C22—42no—
386C22—41no—
387C22—41no—
388C22—42no—
389C22—44no—
390C22—44no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—44no—
395C22—81no—
396C22—82no—
397C22—82no—
398C22—88no—
399C22—88no—
400C22—82no—
401C22—82no—
402C22#503912yes—
403C22#504212yes—
404C22#521511yescharacteristic
405C22#522011yescharacteristic
406C22#523211yescharacteristic
407C4—21no—
408C4—22no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—82no—
413C4—482no—
414C4#503212yes—
415C3#947211yescharacteristic, complemented, Sylow
416C2—21no—
417C2—22no—
418C2—21no—
419C2—22no—
420C2—81no—
421C2—84no—
422C2#7166911yescharacteristic
423C2#7168211yescharacteristic
424C2#7185411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#2408C2
socleC22×C3#386
cosocleC23×C3×C4C23
hypercenter#2408D3
hypocenterC3#9472
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2408

Complements

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

NH ≅ G/Ntype
2. #82302C2semidirect
3. #72128C2semidirect
4. #71619C2semidirect
5. #67625C2semidirect
11. C3×#394C4semidirect
12. C3×#392C4semidirect
55. C22×C3⋊C8D4semidirect
75. #394Dic3semidirect
76. #392Dic3semidirect
415. C3#9472semidirect

Maximal subgroups

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

subgroupindex
C3×#24082
#823022
#721282
#716192
#676252
#94723