Finite Group Databasebeta

← #10692969

Subgroups — #10692969

Order 768.

912 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 912 conjugacy classes of subgroups — too many to draw. Only its 115 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
1#82403C211yesmaximal, characteristic
2#82302C211yesmaximal, characteristic, complemented
3C3×#1799C211yesmaximal, characteristic, complemented
4#69521C212yesmaximal, complemented
5#67625C212yesmaximal, complemented
6#9472—31nomaximal, Sylow
7C23×D12C2211yescharacteristic
8C2×C3×C4×D4C2211yescharacteristic
9C2×#937—21no—
10C2×#930—21no—
11C2×C3×#99—21no—
12C3×#380C2212yes—
13#5212C2211yescharacteristic
14#5207—21no—
15C2×#830C2212yes—
16#5011—24no—
17#4991C412yescomplemented
18C2×#809—24no—
19#4960C412yescomplemented
20#4955—21no—
21#4939—21no—
22#4926—21no—
23C3×#342—21no—
24#2973—31no—
25#2408—31no—
26#1799D311yescharacteristic, complemented
27#1786—32no—
28#1289—32no—
29C24×D3—21no—
30C22×C3×D4D411yescharacteristic
31C23×C3×C4D411yescharacteristic
32C23×C3×C4C2311yescharacteristic
33C22×#67—22no—
34C22×D12—22no—
35C22×D12—21no—
36C22×D12—22no—
37C22×D12—44no—
38C22×D12C2×C412yes—
39C3×C4×D4—22no—
40C3×C4×D4—22no—
41C2×C3×C4⋊C4—22no—
42C2×C3×C4⋊C4C2×C411yescharacteristic
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4C2×C411yescharacteristic
45C2×C3×C22⋊C4D411yescharacteristic
46C2×C3×C42D411yescharacteristic
47C2×#217—21no—
48#937—44no—
49#930—44no—
50C2×(C2×D3)⋊C4—22no—
51C2×(C2×D3)⋊C4—22no—
52C2×(C2×D3)⋊C4—22no—
53C2×C4⋊Dic3—21no—
54C2×#208—22no—
55C22×C3⋊C8D414yescomplemented
56C3×#99—42no—
57C3×#95—22no—
58C3⋊#95C2×C412yes—
59#830—22no—
60#830—22no—
61#830—22no—
62#809—48no—
63C4⋊(C3⋊C8)—22no—
64#803—22no—
65C23×D4—31no—
66C2×#121—61no—
67C2×#120—61no—
68C2×C4×D4C2×D311yescharacteristic
69#435—64no—
70#432—61no—
71C2×#102—64no—
72C2×#99—21no—
73#407—61no—
74C2×C22⋊C8—32no—
75#394—32no—
76#392—32no—
77#390—61no—
78#380—31no—
79#380C2×D312yes—
80#379—61no—
81#342—21no—
82C24×C3C22⋊C411yescharacteristic
83C24×C3C2×D411yescharacteristic
84C23×D3—22no—
85C23×D3—44no—
86C23×D3—42no—
87C23×D3—42no—
88C23×D3—41no—
89C23×D3—41no—
90C23×D3—82no—
91C2×C3×D4—22no—
92C2×C3×D4—21no—
93C2×C3×D4—22no—
94C2×C3×D4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—42no—
98C22×C3×C4—42no—
99C22×C3×C4—42no—
100C22×C3×C4C22⋊C411yescharacteristic
101C22×C3×C4C22⋊C411yescharacteristic
102C22×C3×C4C22⋊C411yescharacteristic
103C22×C3×C4C22×C411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D412yes—
106C22×C3×C4#4112yes—
107C2×#67—44no—
108C2×#67—44no—
109C2×#67—82no—
110C22×Dic3—22no—
111C2×D12—28no—
112C2×D12—24no—
113C2×D12—24no—
114C2×D12—44no—
115C2×D12—48no—
116C2×D12—44no—
117C2×D12—42no—
118C2×D12—41no—
119C2×D12—41no—
120C3×C4⋊C4—22no—
121C3×C4⋊C4—44no—
122C3×C22⋊C4—22no—
123C3×C22⋊C4—22no—
124C3×C22⋊C4—42no—
125C3×C22⋊C4—42no—
126C3×C42—21no—
127C3×C42—21no—
128#217—42no—
129(C2×D3)⋊C4—44no—
130(C2×D3)⋊C4—44no—
131(C2×D3)⋊C4—82no—
132C4⋊Dic3—42no—
133#208—82no—
134C2×C3⋊C8—24no—
135C2×C3⋊C8—44no—
136C25—61no—
137C22×D4—32no—
138C22×D4—62no—
139C22×D4—61no—
140C22×D4—62no—
141C22×D4—62no—
142C22×D4—124no—
143C22×D4#6711yescharacteristic
144C23×C4D1211yescharacteristic
145C23×C4C22×D311yescharacteristic
146C2×M16—62no—
147C22×C8—34no—
148#121—124no—
149#120—124no—
150C4×D4—22no—
151C4×D4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—61no—
154C2×C4⋊C4C4×D311yescharacteristic
155C2×C22⋊C4—22no—
156C2×C22⋊C4—61no—
157C2×C22⋊C4—62no—
158C2×C22⋊C4—62no—
159C2×C22⋊C4—62no—
160C2×C22⋊C4C4×D311yescharacteristic
161C2×C22⋊C4#6711yescharacteristic
162C2×C42D1211yescharacteristic
163C4⋊C8—62no—
164#102—128no—
165#99—42no—
166C22⋊C8—62no—
167C22⋊C8—62no—
168C22⋊C8—62no—
169#97—62no—
170#95—22no—
171#95—32no—
172C23×C3—21no—
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—41no—
176C23×C3—41no—
177C23×C3—41no—
178C23×C3—41no—
179C23×C3C2×C22⋊C411yescharacteristic
180C23×C3#12112yes—
181C23×C3AGL(1,ℤ/8)11yescharacteristic
182C23×C3AGL(1,ℤ/8)12yes—
183C23×C3#13711yescharacteristic
184C22×D3—24no—
185C22×D3—42no—
186C22×D3—48no—
187C22×D3—42no—
188C22×D3—42no—
189C22×D3—42no—
190C22×D3—42no—
191C22×D3—42no—
192C22×D3—88no—
193C22×D3—84no—
194C22×D3—84no—
195C22×D3—84no—
196C22×D3—82no—
197C22×D3—82no—
198C3×D4—42no—
199C3×D4—42no—
200C2×C3×C4—24no—
201C2×C3×C4—22no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—41no—
208C2×C3×C4—42no—
209C2×C3×C4—42no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—81no—
213C2×C3×C4—82no—
214C2×C3×C4—82no—
215C2×C3×C4—82no—
216C2×C3×C4C4×D412yes—
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12311yescharacteristic
219#67—88no—
220C2×Dic3—42no—
221C2×Dic3—42no—
222C2×Dic3—42no—
223D12—28no—
224D12—44no—
225D12—44no—
226D12—44no—
227C3⋊C8—44no—
228C24—62no—
229C24—124no—
230C24—122no—
231C24—121no—
232C24—121no—
233C24—122no—
234C24—242no—
235C24(C2×D3)⋊C411yescharacteristic
236C24C2×#6711yescharacteristic
237C2×D4—21no—
238C2×D4—22no—
239C2×D4—21no—
240C2×D4—22no—
241C2×D4—68no—
242C2×D4—64no—
243C2×D4—64no—
244C2×D4—124no—
245C2×D4—122no—
246C2×D4—124no—
247C2×D4—128no—
248C2×D4—124no—
249C2×D4—124no—
250C2×D4—121no—
251C2×D4—121no—
252C2×D4—242no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—62no—
259C22×C4(C2×D3)⋊C411yescharacteristic
260C22×C4(C2×D3)⋊C411yescharacteristic
261C22×C4(C2×D3)⋊C411yescharacteristic
262C22×C4C2×C4×D311yescharacteristic
263C22×C4C2×D1211yescharacteristic
264C22×C4D3×D412yes—
265C22×C4#23712yes—
266M16—242no—
267C2×C8—64no—
268C2×C8—124no—
269C4⋊C4—22no—
270C4⋊C4—44no—
271C4⋊C4—122no—
272C22⋊C4—22no—
273C22⋊C4—22no—
274C22⋊C4—42no—
275C22⋊C4—42no—
276C22⋊C4—124no—
277C22⋊C4—124no—
278C22⋊C4—122no—
279C22⋊C4—242no—
280C42—21no—
281C42—21no—
282C22×C3—21no—
283C22×C3—22no—
284C22×C3—21no—
285C22×C3—22no—
286C22×C3—41no—
287C22×C3—42no—
288C22×C3—41no—
289C22×C3—42no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3—81no—
294C22×C3#38611yescharacteristic
295C22×C3#41711yescharacteristic
296C22×C3#41811yescharacteristic
297C22×C3#46812yes—
298C22×C3#46912yes—
299C2×D3—44no—
300C2×D3—44no—
301C2×D3—44no—
302C2×D3—82no—
303C2×D3—82no—
304C2×D3—88no—
305C2×D3—88no—
306C2×D3—82no—
307C2×D3—82no—
308C3×C4—22no—
309C3×C4—21no—
310C3×C4—41no—
311C3×C4—41no—
312C3×C4—41no—
313C3×C4—162no—
314C3×C4B(4,𝔽2)12yes—
315Dic3—82no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—64no—
324C23—122no—
325C23—128no—
326C23—122no—
327C23—122no—
328C23—122no—
329C23—122no—
330C23—122no—
331C23—244no—
332C23—244no—
333C23—244no—
334C23—248no—
335C23—242no—
336C23—242no—
337C23C2×(C2×D3)⋊C411yescharacteristic
338C23#93211yescharacteristic
339C23#93912yes—
340C23#94211yescharacteristic
341C23#94912yes—
342D4—42no—
343D4—42no—
344D4—68no—
345D4—124no—
346D4—124no—
347D4—124no—
348D4—248no—
349C2×C4—24no—
350C2×C4—22no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—81no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—122no—
366C2×C4—122no—
367C2×C4—122no—
368C2×C4#67⋊C412yes—
369C2×C4#88211yescharacteristic
370C2×C4#88611yescharacteristic
371C8—124no—
372C2×C3—21no—
373C2×C3—22no—
374C2×C3—21no—
375C2×C3—22no—
376C2×C3—81no—
377C2×C3#188911yescharacteristic
378C2×C3#189511yescharacteristic
379C2×C3#191011yescharacteristic
380D3—84no—
381C22—21no—
382C22—22no—
383C22—21no—
384C22—22no—
385C22—41no—
386C22—41no—
387C22—42no—
388C22—42no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—81no—
393C22—124no—
394C22—124no—
395C22—124no—
396C22—242no—
397C22—242no—
398C22—248no—
399C22—248no—
400C22—242no—
401C22—242no—
402C22#495811yescharacteristic
403C22#496611yescharacteristic
404C22#500111yescharacteristic
405C22#503912yes—
406C22#504912yes—
407C4—21no—
408C4—22no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—162no—
413C4—242no—
414C4#474112yes—
415C3#947211yescharacteristic, complemented, Sylow
416C2—22no—
417C2—21no—
418C2—21no—
419C2—22no—
420C2—81no—
421C2—244no—
422C2#6951711yescharacteristic
423C2#7002211yescharacteristic
424C2#7004711yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1799C2
socleC22×C3#386
prime radical#82403C2
cosocleC23×C3×C4C23
hypercenter#1799D3
hypocenterC3#9472
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
2. #82302C2semidirect
3. C3×#1799C2semidirect
4. #69521C2semidirect
5. #67625C2semidirect
17. #4991C4semidirect
19. #4960C4semidirect
26. #1799D3semidirect
55. C22×C3⋊C8D4semidirect
415. C3#9472semidirect

Maximal subgroups

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

subgroupindex
#824032
#823022
C3×#17992
#695212
#676252
#94723