Finite Group Databasebeta

← #10690975

Subgroups — #10690975

Order 768.

980 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 980 conjugacy classes of subgroups — too many to draw. Only its 135 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#84032C211yesmaximal, characteristic
2C2×#5211C211yesmaximal, characteristic, complemented
3C3×#1779C211yesmaximal, characteristic, complemented
4C2×#4470C212yesmaximal, complemented
5#67675C212yesmaximal, complemented
6#9158—31nomaximal, Sylow
7C24×C3×C4C2211yescharacteristic
8C2×#953—21no—
9C22×#217—21no—
10C2×#930C412yescomplemented
11C2×#930C2211yescharacteristic
12C2×C3×C22⋊C8—22no—
13C2×C3×C22⋊C8C2212yes—
14#5213—21no—
15#5213—22no—
16#5212C412yescomplemented
17#5212C2211yescharacteristic
18#5211—24no—
19C2×C3⋊#95—21no—
20C2×C3⋊#95C2212yes—
21#4470—24no—
22#2847—31no—
23C2×#381—31no—
24#1779D311yescharacteristic, complemented
25C2×#327—32no—
26#1294—32no—
27C25×C3D411yescharacteristic
28C23×C3×C4—22no—
29C23×C3×C4—22no—
30C23×C3×C4—44no—
31C23×C3×C4C2×C412yes—
32C23×C3×C4D411yescharacteristic
33C23×C3×C4D412yes—
34C23×C3×C4C2311yescharacteristic
35C22×#67—22no—
36C22×#67—21no—
37C23×Dic3—22no—
38C22×D12C2×C411yescharacteristic
39C22×C3×C8—22no—
40#953—44no—
41C2×#217—24no—
42C2×#217—22no—
43C2×#217—21no—
44C2×#217—44no—
45#930—22no—
46#930—22no—
47#930—44no—
48C2×(C2×D3)⋊C4—22no—
49C2×(C2×D3)⋊C4—21no—
50C2×C4⋊Dic3C2×C411yescharacteristic
51C2×C4⋊Dic3C2×C412yes—
52C3×C22⋊C8—22no—
53C3×C22⋊C8—22no—
54C3×C22⋊C8—44no—
55C3⋊#95—24no—
56C3⋊#95—22no—
57C3⋊#95—22no—
58C3⋊#95—21no—
59C3⋊#95D414yescomplemented
60C24×C4C2×D311yescharacteristic
61C2×#121—31no—
62C2×#121—32no—
63C2×#120—61no—
64C22×C22⋊C4—61no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8C2×D312yes—
67#386—61no—
68#386—62no—
69#381—64no—
70#380—32no—
71#380—31no—
72C2×#95—32no—
73C2×#95—61no—
74#327—64no—
75C24×C3—24no—
76C24×C3—21no—
77C24×C3—21no—
78C24×C3—22no—
79C24×C3—22no—
80C24×C3—42no—
81C24×C3C22⋊C412yes—
82C24×C3C2×D411yescharacteristic
83C23×D3—21no—
84C22×C3×C4—24no—
85C22×C3×C4—24no—
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4—24no—
89C22×C3×C4—24no—
90C22×C3×C4—42no—
91C22×C3×C4—42no—
92C22×C3×C4—42no—
93C22×C3×C4—42no—
94C22×C3×C4—48no—
95C22×C3×C4—48no—
96C22×C3×C4C22⋊C412yes—
97C22×C3×C4C22×C411yescharacteristic
98C22×C3×C4C2×D412yes—
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4#4112yes—
101C2×#67—44no—
102C2×#67—44no—
103C2×#67—44no—
104C2×#67—41no—
105C2×#67—41no—
106C2×#67—42no—
107C2×#67—42no—
108C22×Dic3—21no—
109C22×Dic3—22no—
110C22×Dic3—24no—
111C22×Dic3—44no—
112C22×Dic3—44no—
113C22×Dic3—42no—
114C22×Dic3—42no—
115C2×D12—22no—
116C2×D12—42no—
117C2×C3×C8—44no—
118C2×C3×C8—42no—
119#217—48no—
120#217—44no—
121#217—42no—
122(C2×D3)⋊C4—44no—
123(C2×D3)⋊C4—42no—
124C4⋊Dic3—22no—
125C4⋊Dic3—42no—
126C25D1211yescharacteristic
127C22×D4—31no—
128C22×D4—62no—
129C22×D4—61no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—44no—
133C23×C4—62no—
134C23×C4C4×D312yes—
135C23×C4D1211yescharacteristic
136C23×C4#6712yes—
137C23×C4C22×D311yescharacteristic
138C22×C8—22no—
139#121—62no—
140#121—62no—
141#121—124no—
142#120—124no—
143C2×C4⋊C4—31no—
144C2×C4⋊C4—32no—
145C2×C22⋊C4—64no—
146C2×C22⋊C4—62no—
147C2×C22⋊C4—61no—
148C2×C22⋊C4—62no—
149C2×C22⋊C4—61no—
150C2×C22⋊C4—124no—
151C22⋊C8—22no—
152C22⋊C8—22no—
153C22⋊C8—44no—
154#95—34no—
155#95—64no—
156#95—62no—
157#95—62no—
158#95—61no—
159C23×C3—22no—
160C23×C3—22no—
161C23×C3—24no—
162C23×C3—24no—
163C23×C3—22no—
164C23×C3—22no—
165C23×C3—22no—
166C23×C3—22no—
167C23×C3—22no—
168C23×C3—22no—
169C23×C3—48no—
170C23×C3—48no—
171C23×C3—42no—
172C23×C3—42no—
173C23×C3—42no—
174C23×C3—42no—
175C23×C3#9914yes—
176C23×C3C2×C22⋊C411yescharacteristic
177C23×C3#12011yescharacteristic
178C23×C3#12311yescharacteristic
179C23×C3#13512yes—
180C23×C3AGL(1,ℤ/8)12yes—
181C22×D3—41no—
182C22×D3—42no—
183C22×D3—41no—
184C22×D3—41no—
185C22×D3—42no—
186C2×C3×C4—22no—
187C2×C3×C4—22no—
188C2×C3×C4—22no—
189C2×C3×C4—22no—
190C2×C3×C4—22no—
191C2×C3×C4—42no—
192C2×C3×C4—42no—
193C2×C3×C4—48no—
194C2×C3×C4—48no—
195C2×C3×C4—42no—
196C2×C3×C4—42no—
197C2×C3×C4C4×D412yes—
198C2×C3×C4#12112yes—
199#67—88no—
200#67—82no—
201#67—82no—
202C2×Dic3—44no—
203C2×Dic3—44no—
204C2×Dic3—41no—
205C2×Dic3—42no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—42no—
209C2×Dic3—42no—
210C2×Dic3—44no—
211C2×Dic3—84no—
212C2×Dic3—84no—
213D12—82no—
214C3×C8—82no—
215C24—24no—
216C24—21no—
217C24—21no—
218C24—22no—
219C24—22no—
220C24—42no—
221C24—61no—
222C24(C2×D3)⋊C412yes—
223C24C2×D1211yescharacteristic
224C2×D4—62no—
225C2×D4—122no—
226C2×D4—124no—
227C2×D4—124no—
228C2×D4—124no—
229C2×D4—121no—
230C2×D4—121no—
231C2×D4—122no—
232C2×D4—122no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—48no—
244C22×C4—48no—
245C22×C4—64no—
246C22×C4—62no—
247C22×C4—61no—
248C22×C4—124no—
249C22×C4—124no—
250C22×C4—122no—
251C22×C4—122no—
252C22×C4(C2×D3)⋊C412yes—
253C22×C4C2×C4×D311yescharacteristic
254C22×C4D3×D412yes—
255C22×C4#23712yes—
256C22×C4C2×#6711yescharacteristic
257C2×C8—44no—
258C2×C8—42no—
259C4⋊C4—62no—
260C4⋊C4—122no—
261C22⋊C4—124no—
262C22⋊C4—128no—
263C22⋊C4—124no—
264C22⋊C4—122no—
265C22⋊C4—122no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—24no—
271C22×C3—24no—
272C22×C3—24no—
273C22×C3—24no—
274C22×C3—24no—
275C22×C3—48no—
276C22×C3—48no—
277C22×C3—42no—
278C22×C3—42no—
279C22×C3—42no—
280C22×C3—42no—
281C22×C3#38611yescharacteristic
282C22×C3C2×#9912yes—
283C22×C3#41611yescharacteristic
284C22×C3#41811yescharacteristic
285C22×C3#42112yes—
286C22×C3#44912yes—
287C22×C3#48212yes—
288C2×D3—81no—
289C2×D3—81no—
290C2×D3—81no—
291C2×D3—82no—
292C2×D3—82no—
293C3×C4—22no—
294C3×C4—22no—
295C3×C4—42no—
296Dic3—84no—
297Dic3—81no—
298Dic3—82no—
299C23—22no—
300C23—22no—
301C23—24no—
302C23—24no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—48no—
310C23—48no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—121no—
316C23—122no—
317C23—122no—
318C23—121no—
319C23—121no—
320C23#80614yes—
321C23#88211yescharacteristic
322C23#88611yescharacteristic
323C23#90412yes—
324C23#90812yes—
325C23C2×(C2×D3)⋊C411yescharacteristic
326D4—242no—
327D4—248no—
328D4—242no—
329D4—242no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—48no—
338C2×C4—48no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—124no—
342C2×C4—124no—
343C2×C4—121no—
344C2×C4—122no—
345C2×C4—122no—
346C2×C4—122no—
347C2×C4—121no—
348C2×C4—124no—
349C2×C4—121no—
350C2×C4—244no—
351C2×C4—244no—
352C2×C4#67⋊C412yes—
353C2×C4#93912yes—
354C8—82no—
355C2×C3—22no—
356C2×C3—22no—
357C2×C3—44no—
358C2×C3#151012yes—
359C2×C3#151612yes—
360C2×C3#187511yescharacteristic
361C2×C3#189711yescharacteristic
362C2×C3#189811yescharacteristic
363D3—161no—
364C22—24no—
365C22—21no—
366C22—21no—
367C22—24no—
368C22—21no—
369C22—21no—
370C22—24no—
371C22—24no—
372C22—24no—
373C22—48no—
374C22—48no—
375C22—42no—
376C22—42no—
377C22—42no—
378C22—42no—
379C22—241no—
380C22—242no—
381C22—241no—
382C22—242no—
383C22—241no—
384C22#473412yes—
385C22#473712yes—
386C22C2×#80612yes—
387C22#495811yescharacteristic
388C22#511311yescharacteristic
389C22#513111yescharacteristic
390C22#513912yes—
391C4—22no—
392C4—22no—
393C4—42no—
394C4—244no—
395C4—241no—
396C4—242no—
397C3#915811yescharacteristic, complemented, Sylow
398C2—22no—
399C2—22no—
400C2—44no—
401C2—481no—
402C2#6949611yescharacteristic
403C2#6993311yescharacteristic
404C2#6994312yes—
405C2#6996611yescharacteristic
406C2#6997812yes—
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1779C2
socleC23×C3C2×C22⋊C4
prime radical#84032C2
cosocleC23×C3×C4C23
hypercenter#1779D3
hypocenterC3#9158
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
2. C2×#5211C2semidirect
3. C3×#1779C2semidirect
4. C2×#4470C2semidirect
5. #67675C2semidirect
10. C2×#930C4semidirect
16. #5212C4semidirect
24. #1779D3semidirect
59. C3⋊#95D4semidirect
397. C3#9158semidirect

Maximal subgroups

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

subgroupindex
#840322
C2×#52112
C3×#17792
C2×#44702
#676752
#91583