Finite Group Databasebeta

← #10690935

Subgroups — #10690935

Order 768.

1088 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 143 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×C4×#217C211yesmaximal, characteristic, complemented
3C3×#1779C211yesmaximal, characteristic, complemented
4C2×#4470C212yesmaximal, complemented
5#67675C212yesmaximal, complemented
6#9150—31nomaximal, Sylow
7C24×C3×C4C2211yescharacteristic
8C2×#953—21no—
9C22×#217—21no—
10C2×#930C412yescomplemented
11C2×#930C2211yescharacteristic
12C22×C4×Dic3—21no—
13C2×C3×C22⋊C8—22no—
14C2×C3×C22⋊C8C2212yes—
15#5213—21no—
16#5213—22no—
17#5212C412yescomplemented
18#5212C2211yescharacteristic
19C4×#217—24no—
20C2×C3⋊#95C2212yes—
21#4470—24no—
22#2847—31no—
23C2×C4×C22⋊C4—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—28no—
43C2×#217—22no—
44C2×#217—21no—
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—
52C2×C4×Dic3—24no—
53C2×C4×Dic3—44no—
54C3×C22⋊C8—22no—
55C3×C22⋊C8—22no—
56C3×C22⋊C8—44no—
57C3⋊#95—22no—
58C3⋊#95—22no—
59C3⋊#95—21no—
60C3⋊#95D414yescomplemented
61C24×C4C2×D311yescharacteristic
62C2×#121—31no—
63C2×#121—32no—
64C2×#120—61no—
65C22×C22⋊C4—61no—
66C22×C42—61no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8C2×D312yes—
69#386—61no—
70#386—62no—
71#380—32no—
72#380—31no—
73C4×C22⋊C4—64no—
74C2×#95—32no—
75#327—64no—
76C24×C3—24no—
77C24×C3—21no—
78C24×C3—21no—
79C24×C3—22no—
80C24×C3—22no—
81C24×C3—42no—
82C24×C3C22⋊C412yes—
83C24×C3C2×D411yescharacteristic
84C23×D3—21no—
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C22×C3×C4—24no—
88C22×C3×C4—24no—
89C22×C3×C4—24no—
90C22×C3×C4—24no—
91C22×C3×C4—24no—
92C22×C3×C4—42no—
93C22×C3×C4—42no—
94C22×C3×C4—42no—
95C22×C3×C4—48no—
96C22×C3×C4—48no—
97C22×C3×C4C22⋊C412yes—
98C22×C3×C4C22×C411yescharacteristic
99C22×C3×C4C2×D412yes—
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4#4112yes—
102C2×#67—44no—
103C2×#67—44no—
104C2×#67—44no—
105C2×#67—41no—
106C2×#67—41no—
107C2×#67—42no—
108C2×#67—42no—
109C22×Dic3—28no—
110C22×Dic3—24no—
111C22×Dic3—21no—
112C22×Dic3—22no—
113C22×Dic3—44no—
114C22×Dic3—42no—
115C22×Dic3—42no—
116C2×D12—22no—
117C2×D12—42no—
118C2×C3×C8—44no—
119C2×C3×C8—42no—
120#217—216no—
121#217—44no—
122#217—42no—
123(C2×D3)⋊C4—44no—
124(C2×D3)⋊C4—42no—
125C4⋊Dic3—22no—
126C4⋊Dic3—42no—
127C4×Dic3—48no—
128C25D1211yescharacteristic
129C22×D4—31no—
130C22×D4—62no—
131C22×D4—61no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—44no—
135C23×C4—62no—
136C23×C4C4×D312yes—
137C23×C4D1211yescharacteristic
138C23×C4#6712yes—
139C23×C4C22×D311yescharacteristic
140C22×C8—22no—
141#121—62no—
142#121—62no—
143#121—124no—
144#120—124no—
145C2×C4⋊C4—31no—
146C2×C4⋊C4—32no—
147C2×C22⋊C4—64no—
148C2×C22⋊C4—68no—
149C2×C22⋊C4—62no—
150C2×C22⋊C4—61no—
151C2×C22⋊C4—62no—
152C2×C22⋊C4—61no—
153C2×C42—64no—
154C2×C42—124no—
155C22⋊C8—22no—
156C22⋊C8—22no—
157C22⋊C8—44no—
158#95—34no—
159#95—62no—
160#95—62no—
161#95—61no—
162C23×C3—22no—
163C23×C3—22no—
164C23×C3—24no—
165C23×C3—24no—
166C23×C3—22no—
167C23×C3—22no—
168C23×C3—22no—
169C23×C3—22no—
170C23×C3—22no—
171C23×C3—22no—
172C23×C3—48no—
173C23×C3—48no—
174C23×C3—42no—
175C23×C3—42no—
176C23×C3—42no—
177C23×C3—42no—
178C23×C3#10414yes—
179C23×C3C2×C22⋊C411yescharacteristic
180C23×C3#12011yescharacteristic
181C23×C3#12311yescharacteristic
182C23×C3#13512yes—
183C23×C3AGL(1,ℤ/8)12yes—
184C22×D3—41no—
185C22×D3—42no—
186C22×D3—41no—
187C22×D3—41no—
188C22×D3—42no—
189C2×C3×C4—24no—
190C2×C3×C4—24no—
191C2×C3×C4—22no—
192C2×C3×C4—22no—
193C2×C3×C4—22no—
194C2×C3×C4—22no—
195C2×C3×C4—42no—
196C2×C3×C4—42no—
197C2×C3×C4—48no—
198C2×C3×C4—48no—
199C2×C3×C4#9914yes—
200C2×C3×C4C4×D412yes—
201C2×C3×C4#12112yes—
202#67—88no—
203#67—82no—
204#67—82no—
205C2×Dic3—28no—
206C2×Dic3—44no—
207C2×Dic3—48no—
208C2×Dic3—48no—
209C2×Dic3—44no—
210C2×Dic3—41no—
211C2×Dic3—42no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—42no—
215C2×Dic3—42no—
216D12—82no—
217C3×C8—82no—
218C24—24no—
219C24—21no—
220C24—21no—
221C24—22no—
222C24—22no—
223C24—42no—
224C24—61no—
225C24(C2×D3)⋊C412yes—
226C24C2×D1211yescharacteristic
227C2×D4—62no—
228C2×D4—122no—
229C2×D4—124no—
230C2×D4—124no—
231C2×D4—124no—
232C2×D4—121no—
233C2×D4—121no—
234C2×D4—122no—
235C2×D4—122no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—48no—
247C22×C4—48no—
248C22×C4—68no—
249C22×C4—64no—
250C22×C4—62no—
251C22×C4—61no—
252C22×C4—124no—
253C22×C4—122no—
254C22×C4—122no—
255C22×C4(C2×D3)⋊C412yes—
256C22×C4C2×C4×D311yescharacteristic
257C22×C4D3×D412yes—
258C22×C4#23712yes—
259C22×C4C2×#6711yescharacteristic
260C2×C8—44no—
261C2×C8—42no—
262C4⋊C4—62no—
263C4⋊C4—122no—
264C22⋊C4—616no—
265C22⋊C4—124no—
266C22⋊C4—122no—
267C22⋊C4—124no—
268C22⋊C4—122no—
269C42—128no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—24no—
275C22×C3—24no—
276C22×C3—24no—
277C22×C3—24no—
278C22×C3—24no—
279C22×C3—48no—
280C22×C3—48no—
281C22×C3—42no—
282C22×C3—42no—
283C22×C3—42no—
284C22×C3—42no—
285C22×C3#38611yescharacteristic
286C22×C3C2×#9912yes—
287C22×C3#41611yescharacteristic
288C22×C3#41811yescharacteristic
289C22×C3C2×#10412yes—
290C22×C3#44912yes—
291C22×C3#48212yes—
292C2×D3—81no—
293C2×D3—81no—
294C2×D3—81no—
295C2×D3—82no—
296C2×D3—82no—
297C3×C4—22no—
298C3×C4—22no—
299C3×C4—24no—
300Dic3—48no—
301Dic3—81no—
302Dic3—82no—
303C23—22no—
304C23—22no—
305C23—24no—
306C23—24no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—48no—
314C23—48no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—121no—
320C23—122no—
321C23—122no—
322C23—121no—
323C23—121no—
324C23#82514yes—
325C23#88211yescharacteristic
326C23#88611yescharacteristic
327C23#90412yes—
328C23#90812yes—
329C23C2×(C2×D3)⋊C411yescharacteristic
330D4—242no—
331D4—248no—
332D4—242no—
333D4—242no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—48no—
343C2×C4—48no—
344C2×C4—68no—
345C2×C4—124no—
346C2×C4—128no—
347C2×C4—128no—
348C2×C4—124no—
349C2×C4—121no—
350C2×C4—122no—
351C2×C4—122no—
352C2×C4—122no—
353C2×C4—121no—
354C2×C4—121no—
355C2×C4#80614yes—
356C2×C4#67⋊C412yes—
357C2×C4#93912yes—
358C8—82no—
359C2×C3—22no—
360C2×C3—22no—
361C2×C3—44no—
362C2×C3#150112yes—
363C2×C3#150712yes—
364C2×C3#187511yescharacteristic
365C2×C3#189611yescharacteristic
366C2×C3#189811yescharacteristic
367D3—161no—
368C22—24no—
369C22—21no—
370C22—21no—
371C22—24no—
372C22—21no—
373C22—21no—
374C22—24no—
375C22—24no—
376C22—24no—
377C22—48no—
378C22—48no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—241no—
384C22—242no—
385C22—241no—
386C22—242no—
387C22—241no—
388C22#473412yes—
389C22#473712yes—
390C22C2×#80612yes—
391C22#495811yescharacteristic
392C22#511311yescharacteristic
393C22#513111yescharacteristic
394C22C2×#82512yes—
395C4—22no—
396C4—24no—
397C4—22no—
398C4—128no—
399C4—241no—
400C4—242no—
401C3#915011yescharacteristic, complemented, Sylow
402C2—22no—
403C2—22no—
404C2—44no—
405C2—481no—
406C2#6949611yescharacteristic
407C2#6993311yescharacteristic
408C2#6994212yes—
409C2#6996511yescharacteristic
410C2#6997712yes—
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#9150
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×C4×#217C2semidirect
3. C3×#1779C2semidirect
4. C2×#4470C2semidirect
5. #67675C2semidirect
10. C2×#930C4semidirect
17. #5212C4semidirect
24. #1779D3semidirect
60. C3⋊#95D4semidirect
401. C3#9150semidirect

Maximal subgroups

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

subgroupindex
#840322
C2×C4×#2172
C3×#17792
C2×#44702
#676752
#91503