Finite Group Databasebeta

← #10690216

Subgroups — #10690216

Order 768.

812 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 147 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×#2315C211yesmaximal, characteristic
2C2×#4949C211yesmaximal, characteristic
3C2×#4525C211yesmaximal, characteristic
4#69502C212yesmaximal, complemented
5#67624C212yesmaximal, complemented
6#9021—31nomaximal, Sylow
7C22×C3×C22⋊C4—21no—
8C22×C3×C22⋊C4C2211yescharacteristic
9C22×#217—21no—
10C22×C4⋊Dic3C2211yescharacteristic
11C22×Dic3⋊C4—21no—
12C3×#385—24no—
13C3×#380—21no—
14C3×#380C2212yes—
15C2×C3⋊#95C2211yescharacteristic
16C2×#830C2212yes—
17#4949—22no—
18#4949C414yescomplemented
19#4529—24no—
20#4525—22no—
21C2×#399—31no—
22#2315D311yescharacteristic
23#1824—32no—
24C2×#323—31no—
25#1294—32no—
26C25×C3D411yescharacteristic
27C23×C3×C4—22no—
28C23×C3×C4D411yescharacteristic
29C23×C3×C4C2311yescharacteristic
30C23×Dic3—21no—
31C23×Dic3—21no—
32C2×C3×C4⋊C4—22no—
33C2×C3×C4⋊C4D414yes—
34C2×C3×C22⋊C4—22no—
35C2×C3×C22⋊C4—22no—
36C2×C3×C22⋊C4—22no—
37C2×C3×C22⋊C4—22no—
38C2×C3×C22⋊C4—42no—
39C2×C3×C22⋊C4—42no—
40C2×C3×C22⋊C4C2×C412yes—
41C2×C3×C22⋊C4D412yes—
42C2×#217—24no—
43C2×#217—44no—
44C2×C4⋊Dic3—22no—
45C2×C4⋊Dic3—22no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3C2×C412yescomplemented
48C2×Dic3⋊C4—24no—
49C2×Dic3⋊C4—44no—
50C22×C3⋊C8—22no—
51C3×#95—22no—
52C3×#95—22no—
53C3⋊#95—21no—
54C3⋊#95C2×C412yescomplemented
55#830—22no—
56#830—22no—
57C22×C4⋊C4—31no—
58C22×C4⋊C4—61no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—61no—
61C22×C22⋊C4C2×D311yescharacteristic
62C2×C22⋊C8—32no—
63#399—34no—
64#399—62no—
65#385—24no—
66#380—21no—
67#380C2×D312yes—
68C2×#95—31no—
69#340—64no—
70#323—62no—
71C24×C3—21no—
72C24×C3—21no—
73C24×C3—21no—
74C24×C3—21no—
75C24×C3—42no—
76C24×C3—42no—
77C24×C3C22⋊C412yes—
78C24×C3SD1612yes—
79C24×C3Q1612yes—
80C24×C3C2×D411yescharacteristic
81C22×C3×C4—21no—
82C22×C3×C4—21no—
83C22×C3×C4—24no—
84C22×C3×C4—42no—
85C22×C3×C4—42no—
86C22×C3×C4—44no—
87C22×C3×C4—82no—
88C22×C3×C4—82no—
89C22×C3×C4C22⋊C412yes—
90C22×C3×C4C22×C411yescharacteristic
91C22×C3×C4C2×D411yescharacteristic
92C22×C3×C4C2×D412yes—
93C22×C3×C4#4112yes—
94C22×Dic3—24no—
95C22×Dic3—22no—
96C22×Dic3—22no—
97C22×Dic3—42no—
98C22×Dic3—41no—
99C22×Dic3—41no—
100C22×Dic3—42no—
101C22×Dic3—42no—
102C22×Dic3—41no—
103C22×Dic3—41no—
104C3×C4⋊C4—44no—
105C3×C4⋊C4—82no—
106C3×C22⋊C4—24no—
107C3×C22⋊C4—24no—
108C3×C22⋊C4—44no—
109C3×C22⋊C4—42no—
110C3×C22⋊C4—42no—
111#217—48no—
112C4⋊Dic3—24no—
113C4⋊Dic3—42no—
114Dic3⋊C4—48no—
115C2×C3⋊C8—44no—
116C2×C3⋊C8—42no—
117C25#6711yescharacteristic
118C23×C4—22no—
119C23×C4—61no—
120C23×C4—61no—
121C23×C4#6711yescharacteristic
122C23×C4C22×D311yescharacteristic
123C22×C8—62no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—32no—
126C2×C4⋊C4—62no—
127C2×C4⋊C4—62no—
128C2×C4⋊C4—64no—
129C2×C4⋊C4—61no—
130C2×C4⋊C4—124no—
131C2×C4⋊C4#6714yes—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—64no—
139C2×C22⋊C4—124no—
140C2×C22⋊C4C4×D312yes—
141C2×C22⋊C4D1212yes—
142C22⋊C8—62no—
143C22⋊C8—62no—
144#95—22no—
145#95—22no—
146#95—32no—
147#95—61no—
148C23×C3—22no—
149C23×C3—22no—
150C23×C3—22no—
151C23×C3—22no—
152C23×C3—22no—
153C23×C3—22no—
154C23×C3—22no—
155C23×C3—22no—
156C23×C3—42no—
157C23×C3—42no—
158C23×C3—44no—
159C23×C3—44no—
160C23×C3—42no—
161C23×C3—42no—
162C23×C3—82no—
163C23×C3—82no—
164C23×C3—81no—
165C23×C3—81no—
166C23×C3#10314yes—
167C23×C3C2×C22⋊C411yescharacteristic
168C23×C3#12011yescharacteristic
169C23×C3#12311yescharacteristic
170C23×C3C2×SD1611yescharacteristic
171C23×C3C2×Q1611yescharacteristic
172C23×C3AGL(1,ℤ/8)11yescharacteristic
173C23×C3#13711yescharacteristic
174C2×C3×C4—24no—
175C2×C3×C4—22no—
176C2×C3×C4—41no—
177C2×C3×C4—41no—
178C2×C3×C4—42no—
179C2×C3×C4—42no—
180C2×C3×C4—42no—
181C2×C3×C4—44no—
182C2×C3×C4—44no—
183C2×C3×C4—44no—
184C2×C3×C4—81no—
185C2×C3×C4—88no—
186C2×C3×C4C4×D412yes—
187C2×C3×C4#12112yes—
188C2×Dic3—24no—
189C2×Dic3—44no—
190C2×Dic3—44no—
191C2×Dic3—42no—
192C2×Dic3—42no—
193C2×Dic3—42no—
194C2×Dic3—42no—
195C2×Dic3—42no—
196C2×Dic3—82no—
197C2×Dic3—82no—
198C3⋊C8—82no—
199C24—21no—
200C24—21no—
201C24—21no—
202C24—21no—
203C24—42no—
204C24—42no—
205C24(C2×D3)⋊C412yes—
206C24#21412yes—
207C24#21612yes—
208C24C2×#6711yescharacteristic
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—24no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—44no—
215C22×C4—64no—
216C22×C4—62no—
217C22×C4—62no—
218C22×C4—82no—
219C22×C4—82no—
220C22×C4—122no—
221C22×C4—121no—
222C22×C4—121no—
223C22×C4—122no—
224C22×C4—122no—
225C22×C4—121no—
226C22×C4—121no—
227C22×C4(C2×D3)⋊C412yes—
228C22×C4C2×C4×D311yescharacteristic
229C22×C4C2×D1211yescharacteristic
230C22×C4#23512yes—
231C22×C4C2×#6712yes—
232C2×C8—124no—
233C2×C8—122no—
234C4⋊C4—44no—
235C4⋊C4—64no—
236C4⋊C4—82no—
237C4⋊C4—128no—
238C4⋊C4—122no—
239C22⋊C4—24no—
240C22⋊C4—24no—
241C22⋊C4—44no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—128no—
245C22×C3—22no—
246C22×C3—22no—
247C22×C3—22no—
248C22×C3—22no—
249C22×C3—21no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—24no—
254C22×C3—44no—
255C22×C3—44no—
256C22×C3—44no—
257C22×C3—42no—
258C22×C3—42no—
259C22×C3—82no—
260C22×C3—82no—
261C22×C3—82no—
262C22×C3#38611yescharacteristic
263C22×C3C2×#10311yescharacteristic
264C22×C3#41711yescharacteristic
265C22×C3#45011yescharacteristic
266C22×C3#45111yescharacteristic
267C22×C3#45312yes—
268C22×C3#45812yes—
269C22×C3#48311yescharacteristic
270C22×C3#48411yescharacteristic
271C3×C4—22no—
272C3×C4—22no—
273C3×C4—42no—
274C3×C4—84no—
275Dic3—44no—
276Dic3—82no—
277C23—22no—
278C23—22no—
279C23—22no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—42no—
286C23—42no—
287C23—44no—
288C23—44no—
289C23—42no—
290C23—42no—
291C23—82no—
292C23—82no—
293C23—81no—
294C23—81no—
295C23#81014yes—
296C23C2×(C2×D3)⋊C411yescharacteristic
297C23#92911yescharacteristic
298C23#93211yescharacteristic
299C23C2×#21411yescharacteristic
300C23#94211yescharacteristic
301C23C2×#21611yescharacteristic
302C23#95311yescharacteristic
303C2×C4—24no—
304C2×C4—22no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—64no—
314C2×C4—81no—
315C2×C4—88no—
316C2×C4—124no—
317C2×C4—124no—
318C2×C4—122no—
319C2×C4—122no—
320C2×C4—122no—
321C2×C4—122no—
322C2×C4—122no—
323C2×C4—242no—
324C2×C4—242no—
325C2×C4C4×#6712yes—
326C2×C4#93012yes—
327C8—242no—
328C2×C3—21no—
329C2×C3—21no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3—44no—
333C2×C3#161312yes—
334C2×C3#161412yes—
335C2×C3#187211yescharacteristic
336C2×C3#188711yescharacteristic
337C2×C3#189011yescharacteristic
338C22—22no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—24no—
347C22—44no—
348C22—44no—
349C22—44no—
350C22—42no—
351C22—42no—
352C22—82no—
353C22—82no—
354C22—82no—
355C22#496611yescharacteristic
356C22C2×#81011yescharacteristic
357C22#497011yescharacteristic
358C22#497111yescharacteristic
359C22#521311yescharacteristic
360C22#521811yescharacteristic
361C22#522811yescharacteristic
362C22#524112yes—
363C22#524312yes—
364C4—22no—
365C4—22no—
366C4—42no—
367C4—84no—
368C4—124no—
369C4—242no—
370C3#902111yescharacteristic, complemented, Sylow
371C2—21no—
372C2—21no—
373C2—21no—
374C2—21no—
375C2—44no—
376C2#7155911yescharacteristic
377C2#7165011yescharacteristic
378C2#7165511yescharacteristic
379C2#7184412yes—
380C2#7189112yes—
subgroupG/H
centerC23#953
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2315C2
socleC23×C3#120
prime radicalC25×C3D4
cosocleC23×C3×C4C23
hypercenter#2315D3
hypocenterC3#9021
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2315

Complements

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

NH ≅ G/Ntype
4. #69502C2semidirect
5. #67624C2semidirect
18. #4949C4semidirect
47. C2×C4⋊Dic3C2×C4semidirect
54. C3⋊#95C2×C4semidirect
370. C3#9021semidirect

Maximal subgroups

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

subgroupindex
C3×#23152
C2×#49492
C2×#45252
#695022
#676242
#90213