Finite Group Databasebeta

← #10698070

Subgroups — #10698070

Order 768.

910 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 103 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#78617C211yesmaximal, characteristic, complemented
2#74049C212yesmaximal
3#69955C211yesmaximal, characteristic
4#69510C212yesmaximal, complemented
5C3×#1289C211yesmaximal, characteristic, complemented
6#10324—31nomaximal, Sylow
7C23×C4×D3C2211yescharacteristic
8C2×#924C2212yes—
9C2×#897—22no—
10C2×C3×C22⋊C8C2211yescharacteristic
11C3×#380C2212yes—
12C2×#830C2211yescharacteristic
13#5116—22no—
14C2×(C2×D3)⋊C8—22no—
15#4987—22no—
16#4955—22no—
17#4950—22no—
18#4739—22no—
19#4722—22no—
20#4668—24no—
21#4452—24no—
22#3189—31no—
23#2848—32no—
24#1779—31no—
25#1520—32no—
26#1289D311yescharacteristic, complemented
27C24×D3D411yescharacteristic
28C23×C3×C4C2311yescharacteristic
29C23×Dic3D411yescharacteristic
30C22×C4×D3—22no—
31C22×C4×D3—21no—
32C22×C4×D3—22no—
33C22×C4×D3—21no—
34C22×C4×D3—44no—
35C22×Dic6D412yes—
36C22×C3×C8D412yes—
37C2×C3×C4⋊C4—22no—
38C2×C3×C22⋊C4—22no—
39C2×#217—22no—
40C2×(C2×D3)⋊C4—24no—
41C2×(C2×D3)⋊C4—24no—
42C2×C4⋊Dic3—24no—
43C2×C4⋊Dic3D412yes—
44#924—22no—
45#924—22no—
46#924—22no—
47#924—22no—
48C2×Dic3⋊C4—22no—
49C22×C3⋊C8—21no—
50C2×Dic12—22no—
51#897—48no—
52C3×C22⋊C8—22no—
53C3×#95D414yes—
54C3⋊#95—24no—
55#830—22no—
56(C2×D3)⋊C8—44no—
57#818—22no—
58#816—24no—
59C24×C4—31no—
60C2×C22⋊Q8—32no—
61C2×C22⋊Q8—62no—
62#484—62no—
63#467—62no—
64#451—62no—
65C2×C22⋊C8—31no—
66C2×C22⋊C8—62no—
67C2×C22⋊C8C2×D311yescharacteristic
68#385—62no—
69#385—64no—
70#380—62no—
71#380C2×D312yes—
72#379—62no—
73#328—64no—
74C24×C3C2×D411yescharacteristic
75C23×D3—22no—
76C23×D3—21no—
77C23×D3—21no—
78C23×D3—22no—
79C23×D3—42no—
80C23×D3—42no—
81C23×D3Q1612yescomplemented
82C22×C3×C4—24no—
83C22×C3×C4—42no—
84C22×C3×C4C2×D411yescharacteristic
85C22×C3×C4C2×D412yes—
86C22×C3×C4C2×D412yes—
87C22×C3×C4#4111yescharacteristic
88C22×Dic3—22no—
89C22×Dic3—21no—
90C22×Dic3—22no—
91C22×Dic3—22no—
92C22×Dic3—42no—
93C2×C4×D3—24no—
94C2×C4×D3—28no—
95C2×C4×D3—44no—
96C2×C4×D3—42no—
97C2×C4×D3—42no—
98C2×C4×D3—44no—
99C2×C4×D3—44no—
100C2×C4×D3—42no—
101C2×C4×D3—41no—
102C2×C4×D3—42no—
103C2×C4×D3—41no—
104C2×Dic6—22no—
105C2×Dic6—22no—
106C2×Dic6—44no—
107C2×C3×C8—22no—
108C2×C3×C8—42no—
109C3×C4⋊C4—44no—
110C3×C22⋊C4—42no—
111C3×C22⋊C4—42no—
112#217—42no—
113#217—42no—
114(C2×D3)⋊C4—48no—
115(C2×D3)⋊C4—48no—
116C4⋊Dic3—22no—
117C4⋊Dic3—22no—
118C4⋊Dic3—48no—
119Dic3⋊C4—42no—
120Dic3⋊C4—42no—
121C2×C3⋊C8—42no—
122C2×C3⋊C8—41no—
123Dic12—82no—
124C25—31no—
125C22×Q8—32no—
126C23×C4—31no—
127C23×C4—61no—
128C23×C4—62no—
129C23×C4—62no—
130C23×C4—61no—
131C23×C4—124no—
132C23×C4C22×D311yescharacteristic
133C2×Q16—62no—
134C22×C8—61no—
135C22×C8D1212yes—
136C22⋊Q8—62no—
137C22⋊Q8—62no—
138C22⋊Q8—62no—
139C22⋊Q8—62no—
140C22⋊Q8—128no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—32no—
143C2×C4⋊C4—64no—
144C2×C4⋊C4—62no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—64no—
147C2×C22⋊C4—64no—
148C2×C22⋊C4—62no—
149#107—62no—
150#103—64no—
151C22⋊C8—22no—
152C22⋊C8—62no—
153C22⋊C8—124no—
154#95—64no—
155#95D1214yes—
156C23×C3—41no—
157C23×C3—41no—
158C23×C3#12012yes—
159C23×C3#12111yescharacteristic
160C23×C3C2×Q1611yescharacteristic
161C23×C3#13511yescharacteristic
162C23×C3#13712yes—
163C22×D3—24no—
164C22×D3—24no—
165C22×D3—22no—
166C22×D3—22no—
167C22×D3—24no—
168C22×D3—22no—
169C22×D3—44no—
170C22×D3—44no—
171C22×D3—48no—
172C22×D3—42no—
173C22×D3—41no—
174C22×D3—42no—
175C22×D3—41no—
176C22×D3—41no—
177C22×D3—42no—
178C22×D3—41no—
179C2×C3×C4—22no—
180C2×C3×C4—24no—
181C2×C3×C4—42no—
182C2×C3×C4—41no—
183C2×C3×C4—44no—
184C2×C3×C4—44no—
185C2×C3×C4—44no—
186C2×C3×C4#12011yescharacteristic
187C2×C3×C4#12112yes—
188C2×C3×C4#12711yescharacteristic
189C2×Dic3—24no—
190C2×Dic3—22no—
191C2×Dic3—42no—
192C2×Dic3—41no—
193C2×Dic3—41no—
194C2×Dic3—44no—
195C2×Dic3—44no—
196C2×Dic3—42no—
197C2×Dic3—42no—
198C4×D3—44no—
199C4×D3—48no—
200C4×D3—44no—
201Dic6—44no—
202Dic6—82no—
203C3×C8—42no—
204C3⋊C8—81no—
205C24—32no—
206C24—62no—
207C24—61no—
208C24—61no—
209C24—62no—
210C24—122no—
211C24—122no—
212C24D3×D411yescharacteristic
213C2×Q8—62no—
214C2×Q8—62no—
215C2×Q8—124no—
216C22×C4—24no—
217C22×C4—42no—
218C22×C4—64no—
219C22×C4—68no—
220C22×C4—62no—
221C22×C4—61no—
222C22×C4—62no—
223C22×C4—62no—
224C22×C4—124no—
225C22×C4—122no—
226C22×C4—124no—
227C22×C4—122no—
228C22×C4—124no—
229C22×C4—122no—
230C22×C4—122no—
231C22×C4—121no—
232C22×C4—121no—
233C22×C4—122no—
234C22×C4C2×D1211yescharacteristic
235C22×C4C2×D1212yes—
236C22×C4D3×D412yes—
237C22×C4#23911yescharacteristic
238Q16—242no—
239C2×C8—22no—
240C2×C8—42no—
241C2×C8—122no—
242C2×C8—121no—
243C4⋊C4—44no—
244C4⋊C4—62no—
245C4⋊C4—62no—
246C4⋊C4—128no—
247C4⋊C4—122no—
248C4⋊C4—122no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—128no—
252C22⋊C4—128no—
253C22⋊C4—122no—
254C22⋊C4—122no—
255C22×C3—22no—
256C22×C3—21no—
257C22×C3—22no—
258C22×C3—21no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—42no—
262C22×C3#39211yescharacteristic
263C22×C3#45112yes—
264C22×C3#45212yes—
265C22×C3#46711yescharacteristic
266C22×C3#47011yescharacteristic
267C2×D3—28no—
268C2×D3—24no—
269C2×D3—44no—
270C2×D3—44no—
271C2×D3—44no—
272C2×D3—42no—
273C2×D3—42no—
274C2×D3—42no—
275C2×D3—42no—
276C2×D3—42no—
277C2×D3—42no—
278C3×C4—21no—
279C3×C4—22no—
280C3×C4—84no—
281C3×C4B(4,𝔽2)12yes—
282Dic3—42no—
283Dic3—82no—
284Dic3—82no—
285C23—41no—
286C23—41no—
287C23—64no—
288C23—64no—
289C23—62no—
290C23—62no—
291C23—62no—
292C23—64no—
293C23—122no—
294C23—121no—
295C23—122no—
296C23—121no—
297C23—124no—
298C23—124no—
299C23—128no—
300C23—121no—
301C23—121no—
302C23—122no—
303C23#88212yes—
304C23#89511yescharacteristic
305C23#91211yescharacteristic
306C23#91512yes—
307C23D3×Q1611yescharacteristic
308Q8—124no—
309Q8—242no—
310C2×C4—24no—
311C2×C4—22no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—64no—
318C2×C4—62no—
319C2×C4—124no—
320C2×C4—128no—
321C2×C4—122no—
322C2×C4—121no—
323C2×C4—121no—
324C2×C4—124no—
325C2×C4—124no—
326C2×C4—124no—
327C2×C4—122no—
328C2×C4—122no—
329C2×C4#87411yescharacteristic
330C2×C4#88211yescharacteristic
331C2×C4#89512yes—
332C8—42no—
333C8—241no—
334C2×C3—22no—
335C2×C3—22no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3#200111yescharacteristic
339C2×C3#201011yescharacteristic
340C2×C3#202111yescharacteristic
341D3—44no—
342C22—22no—
343C22—21no—
344C22—22no—
345C22—21no—
346C22—41no—
347C22—42no—
348C22—41no—
349C22—64no—
350C22—68no—
351C22—124no—
352C22—124no—
353C22—122no—
354C22—124no—
355C22—122no—
356C22—122no—
357C22—122no—
358C22—122no—
359C22—122no—
360C22#467111yescharacteristic
361C22#478312yes—
362C22#480912yes—
363C22#486711yescharacteristic
364C22#488711yescharacteristic
365C4—21no—
366C4—22no—
367C4—84no—
368C4—122no—
369C4—242no—
370C4—242no—
371C4#474112yes—
372C3#1032411yescharacteristic, complemented, Sylow
373C2—22no—
374C2—21no—
375C2—22no—
376C2—21no—
377C2—124no—
378C2#6870311yescharacteristic
379C2#6888811yescharacteristic
380C2#6896511yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1289C2
socleC22×C3#392
prime radicalC24×D3D4
hypercenter#1289D3
hypocenterC3#10324
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1289

Complements

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

NH ≅ G/Ntype
1. #78617C2semidirect
4. #69510C2semidirect
5. C3×#1289C2semidirect
26. #1289D3semidirect
81. C23×D3Q16semidirect
372. C3#10324semidirect

Maximal subgroups

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

subgroupindex
#786172
#740492
#699552
#695102
C3×#12892
#103243