Finite Group Databasebeta

← #10698069

Subgroups — #10698069

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#84031C212yesmaximal
2#78617C211yesmaximal, characteristic, complemented
3C3×#1779C211yesmaximal, characteristic
4#69946C212yesmaximal, complemented
5#67676C211yesmaximal, characteristic, complemented
6#10324—31nomaximal, Sylow
7C24×C3×C4C2211yescharacteristic
8C2×#924—22no—
9C2×#924C2212yes—
10C2×C3×C22⋊C8—22no—
11C2×C3×C22⋊C8C2211yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13#5247—22no—
14#5212—22no—
15#5212C2212yes—
16#5210—22no—
17#5210—24no—
18#5116—22no—
19#4739—22no—
20#4722—22no—
21#4467—24no—
22#3189—31no—
23#2848—32no—
24#1779D311yescharacteristic
25#1520—32no—
26#1289—31no—
27C25×C3D411yescharacteristic
28C23×C3×C4—22no—
29C23×C3×C4—21no—
30C23×C3×C4—22no—
31C23×C3×C4—21no—
32C23×C3×C4—44no—
33C23×C3×C4D411yescharacteristic
34C23×C3×C4C2311yescharacteristic
35C22×Dic6D412yes—
36C22×C3×C8—21no—
37C22×C3×C8D412yes—
38C2×#217—24no—
39C2×#217—24no—
40C2×#217—22no—
41C2×#217—22no—
42C2×C4⋊Dic3—22no—
43C2×C4⋊Dic3D412yes—
44#924—22no—
45#924—22no—
46#924—22no—
47#924—22no—
48#924—48no—
49C2×Dic3⋊C4—24no—
50C2×Dic3⋊C4—22no—
51C2×Dic12—22no—
52C3×C22⋊C8—22no—
53C3×C22⋊C8—22no—
54C3×C22⋊C8—44no—
55C3⋊#95—24no—
56C3⋊#95D414yes—
57#818—22no—
58#816—24no—
59C24×C4C2×D311yescharacteristic
60C2×C22⋊Q8—32no—
61C2×C22⋊Q8—62no—
62#484—62no—
63#467—62no—
64#451—62no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8C2×D311yescharacteristic
67C2×C22⋊C8C2×D311yescharacteristic
68#385—64no—
69#385—62no—
70#380—32no—
71#380—62no—
72#379—62no—
73#328—64no—
74C24×C3—22no—
75C24×C3—21no—
76C24×C3—21no—
77C24×C3—22no—
78C24×C3—42no—
79C24×C3—42no—
80C24×C3Q1612yescomplemented
81C24×C3C2×D411yescharacteristic
82C22×C3×C4—24no—
83C22×C3×C4—28no—
84C22×C3×C4—22no—
85C22×C3×C4—21no—
86C22×C3×C4—42no—
87C22×C3×C4—42no—
88C22×C3×C4—44no—
89C22×C3×C4—42no—
90C22×C3×C4—44no—
91C22×C3×C4—42no—
92C22×C3×C4—44no—
93C22×C3×C4—42no—
94C22×C3×C4—41no—
95C22×C3×C4—42no—
96C22×C3×C4—41no—
97C22×C3×C4C2×D412yes—
98C22×C3×C4C2×D412yes—
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4#4111yescharacteristic
101C22×Dic3—22no—
102C22×Dic3—24no—
103C22×Dic3—22no—
104C2×Dic6—22no—
105C2×Dic6—22no—
106C2×Dic6—44no—
107C2×C3×C8—22no—
108C2×C3×C8—42no—
109C2×C3×C8—42no—
110C2×C3×C8—41no—
111#217—48no—
112#217—48no—
113#217—42no—
114#217—42no—
115#217—42no—
116#217—42no—
117C4⋊Dic3—22no—
118C4⋊Dic3—22no—
119C4⋊Dic3—44no—
120Dic3⋊C4—48no—
121Dic3⋊C4—42no—
122Dic3⋊C4—42no—
123Dic12—82no—
124C25D1211yescharacteristic
125C22×Q8—32no—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—44no—
131C23×C4D1211yescharacteristic
132C23×C4C22×D311yescharacteristic
133C2×Q16—62no—
134C22×C8—21no—
135C22×C8#6712yes—
136C22⋊Q8—62no—
137C22⋊Q8—62no—
138C22⋊Q8—62no—
139C22⋊Q8—62no—
140C22⋊Q8—128no—
141C2×C4⋊C4—32no—
142C2×C4⋊C4—64no—
143C2×C4⋊C4—62no—
144C2×C4⋊C4—62no—
145C2×C22⋊C4—64no—
146C2×C22⋊C4—64no—
147C2×C22⋊C4—62no—
148C2×C22⋊C4—62no—
149#107—62no—
150#103—64no—
151C22⋊C8—22no—
152C22⋊C8—22no—
153C22⋊C8—44no—
154#95—34no—
155#95—64no—
156C23×C3—24no—
157C23×C3—24no—
158C23×C3—22no—
159C23×C3—22no—
160C23×C3—24no—
161C23×C3—22no—
162C23×C3—41no—
163C23×C3—44no—
164C23×C3—41no—
165C23×C3—42no—
166C23×C3—41no—
167C23×C3—42no—
168C23×C3—41no—
169C23×C3—44no—
170C23×C3—48no—
171C23×C3—41no—
172C23×C3—42no—
173C23×C3—41no—
174C23×C3#12012yes—
175C23×C3#12111yescharacteristic
176C23×C3C2×Q1611yescharacteristic
177C23×C3#13511yescharacteristic
178C23×C3#13712yes—
179C2×C3×C4—24no—
180C2×C3×C4—22no—
181C2×C3×C4—24no—
182C2×C3×C4—22no—
183C2×C3×C4—42no—
184C2×C3×C4—41no—
185C2×C3×C4—44no—
186C2×C3×C4—48no—
187C2×C3×C4—42no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—44no—
191C2×C3×C4#12011yescharacteristic
192C2×C3×C4#12112yes—
193C2×C3×C4#12711yescharacteristic
194C2×Dic3—44no—
195C2×Dic3—44no—
196C2×Dic3—44no—
197C2×Dic3—44no—
198C2×Dic3—42no—
199C2×Dic3—42no—
200C2×Dic3—44no—
201Dic6—44no—
202Dic6—82no—
203C3×C8—42no—
204C3×C8—81no—
205C24—22no—
206C24—21no—
207C24—21no—
208C24—22no—
209C24—42no—
210C24—42no—
211C24Dic1212yescomplemented
212C24C2×D1211yescharacteristic
213C2×Q8—62no—
214C2×Q8—62no—
215C2×Q8—124no—
216C22×C4—24no—
217C22×C4—28no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—44no—
223C22×C4—42no—
224C22×C4—44no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—44no—
228C22×C4—42no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—64no—
232C22×C4—62no—
233C22×C4—62no—
234C22×C4#23511yescharacteristic
235C22×C4D3×D412yes—
236C22×C4D3×D412yes—
237C22×C4C2×#6711yescharacteristic
238Q16—242no—
239C2×C8—22no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—41no—
243C4⋊C4—62no—
244C4⋊C4—62no—
245C4⋊C4—128no—
246C4⋊C4—124no—
247C4⋊C4—122no—
248C4⋊C4—122no—
249C22⋊C4—128no—
250C22⋊C4—128no—
251C22⋊C4—122no—
252C22⋊C4—122no—
253C22⋊C4—122no—
254C22⋊C4—122no—
255C22×C3—22no—
256C22×C3—21no—
257C22×C3—22no—
258C22×C3—21no—
259C22×C3—28no—
260C22×C3—24no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3—42no—
264C22×C3—44no—
265C22×C3—44no—
266C22×C3—44no—
267C22×C3—42no—
268C22×C3—42no—
269C22×C3—42no—
270C22×C3—42no—
271C22×C3—42no—
272C22×C3—42no—
273C22×C3#39211yescharacteristic
274C22×C3#45112yes—
275C22×C3#45212yes—
276C22×C3#46711yescharacteristic
277C22×C3#47011yescharacteristic
278C3×C4—21no—
279C3×C4—22no—
280C3×C4—42no—
281C3×C4B(4,𝔽2)12yes—
282Dic3—84no—
283Dic3—82no—
284Dic3—82no—
285C23—24no—
286C23—24no—
287C23—22no—
288C23—22no—
289C23—24no—
290C23—22no—
291C23—41no—
292C23—44no—
293C23—41no—
294C23—42no—
295C23—41no—
296C23—42no—
297C23—41no—
298C23—44no—
299C23—48no—
300C23—41no—
301C23—41no—
302C23—42no—
303C23#88212yes—
304C23#90411yescharacteristic
305C23C2×Dic1211yescharacteristic
306C23#90912yes—
307C23#93011yescharacteristic
308Q8—124no—
309Q8—242no—
310C2×C4—24no—
311C2×C4—22no—
312C2×C4—24no—
313C2×C4—22no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—48no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—44no—
322C2×C4—124no—
323C2×C4—124no—
324C2×C4—124no—
325C2×C4—124no—
326C2×C4—122no—
327C2×C4—122no—
328C2×C4—124no—
329C2×C4#88412yes—
330C2×C4#93711yescharacteristic
331C2×C4#94011yescharacteristic
332C8—42no—
333C8—81no—
334C2×C3—22no—
335C2×C3—22no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3—44no—
339C2×C3#200111yescharacteristic
340C2×C3#201011yescharacteristic
341C2×C3#202111yescharacteristic
342C22—22no—
343C22—21no—
344C22—22no—
345C22—21no—
346C22—28no—
347C22—24no—
348C22—41no—
349C22—41no—
350C22—42no—
351C22—44no—
352C22—44no—
353C22—44no—
354C22—42no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—42no—
360C22#473312yes—
361C22#473912yes—
362C22#496011yescharacteristic
363C22#511611yescharacteristic
364C22#513711yescharacteristic
365C4—21no—
366C4—22no—
367C4—42no—
368C4—244no—
369C4—242no—
370C4—242no—
371C4#503212yes—
372C3#1032411yescharacteristic, complemented, Sylow
373C2—22no—
374C2—22no—
375C2—21no—
376C2—21no—
377C2—44no—
378C2#6950011yescharacteristic
379C2#6996411yescharacteristic
380C2#6999311yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1779C2
socleC22×C3#392
prime radicalC25×C3D4
hypercenter#1779D3
hypocenterC3#10324
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
2. #78617C2semidirect
4. #69946C2semidirect
5. #67676C2semidirect
80. C24×C3Q16semidirect
211. C24Dic12semidirect
372. C3#10324semidirect

Maximal subgroups

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

subgroupindex
#840312
#786172
C3×#17792
#699462
#676762
#103243