Finite Group Databasebeta

← #10698066

Subgroups — #10698066

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
1C3×#3189C211yesmaximal, characteristic, complemented
2#82834C212yesmaximal
3#71690C212yesmaximal, complemented
4#69955C211yesmaximal, characteristic
5#67676C211yesmaximal, characteristic, complemented
6#10324—31nomaximal, Sylow
7C23×C4×D3C2211yescharacteristic
8C2×C3×C22⋊Q8C2212yes—
9C2×#946—22no—
10C3×#484—22no—
11C3×#467—22no—
12C3×#451—22no—
13C2×C3×C22⋊C8C2211yescharacteristic
14#5212C2212yes—
15C2×#830C2211yescharacteristic
16C2×(C2×D3)⋊C8—22no—
17#4989—24no—
18#4987—22no—
19#4955—22no—
20#4946—22no—
21#4538—24no—
22#3189D311yescharacteristic, complemented
23#2848—32no—
24#1779—31no—
25#1520—32no—
26#1289—31no—
27C24×D3D411yescharacteristic
28C22×C3×Q8D412yes—
29C23×C3×C4C2311yescharacteristic
30C23×Dic3D411yescharacteristic
31C22×C4×D3—22no—
32C22×C4×D3—21no—
33C22×C4×D3—22no—
34C22×C4×D3—21no—
35C22×C4×D3—44no—
36C2×C3×Q16—22no—
37C22×C3×C8D412yes—
38C3×C22⋊Q8—22no—
39C3×C22⋊Q8—22no—
40C3×C22⋊Q8—22no—
41C3×C22⋊Q8—22no—
42C2×C3×C4⋊C4—22no—
43C2×C3×C4⋊C4D412yes—
44C2×C3×C22⋊C4—22no—
45C2×#217—22no—
46#946—48no—
47C2×(C2×D3)⋊C4—24no—
48C2×(C2×D3)⋊C4—24no—
49C2×C4⋊Dic3—22no—
50C2×Dic3⋊C4—24no—
51C22×C3⋊C8—21no—
52C3×#107—22no—
53C3×#103—24no—
54C3×C22⋊C8—22no—
55C3⋊#95—24no—
56C3⋊#95D414yes—
57#830—22no—
58(C2×D3)⋊C8—44no—
59C24×C4—31no—
60C2×C22⋊Q8—62no—
61C2×C22⋊Q8C2×D312yes—
62#484—22no—
63#467—22no—
64#451—22no—
65C2×C22⋊C8—31no—
66C2×C22⋊C8—62no—
67C2×C22⋊C8C2×D311yescharacteristic
68#385—64no—
69#385—62no—
70#380—32no—
71#380—62no—
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
82C2×C3×Q8—22no—
83C2×C3×Q8—22no—
84C2×C3×Q8—44no—
85C22×C3×C4—22no—
86C22×C3×C4—22no—
87C22×C3×C4—42no—
88C22×C3×C4C2×D412yes—
89C22×C3×C4C2×D412yes—
90C22×C3×C4C2×D411yescharacteristic
91C22×C3×C4#4111yescharacteristic
92C22×Dic3—22no—
93C22×Dic3—21no—
94C22×Dic3—24no—
95C22×Dic3—42no—
96C2×C4×D3—24no—
97C2×C4×D3—28no—
98C2×C4×D3—44no—
99C2×C4×D3—42no—
100C2×C4×D3—42no—
101C2×C4×D3—44no—
102C2×C4×D3—44no—
103C2×C4×D3—42no—
104C2×C4×D3—41no—
105C2×C4×D3—42no—
106C2×C4×D3—41no—
107C3×Q16—82no—
108C2×C3×C8—22no—
109C2×C3×C8—42no—
110C3×C4⋊C4—22no—
111C3×C4⋊C4—22no—
112C3×C4⋊C4—42no—
113C3×C4⋊C4—42no—
114C3×C22⋊C4—42no—
115C3×C22⋊C4—42no—
116#217—42no—
117#217—42no—
118(C2×D3)⋊C4—48no—
119(C2×D3)⋊C4—48no—
120C4⋊Dic3—44no—
121Dic3⋊C4—48no—
122C2×C3⋊C8—42no—
123C2×C3⋊C8—41no—
124C25—31no—
125C22×Q8#6712yes—
126C23×C4—31no—
127C23×C4—62no—
128C23×C4—61no—
129C23×C4—62no—
130C23×C4—61no—
131C23×C4—124no—
132C23×C4C22×D311yescharacteristic
133C2×Q16—22no—
134C22×C8—61no—
135C22×C8#6712yes—
136C22⋊Q8—22no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—128no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—64no—
143C2×C4⋊C4—62no—
144C2×C4⋊C4#6712yes—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—64no—
147C2×C22⋊C4—64no—
148C2×C22⋊C4—62no—
149#107—22no—
150#103—24no—
151C22⋊C8—22no—
152C22⋊C8—62no—
153C22⋊C8—124no—
154#95—34no—
155#95—64no—
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—
179C3×Q8—44no—
180C3×Q8—82no—
181C2×C3×C4—22no—
182C2×C3×C4—24no—
183C2×C3×C4—42no—
184C2×C3×C4—41no—
185C2×C3×C4—44no—
186C2×C3×C4—44no—
187C2×C3×C4—42no—
188C2×C3×C4—42no—
189C2×C3×C4#12011yescharacteristic
190C2×C3×C4#12112yes—
191C2×C3×C4#12711yescharacteristic
192C2×Dic3—24no—
193C2×Dic3—22no—
194C2×Dic3—42no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—44no—
198C2×Dic3—44no—
199C2×Dic3—44no—
200C4×D3—44no—
201C4×D3—48no—
202C4×D3—44no—
203C3×C8—42no—
204C3⋊C8—81no—
205C24—32no—
206C24—62no—
207C24—61no—
208C24—62no—
209C24—61no—
210C24—122no—
211C24—122no—
212C24D3×D411yescharacteristic
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—44no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—42no—
219C22×C4—64no—
220C22×C4—68no—
221C22×C4—62no—
222C22×C4—61no—
223C22×C4—64no—
224C22×C4—124no—
225C22×C4—122no—
226C22×C4—124no—
227C22×C4—122no—
228C22×C4—124no—
229C22×C4—122no—
230C22×C4—122no—
231C22×C4—122no—
232C22×C4—121no—
233C22×C4—121no—
234C22×C4D3×D412yes—
235C22×C4#23711yescharacteristic
236C22×C4C2×#6712yes—
237C22×C4C2×#6711yescharacteristic
238Q16—82no—
239C2×C8—22no—
240C2×C8—42no—
241C2×C8—122no—
242C2×C8—121no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—124no—
248C4⋊C4—128no—
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—82no—
281C3×C4—82no—
282C3×C4B(4,𝔽2)12yes—
283Dic3—42no—
284Dic3—84no—
285C23—41no—
286C23—41no—
287C23—64no—
288C23—64no—
289C23—62no—
290C23—62no—
291C23—64no—
292C23—62no—
293C23—122no—
294C23—121no—
295C23—122no—
296C23—121no—
297C23—124no—
298C23—128no—
299C23—124no—
300C23—122no—
301C23—121no—
302C23—121no—
303C23D3×Q1611yescharacteristic
304C23#91812yes—
305C23#91911yescharacteristic
306C23#93712yes—
307C23#93811yescharacteristic
308Q8—44no—
309Q8—82no—
310C2×C4—24no—
311C2×C4—22no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—64no—
319C2×C4—62no—
320C2×C4—124no—
321C2×C4—128no—
322C2×C4—122no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4—124no—
326C2×C4—124no—
327C2×C4—124no—
328C2×C4—124no—
329C2×C4#93912yes—
330C2×C4#94011yescharacteristic
331C2×C4#95311yescharacteristic
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—68no—
350C22—64no—
351C22—124no—
352C22—124no—
353C22—124no—
354C22—122no—
355C22—122no—
356C22—122no—
357C22—122no—
358C22—122no—
359C22—122no—
360C22#518612yes—
361C22#518712yes—
362C22#518811yescharacteristic
363C22#518911yescharacteristic
364C22#522311yescharacteristic
365C4—21no—
366C4—22no—
367C4—82no—
368C4—82no—
369C4—122no—
370C4—244no—
371C4#524412yes—
372C3#1032411yescharacteristic, complemented, Sylow
373C2—22no—
374C2—22no—
375C2—21no—
376C2—21no—
377C2—124no—
378C2#7185111yescharacteristic
379C2#7207111yescharacteristic
380C2#7211811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#3189C2
socleC22×C3#392
prime radicalC24×D3D4
hypercenter#3189D3
hypocenterC3#10324
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3189

Complements

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

NH ≅ G/Ntype
1. C3×#3189C2semidirect
3. #71690C2semidirect
5. #67676C2semidirect
22. #3189D3semidirect
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
C3×#31892
#828342
#716902
#699552
#676762
#103243