Finite Group Databasebeta

← #10691253

Subgroups — #10691253

Order 768.

622 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 127 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#82199C211yesmaximal, characteristic
2#82156C211yesmaximal, characteristic, complemented
3C2×C3×#324C211yesmaximal, characteristic, complemented
4#69944C212yesmaximal
5#67676C212yesmaximal, complemented
6#9204—31nomaximal, Sylow
7C22×C3×C4⋊C4C2211yescharacteristic
8C2×#930C2211yescharacteristic
9C2×#929—21no—
10C2×#926—21no—
11C2×C3×C22⋊C8C2212yes—
12#5213—21no—
13#5212—21no—
14#5212C2211yescharacteristic
15#5212C2212yes—
16#4994—22no—
17#4991C412yescomplemented
18#4984—24no—
19#4981C412yescomplemented
20C3×#324—22no—
21#4470—24no—
22#2618—31no—
23#2316—31no—
24#1519—32no—
25C2×#324D311yescharacteristic, complemented
26#1289—32no—
27C23×C3×C4D411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4C2311yescharacteristic
30C22×#67—21no—
31C22×D12C2×C411yescharacteristic
32C22×C3×C8—22no—
33C2×C3×C4⋊C4—22no—
34C2×C3×C4⋊C4—22no—
35C2×C3×C4⋊C4C2×C412yes—
36C2×C3×C4⋊C4D412yes—
37C2×#217—22no—
38C2×#217—21no—
39#930—22no—
40#930—22no—
41#929—42no—
42#929—42no—
43C2×(C2×D3)⋊C4—22no—
44C2×(C2×D3)⋊C4—21no—
45C2×(C2×D3)⋊C4—22no—
46#926—42no—
47#926—42no—
48C2×C4⋊Dic3—22no—
49C2×C4⋊Dic3—22no—
50C2×C4⋊Dic3—22no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3C2×C411yescharacteristic
53C2×Dic3⋊C4—22no—
54C2×C4×Dic3—22no—
55C3×C22⋊C8—22no—
56C3×C22⋊C8—22no—
57C3⋊#95—22no—
58C3⋊#95—22no—
59C3⋊#95C2×C412yes—
60C3⋊#95D414yes—
61C2×#123—61no—
62C2×#121—31no—
63C2×#117—61no—
64C22×C4⋊C4C2×D311yescharacteristic
65C2×C22⋊C8C2×D312yes—
66#398—32no—
67#396—62no—
68#394—32no—
69#389—64no—
70#386—61no—
71#380—31no—
72#380—32no—
73#380—61no—
74#327—64no—
75#324—22no—
76C24×C3C2×D411yescharacteristic
77C23×D3—21no—
78C22×C3×C4—21no—
79C22×C3×C4—21no—
80C22×C3×C4—21no—
81C22×C3×C4—21no—
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—81no—
85C22×C3×C4—81no—
86C22×C3×C4C22⋊C412yes—
87C22×C3×C4C22⋊C412yes—
88C22×C3×C4D812yes—
89C22×C3×C4SD1612yes—
90C22×C3×C4C22×C411yescharacteristic
91C22×C3×C4C2×D412yes—
92C22×C3×C4C2×D411yescharacteristic
93C22×C3×C4#4112yes—
94C2×#67—41no—
95C2×#67—41no—
96C2×#67—41no—
97C2×#67—41no—
98C2×#67—41no—
99C2×#67—41no—
100C22×Dic3—21no—
101C22×Dic3—22no—
102C22×Dic3—24no—
103C2×D12—22no—
104C2×D12—81no—
105C2×C3×C8—42no—
106C2×C3×C8—82no—
107C3×C4⋊C4—24no—
108C3×C4⋊C4—24no—
109#217—42no—
110#217—41no—
111#217—41no—
112#217—42no—
113(C2×D3)⋊C4—44no—
114(C2×D3)⋊C4—42no—
115(C2×D3)⋊C4—82no—
116C4⋊Dic3—22no—
117C4⋊Dic3—44no—
118C4⋊Dic3—82no—
119C4⋊Dic3—82no—
120C4⋊Dic3—81no—
121Dic3⋊C4—44no—
122C4×Dic3—44no—
123C22×D4—31no—
124C22×D4—61no—
125C23×C4D1211yescharacteristic
126C23×C4D1211yescharacteristic
127C23×C4C22×D311yescharacteristic
128C22×C8—22no—
129#123—122no—
130#123—122no—
131#121—62no—
132#121—62no—
133#117—122no—
134#117—122no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—31no—
138C2×C4⋊C4—62no—
139C2×C4⋊C4—62no—
140C2×C4⋊C4—62no—
141C2×C4⋊C4—62no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4C4×D312yes—
144C2×C4⋊C4#6712yes—
145C2×C22⋊C4—62no—
146C2×C22⋊C4—61no—
147C2×C22⋊C4—62no—
148C2×C22⋊C4—61no—
149C2×C22⋊C4—62no—
150C2×C42—62no—
151C22⋊C8—22no—
152C22⋊C8—22no—
153#95—32no—
154#95—34no—
155#95—62no—
156#95—62no—
157C23×C3—41no—
158C23×C3—41no—
159C23×C3C2×C22⋊C411yescharacteristic
160C23×C3#12011yescharacteristic
161C23×C3#12311yescharacteristic
162C23×C3C2×D811yescharacteristic
163C23×C3C2×SD1611yescharacteristic
164C23×C3AGL(1,ℤ/8)11yescharacteristic
165C23×C3#13711yescharacteristic
166C22×D3—42no—
167C22×D3—42no—
168C22×D3—41no—
169C22×D3—41no—
170C22×D3—41no—
171C2×C3×C4—22no—
172C2×C3×C4—22no—
173C2×C3×C4—22no—
174C2×C3×C4—22no—
175C2×C3×C4—22no—
176C2×C3×C4—22no—
177C2×C3×C4—22no—
178C2×C3×C4—42no—
179C2×C3×C4—41no—
180C2×C3×C4—42no—
181C2×C3×C4—41no—
182C2×C3×C4—42no—
183C2×C3×C4—42no—
184C2×C3×C4—81no—
185C2×C3×C4#10214yes—
186C2×C3×C4C4×D412yes—
187C2×C3×C4#12112yes—
188#67—81no—
189#67—81no—
190#67—81no—
191#67—81no—
192C2×Dic3—44no—
193C2×Dic3—42no—
194C2×Dic3—44no—
195C2×Dic3—42no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—42no—
199C2×Dic3—41no—
200C2×Dic3—44no—
201D12—82no—
202C3×C8—82no—
203C24—61no—
204C24C2×D1211yescharacteristic
205C2×D4—62no—
206C2×D4—121no—
207C2×D4—121no—
208C2×D4—121no—
209C2×D4—121no—
210C2×D4—121no—
211C2×D4—121no—
212C2×D4—241no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—64no—
220C22×C4—62no—
221C22×C4—61no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4#20412yes—
225C22×C4D2412yes—
226C22×C4(C2×D3)⋊C412yes—
227C22×C4(C2×D3)⋊C412yes—
228C22×C4C2×C4×D311yescharacteristic
229C22×C4D3×D412yes—
230C22×C4#23712yes—
231C22×C4C2×#6711yescharacteristic
232C2×C8—42no—
233C2×C8—82no—
234C4⋊C4—24no—
235C4⋊C4—24no—
236C4⋊C4—62no—
237C4⋊C4—124no—
238C4⋊C4—124no—
239C4⋊C4—242no—
240C4⋊C4—241no—
241C4⋊C4—242no—
242C22⋊C4—121no—
243C22⋊C4—122no—
244C22⋊C4—121no—
245C22⋊C4—122no—
246C22⋊C4—124no—
247C22⋊C4—122no—
248C22⋊C4—242no—
249C42—124no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—41no—
257C22×C3—41no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3#38611yescharacteristic
261C22×C3C2×#10211yescharacteristic
262C22×C3#41711yescharacteristic
263C22×C3#44711yescharacteristic
264C22×C3#44811yescharacteristic
265C22×C3#48011yescharacteristic
266C22×C3#48111yescharacteristic
267C2×D3—82no—
268C2×D3—82no—
269C2×D3—81no—
270C2×D3—81no—
271C2×D3—81no—
272C3×C4—24no—
273C3×C4—22no—
274C3×C4—22no—
275C3×C4—42no—
276Dic3—84no—
277Dic3—82no—
278Dic3—81no—
279C23—41no—
280C23—41no—
281C23—122no—
282C23—122no—
283C23—121no—
284C23—121no—
285C23—121no—
286C23#88211yescharacteristic
287C23#88611yescharacteristic
288C23C2×#20411yescharacteristic
289C23C2×D2411yescharacteristic
290C23#90811yescharacteristic
291C23#90911yescharacteristic
292C23C2×(C2×D3)⋊C411yescharacteristic
293D4—242no—
294D4—241no—
295D4—241no—
296D4—241no—
297D4—241no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—81no—
312C2×C4—122no—
313C2×C4—124no—
314C2×C4—124no—
315C2×C4—122no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4—124no—
319C2×C4—122no—
320C2×C4—121no—
321C2×C4#82114yes—
322C2×C4#67⋊C412yes—
323C2×C4#93912yes—
324C8—82no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—21no—
330C2×C3—21no—
331C2×C3#187411yescharacteristic
332C2×C3#191211yescharacteristic
333C2×C3#191311yescharacteristic
334D3—161no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—242no—
346C22—242no—
347C22—241no—
348C22—241no—
349C22—241no—
350C22#473211yescharacteristic
351C22#473511yescharacteristic
352C22#473611yescharacteristic
353C22#473811yescharacteristic
354C22#495811yescharacteristic
355C22C2×#82111yescharacteristic
356C22#513311yescharacteristic
357C4—24no—
358C4—22no—
359C4—22no—
360C4—42no—
361C4—244no—
362C4—242no—
363C4—241no—
364C3#920411yescharacteristic, complemented, Sylow
365C2—21no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—21no—
370C2—21no—
371C2—481no—
372C2#6993211yescharacteristic
373C2#7002411yescharacteristic
374C2#7010511yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#324C2
socleC22×C3#386
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenterC2×#324D3
hypocenterC3#9204
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. #82156C2semidirect
3. C2×C3×#324C2semidirect
5. #67676C2semidirect
17. #4991C4semidirect
19. #4981C4semidirect
25. C2×#324D3semidirect
364. C3#9204semidirect

Maximal subgroups

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

subgroupindex
#821992
#821562
C2×C3×#3242
#699442
#676762
#92043