Finite Group Databasebeta

← #10692839

Subgroups — #10692839

Order 768.

516 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 95 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#82257C211yesmaximal, characteristic
2#82209C211yesmaximal, characteristic, complemented
3C3×#1820C211yesmaximal, characteristic, complemented
4#69531C212yesmaximal
5#67628C212yesmaximal, complemented
6#9446—31nomaximal, Sylow
7C2×C3×#117C2211yescharacteristic
8C2×#930C2211yescharacteristic
9C3×#381C2212yes—
10C4×#217C2211yescharacteristic
11#5207C2212yes—
12#5000—24no—
13#4988C411yescharacteristic, complemented
14#4988C411yescharacteristic, complemented
15#4977C411yescharacteristic, complemented
16#4975C411yescharacteristic, complemented
17#4959—21no—
18#4959—21no—
19#4940—21no—
20#4940—21no—
21#4924—21no—
22C4×#208—21no—
23C3×#342—21no—
24C3×#342—21no—
25#4451—24no—
26#2921—31no—
27#2500—31no—
28#1820D311yescharacteristic, complemented
29#1510—32no—
30#1284—32no—
31C23×C3×C4C2311yescharacteristic
32C22×#67—21no—
33C22×D12C2×C411yescharacteristic
34C3×#117—22no—
35C3×#117—22no—
36C2×C3×C4⋊C4C2×C411yescharacteristic
37C2×C3×C4⋊C4C2×C411yescharacteristic
38C2×C3×C22⋊C4—22no—
39C2×C3×C22⋊C4D411yescharacteristic
40C2×C3×C22⋊C4D411yescharacteristic
41C2×C3×C42D411yescharacteristic
42C2×C3×C42D411yescharacteristic
43C2×#217—21no—
44#930—22no—
45#930—22no—
46C2×(C2×D3)⋊C4—21no—
47C2×(C2×D3)⋊C4—21no—
48C2×(C2×D3)⋊C4—21no—
49C2×(C2×D3)⋊C4—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×C4⋊Dic3C2×C411yescharacteristic
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4—21no—
54C2×C4×Dic3C2×C411yescharacteristic
55C2×C4×Dic3C2×C411yescharacteristic
56C2×#208—22no—
57C3×#95—22no—
58C3×#95—22no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62#830—22no—
63#830D414yes—
64C4⋊(C3⋊C8)—24no—
65#803—22no—
66C4×C3⋊C8—22no—
67C2×#121—31no—
68C2×#117C2×D311yescharacteristic
69#433—64no—
70#423—61no—
71#407—32no—
72C4×M16—61no—
73#394—61no—
74#394—61no—
75#390—31no—
76#390—31no—
77#388—61no—
78#388—61no—
79#384—31no—
80#382—31no—
81#381C2×D312yes—
82C4×C22⋊C4—31no—
83#342—21no—
84#342—21no—
85#327—64no—
86C24×C3C2×D411yescharacteristic
87C23×D3—21no—
88C22×C3×C4—41no—
89C22×C3×C4—41no—
90C22×C3×C4—42no—
91C22×C3×C4C22⋊C411yescharacteristic
92C22×C3×C4C22⋊C411yescharacteristic
93C22×C3×C4C22⋊C411yescharacteristic
94C22×C3×C4C22⋊C411yescharacteristic
95C22×C3×C4C22×C411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D412yes—
98C22×C3×C4#4112yes—
99C2×#67—42no—
100C2×#67—81no—
101C2×#67—81no—
102C22×Dic3—21no—
103C22×Dic3—21no—
104C22×Dic3—21no—
105C2×D12—22no—
106C2×D12—41no—
107C2×D12—41no—
108C3×C4⋊C4—22no—
109C3×C4⋊C4—22no—
110C3×C22⋊C4—22no—
111C3×C22⋊C4—22no—
112C3×C22⋊C4—42no—
113C3×C22⋊C4—42no—
114C3×C42—21no—
115C3×C42—21no—
116C3×C42—21no—
117C3×C42—21no—
118#217—42no—
119(C2×D3)⋊C4—42no—
120(C2×D3)⋊C4—42no—
121(C2×D3)⋊C4—41no—
122(C2×D3)⋊C4—41no—
123(C2×D3)⋊C4—81no—
124(C2×D3)⋊C4—81no—
125C4⋊Dic3—22no—
126Dic3⋊C4—81no—
127Dic3⋊C4—81no—
128C4×Dic3—41no—
129C4×Dic3—41no—
130#208—42no—
131#208—42no—
132C2×C3⋊C8—24no—
133C22×D4—31no—
134C22×D4—61no—
135C23×C4C22×D311yescharacteristic
136C2×M16—62no—
137#121—62no—
138#121—62no—
139#117—22no—
140#117—22no—
141C2×C4⋊C4—31no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4C4×D311yescharacteristic
145C2×C4⋊C4C4×D311yescharacteristic
146C2×C22⋊C4—22no—
147C2×C22⋊C4—61no—
148C2×C22⋊C4—61no—
149C2×C22⋊C4—61no—
150C2×C22⋊C4—61no—
151C2×C22⋊C4—61no—
152C2×C22⋊C4—61no—
153C2×C22⋊C4#6711yescharacteristic
154C2×C22⋊C4#6711yescharacteristic
155C2×C42—31no—
156C2×C42—31no—
157C2×C42D1211yescharacteristic
158C2×C42D1211yescharacteristic
159C4⋊C8—64no—
160C22⋊C8—34no—
161C22⋊C8—62no—
162#97—62no—
163C4×C8—62no—
164#95—22no—
165#95—22no—
166#95—61no—
167#95—61no—
168#95—61no—
169C23×C3—21no—
170C23×C3—21no—
171C23×C3—41no—
172C23×C3—41no—
173C23×C3C2×C22⋊C411yescharacteristic
174C23×C3#12112yes—
175C22×D3—41no—
176C22×D3—41no—
177C22×D3—41no—
178C22×D3—41no—
179C22×D3—41no—
180C22×D3—41no—
181C22×D3—41no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—42no—
191C2×C3×C4—42no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—82no—
197C2×C3×C4—82no—
198C2×C3×C4—82no—
199C2×C3×C4C4×D412yes—
200C2×C3×C4#12011yescharacteristic
201C2×C3×C4#12311yescharacteristic
202#67—84no—
203C2×Dic3—22no—
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212D12—82no—
213C3⋊C8—44no—
214C24—61no—
215C24C2×#6711yescharacteristic
216C2×D4—62no—
217C2×D4—121no—
218C2×D4—121no—
219C2×D4—122no—
220C2×D4—241no—
221C2×D4—241no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—42no—
225C22×C4—61no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4(C2×D3)⋊C411yescharacteristic
229C22×C4(C2×D3)⋊C411yescharacteristic
230C22×C4(C2×D3)⋊C411yescharacteristic
231C22×C4(C2×D3)⋊C411yescharacteristic
232C22×C4C2×C4×D311yescharacteristic
233C22×C4C2×D1211yescharacteristic
234C22×C4D3×D412yes—
235C22×C4#23712yes—
236M16—122no—
237M16—122no—
238C2×C8—64no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—62no—
242C4⋊C4—241no—
243C4⋊C4—241no—
244C22⋊C4—22no—
245C22⋊C4—22no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—122no—
249C22⋊C4—122no—
250C22⋊C4—122no—
251C22⋊C4—121no—
252C22⋊C4—121no—
253C22⋊C4—241no—
254C22⋊C4—241no—
255C42—21no—
256C42—21no—
257C42—21no—
258C42—21no—
259C42—121no—
260C42—121no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—22no—
264C22×C3—22no—
265C22×C3—22no—
266C22×C3—81no—
267C22×C3—81no—
268C22×C3#38611yescharacteristic
269C22×C3#41011yescharacteristic
270C22×C3#41011yescharacteristic
271C2×D3—81no—
272C2×D3—81no—
273C2×D3—81no—
274C2×D3—81no—
275C2×D3—81no—
276C2×D3—81no—
277C2×D3—81no—
278C3×C4—21no—
279C3×C4—22no—
280C3×C4—21no—
281C3×C4—41no—
282C3×C4—41no—
283C3×C4—41no—
284C3×C4—41no—
285C3×C4—162no—
286Dic3—42no—
287Dic3—81no—
288Dic3—81no—
289C23—21no—
290C23—21no—
291C23—41no—
292C23—41no—
293C23—121no—
294C23—121no—
295C23—121no—
296C23—121no—
297C23—121no—
298C23—121no—
299C23—121no—
300C23C2×(C2×D3)⋊C411yescharacteristic
301C23#93912yes—
302D4—242no—
303D4—244no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—62no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—121no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—121no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4#67⋊C412yes—
331C2×C4#88211yescharacteristic
332C2×C4#88611yescharacteristic
333C8—124no—
334C2×C3—21no—
335C2×C3—21no—
336C2×C3—42no—
337C2×C3#188411yescharacteristic
338C2×C3#189911yescharacteristic
339C2×C3#191011yescharacteristic
340D3—161no—
341C22—21no—
342C22—21no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22—81no—
347C22—81no—
348C22—241no—
349C22—241no—
350C22—241no—
351C22—241no—
352C22—241no—
353C22—241no—
354C22—241no—
355C22#495811yescharacteristic
356C22#500611yescharacteristic
357C22#500611yescharacteristic
358C4—21no—
359C4—22no—
360C4—21no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—122no—
366C4—162no—
367C4—241no—
368C4—241no—
369C3#944611yescharacteristic, complemented, Sylow
370C2—21no—
371C2—21no—
372C2—42no—
373C2—481no—
374C2#6953511yescharacteristic
375C2#7004511yescharacteristic
376C2#7004711yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1820C2
socleC22×C3#386
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1820D3
hypocenterC3#9446
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1820

Complements

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

NH ≅ G/Ntype
2. #82209C2semidirect
3. C3×#1820C2semidirect
5. #67628C2semidirect
13. #4988C4semidirect
14. #4988C4semidirect
15. #4977C4semidirect
16. #4975C4semidirect
28. #1820D3semidirect
369. C3#9446semidirect

Maximal subgroups

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

subgroupindex
#822572
#822092
C3×#18202
#695312
#676282
#94463