Finite Group Databasebeta

← #10692636

Subgroups — #10692636

Order 768.

644 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 97 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#73550C211yesmaximal, characteristic
2#73435C211yesmaximal, characteristic, complemented
3#69977C211yesmaximal, characteristic
4#69971C211yesmaximal, characteristic, complemented
5#69444C211yesmaximal, characteristic
6C3×#1297C211yesmaximal, characteristic, complemented
7#67618C211yesmaximal, characteristic, complemented
8#9410—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C22×Dic3⋊C4C2211yescharacteristic
11C2×#877—21no—
12C3×#407C2211yescharacteristic
13C3×C4×C22⋊C4C2211yescharacteristic
14C3×C4×C22⋊C4C2211yescharacteristic
15C4×#217C2211yescharacteristic
16#5207C2211yescharacteristic
17#4958—21no—
18#4958—21no—
19Dic3×C22⋊C4—21no—
20#4940—22no—
21C4×Dic3⋊C4—22no—
22#4672C412yescomplemented
23#4659C412yescomplemented
24#4649—22no—
25#4470—22no—
26#4469—22no—
27#4451—22no—
28#2591—31no—
29#2274—31no—
30#1507—31no—
31#1507—31no—
32#1463—31no—
33#1297—31no—
34#1297D311yescharacteristic, complemented
35C23×C3×C4C2311yescharacteristic
36C22×#67—21no—
37C23×Dic3D411yescharacteristic
38C23×Dic3D411yescharacteristic
39C22×D12C2×C411yescharacteristic
40C2×C3×M16—21no—
41C2×C3×C22⋊C4—21no—
42C2×C3×C22⋊C4—21no—
43C2×C3×C42—21no—
44C2×C3×C42—21no—
45C2×#217—21no—
46#930—21no—
47#930—21no—
48#930—21no—
49#930—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—22no—
52C2×(C2×D3)⋊C4—22no—
53C2×C4⋊Dic3C2×C411yescharacteristic
54C2×Dic3⋊C4—22no—
55C2×Dic3⋊C4—21no—
56C2×Dic3⋊C4—21no—
57C2×Dic3⋊C4—21no—
58C2×Dic3⋊C4—21no—
59C2×Dic3⋊C4C2×C412yes—
60C2×Dic3⋊C4D412yes—
61C2×C4×Dic3—22no—
62C2×C4×Dic3—22no—
63C2×C4×Dic3—21no—
64C2×#208—21no—
65#877—28no—
66C3×C22⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×#95C2×C412yes—
69C3×#95D412yes—
70C3⋊#95—22no—
71C3⋊#95—22no—
72C3⋊#95—22no—
73C3⋊#95D412yes—
74#830—21no—
75#830—21no—
76C2×#121—31no—
77C2×#117—61no—
78C22×C4⋊C4—31no—
79#407—31no—
80#407C2×D311yescharacteristic
81#400—32no—
82#394—32no—
83#388—62no—
84#386—61no—
85#386—61no—
86C4×C4⋊C4—62no—
87C4×C4⋊C4—62no—
88C4×C22⋊C4—31no—
89C4×C22⋊C4—61no—
90C4×C22⋊C4C2×D311yescharacteristic
91C4×C22⋊C4C2×D311yescharacteristic
92#327—62no—
93#327—62no—
94#324—62no—
95C24×C3C2×D411yescharacteristic
96C23×D3—21no—
97C22×C3×C4—22no—
98C22×C3×C4—22no—
99C22×C3×C4—81no—
100C22×C3×C4C22×C411yescharacteristic
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4#4111yescharacteristic
105C22×C3×C4#4111yescharacteristic
106C2×#67—44no—
107C2×#67—41no—
108C2×#67—41no—
109C22×Dic3—21no—
110C22×Dic3—21no—
111C22×Dic3—24no—
112C22×Dic3—21no—
113C22×Dic3—22no—
114C22×Dic3—21no—
115C22×Dic3—21no—
116C22×Dic3—81no—
117C22×Dic3C22⋊C412yes—
118C22×Dic3C22⋊C412yes—
119C2×D12—21no—
120C2×D12—21no—
121C2×D12—81no—
122C3×M16—81no—
123C2×C3×C8—41no—
124C3×C22⋊C4—24no—
125C3×C22⋊C4—42no—
126C3×C42—81no—
127C3×C42—81no—
128#217—24no—
129(C2×D3)⋊C4—41no—
130(C2×D3)⋊C4—41no—
131(C2×D3)⋊C4—44no—
132(C2×D3)⋊C4—82no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135Dic3⋊C4—28no—
136Dic3⋊C4—22no—
137Dic3⋊C4—22no—
138Dic3⋊C4—22no—
139Dic3⋊C4—22no—
140C4×Dic3—28no—
141C4×Dic3—81no—
142C4×Dic3—82no—
143#208—81no—
144C2×C3⋊C8—41no—
145C22×D4—31no—
146C22×D4—61no—
147C23×C4—31no—
148C23×C4—31no—
149C23×C4C22×D311yescharacteristic
150C2×M16—21no—
151C2×M16—61no—
152#121—61no—
153#121—61no—
154#121—61no—
155#121—61no—
156#117—68no—
157C2×C4⋊C4—32no—
158C2×C4⋊C4—32no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—62no—
161C2×C4⋊C4—61no—
162C2×C4⋊C4—61no—
163C2×C4⋊C4—61no—
164C2×C4⋊C4—61no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—61no—
168C2×C22⋊C4—62no—
169C2×C22⋊C4—61no—
170C2×C22⋊C4—62no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—62no—
174C2×C42—62no—
175C2×C42—61no—
176C22⋊C8—21no—
177C22⋊C8—21no—
178C22⋊C8—61no—
179C22⋊C8—61no—
180#95—32no—
181#95—62no—
182#95—62no—
183#95—62no—
184#95C4×D312yes—
185#95D1212yes—
186C23×C3—22no—
187C23×C3—22no—
188C23×C3—41no—
189C23×C3C2×C22⋊C411yescharacteristic
190C23×C3#12011yescharacteristic
191C23×C3#12311yescharacteristic
192C22×D3—42no—
193C22×D3—42no—
194C22×D3—41no—
195C22×D3—42no—
196C2×C3×C4—24no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—42no—
204C2×C3×C4—42no—
205C2×C3×C4—42no—
206C2×C3×C4—42no—
207C2×C3×C4—42no—
208C2×C3×C4C4×D411yescharacteristic
209C2×C3×C4C4×D411yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12111yescharacteristic
212#67—82no—
213#67—82no—
214C2×Dic3—22no—
215C2×Dic3—22no—
216C2×Dic3—24no—
217C2×Dic3—22no—
218C2×Dic3—22no—
219C2×Dic3—24no—
220C2×Dic3—22no—
221C2×Dic3—42no—
222C2×Dic3—42no—
223C2×Dic3—42no—
224C2×Dic3—42no—
225C2×Dic3—42no—
226C2×Dic3—41no—
227C2×Dic3—81no—
228C2×Dic3#10414yes—
229D12—81no—
230D12—81no—
231C3×C8—81no—
232C3⋊C8—81no—
233C24—61no—
234C24D3×D411yescharacteristic
235C2×D4—61no—
236C2×D4—61no—
237C2×D4—124no—
238C2×D4—121no—
239C2×D4—121no—
240C2×D4—241no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—32no—
244C22×C4—32no—
245C22×C4—64no—
246C22×C4—61no—
247C22×C4—61no—
248C22×C4—62no—
249C22×C4—61no—
250C22×C4—61no—
251C22×C4—61no—
252C22×C4—81no—
253C22×C4—241no—
254C22×C4C2×C4×D311yescharacteristic
255C22×C4C2×D1211yescharacteristic
256C22×C4#23511yescharacteristic
257C22×C4D3×D411yescharacteristic
258C22×C4D3×D411yescharacteristic
259C22×C4#23911yescharacteristic
260M16—81no—
261M16—241no—
262C2×C8—41no—
263C2×C8—121no—
264C4⋊C4—68no—
265C4⋊C4—62no—
266C4⋊C4—62no—
267C4⋊C4—62no—
268C4⋊C4—62no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C22⋊C4—24no—
272C22⋊C4—42no—
273C22⋊C4—64no—
274C22⋊C4—124no—
275C22⋊C4—121no—
276C22⋊C4—121no—
277C22⋊C4—242no—
278C42—68no—
279C42—81no—
280C42—81no—
281C42—241no—
282C42—242no—
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—44no—
291C22×C3#38611yescharacteristic
292C22×C3C2×#10412yes—
293C22×C3#45412yes—
294C2×D3—84no—
295C2×D3—81no—
296C2×D3—81no—
297C2×D3—81no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—44no—
303C3×C4—82no—
304Dic3—24no—
305Dic3—44no—
306Dic3—42no—
307Dic3—81no—
308C23—22no—
309C23—22no—
310C23—41no—
311C23—122no—
312C23—122no—
313C23—121no—
314C23—122no—
315C23D3×C22⋊C411yescharacteristic
316C23#88211yescharacteristic
317C23#89411yescharacteristic
318D4—241no—
319D4—241no—
320D4—242no—
321D4—242no—
322C2×C4—24no—
323C2×C4—34no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—62no—
336C2×C4—62no—
337C2×C4—64no—
338C2×C4—62no—
339C2×C4—62no—
340C2×C4—64no—
341C2×C4—62no—
342C2×C4—122no—
343C2×C4—122no—
344C2×C4—122no—
345C2×C4—122no—
346C2×C4—122no—
347C2×C4—121no—
348C2×C4—241no—
349C2×C4C4×D1211yescharacteristic
350C2×C4#88411yescharacteristic
351C2×C4#89311yescharacteristic
352C2×C4#89511yescharacteristic
353C8—81no—
354C8—241no—
355C2×C3—22no—
356C2×C3—22no—
357C2×C3—41no—
358C2×C3#188811yescharacteristic
359C2×C3#189611yescharacteristic
360C2×C3#191311yescharacteristic
361D3—161no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—44no—
370C22—244no—
371C22—241no—
372C22—241no—
373C22—241no—
374C22#466511yescharacteristic
375C22D3×#10412yes—
376C22#482712yes—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—44no—
382C4—64no—
383C4—82no—
384C4—124no—
385C4—122no—
386C4—241no—
387C3#941011yescharacteristic, complemented, Sylow
388C2—22no—
389C2—22no—
390C2—41no—
391C2—481no—
392C2#6875611yescharacteristic
393C2#6876611yescharacteristic
394C2#6896411yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1297C2
socleC22×C3#386
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1297D3
hypocenterC3#9410
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1297

Complements

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

NH ≅ G/Ntype
2. #73435C2semidirect
4. #69971C2semidirect
6. C3×#1297C2semidirect
7. #67618C2semidirect
22. #4672C4semidirect
23. #4659C4semidirect
34. #1297D3semidirect
387. C3#9410semidirect

Maximal subgroups

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

subgroupindex
#735502
#734352
#699772
#699712
#694442
C3×#12972
#676182
#94103