Finite Group Databasebeta

← #10690723

Subgroups — #10690723

Order 768.

674 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2321C211yesmaximal, characteristic
2#82301C211yesmaximal, characteristic, complemented
3#71600C211yesmaximal, characteristic, complemented
4C2×#4525C212yesmaximal, complemented
5#67617C212yesmaximal, complemented
6#9107—31nomaximal, Sylow
7C2×C3×#120C411yescharacteristic
8C2×C3×C4×D4C2211yescharacteristic
9C22×C3×C22⋊C4C2211yescharacteristic
10C3×#390—21no—
11C3×#388—21no—
12C3×#388—21no—
13C3×#386—21no—
14C3×#386C411yescharacteristic
15C3×#385C411yescharacteristic
16C3×#379C411yescharacteristic
17C3×C4×C22⋊C4C2211yescharacteristic
18C2×#830—21no—
19C2×#830C2212yes—
20C2×#830C2212yes—
21D4×C3⋊C8—24no—
22#4925—21no—
23#4528—21no—
24#4525—21no—
25#4525—24no—
26#2972—31no—
27#2321D311yescharacteristic
28#1459—31no—
29C2×#323—32no—
30#1282—32no—
31C25×C3C2×C411yescharacteristic
32C22×C3×D4—21no—
33C22×C3×D4C2×C411yescharacteristic
34C23×C3×C4C2×C411yescharacteristic
35C23×C3×C4D411yescharacteristic
36C23×C3×C4C2311yescharacteristic
37C3×#120—22no—
38C3×#120C814yescomplemented
39C3×C4×D4—22no—
40C3×C4×D4—22no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4D411yescharacteristic
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4—22no—
47C2×C3×C22⋊C4—22no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4C2×C411yescharacteristic
52C2×C3×C22⋊C4C2×C411yescharacteristic
53C2×C3×C22⋊C4C2×C411yescharacteristic
54C2×C3×C22⋊C4D411yescharacteristic
55C2×C3×C42—21no—
56C2×C3×C42D411yescharacteristic
57C22×C3⋊C8—22no—
58C22×C3⋊C8—22no—
59C22×C3⋊C8—22no—
60C3×#95—21no—
61C3×#95—21no—
62C3×#95—21no—
63#830—22no—
64#830—22no—
65#830—22no—
66#830—22no—
67#830D414yescomplemented
68C4⋊(C3⋊C8)—22no—
69C4×C3⋊C8—22no—
70C2×#120Dic311yescharacteristic
71C2×C4×D4C2×D311yescharacteristic
72C22×C22⋊C4C2×D311yescharacteristic
73C8×D4—64no—
74#431—61no—
75C2×C22⋊C8—32no—
76C2×C22⋊C8—32no—
77C2×C22⋊C8—61no—
78#390—21no—
79#388—21no—
80#388—21no—
81#386—21no—
82#386Dic311yescharacteristic
83#385Dic311yescharacteristic
84#379Dic311yescharacteristic
85C4×C22⋊C4C2×D311yescharacteristic
86#324—61no—
87#323—64no—
88#323—61no—
89C24×C3—21no—
90C24×C3—22no—
91C24×C3—21no—
92C24×C3—22no—
93C24×C3—81no—
94C24×C3—81no—
95C24×C3C22⋊C411yescharacteristic
96C24×C3C2×C812yescomplemented
97C24×C3C22×C411yescharacteristic
98C2×C3×D4—22no—
99C2×C3×D4—22no—
100C2×C3×D4—24no—
101C2×C3×D4—81no—
102C2×C3×D4—81no—
103C2×C3×D4C2×C812yes—
104C2×C3×D4M1612yes—
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4—21no—
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4—22no—
111C22×C3×C4—81no—
112C22×C3×C4—81no—
113C22×C3×C4—81no—
114C22×C3×C4C22⋊C411yescharacteristic
115C22×C3×C4C22⋊C411yescharacteristic
116C22×C3×C4C22⋊C411yescharacteristic
117C22×C3×C4C2×D412yes—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4#4112yes—
121C3×C4⋊C4—22no—
122C3×C4⋊C4—81no—
123C3×C22⋊C4—22no—
124C3×C22⋊C4—21no—
125C3×C22⋊C4—24no—
126C3×C22⋊C4—22no—
127C3×C22⋊C4—22no—
128C3×C22⋊C4—41no—
129C3×C22⋊C4—42no—
130C3×C22⋊C4—41no—
131C3×C22⋊C4—81no—
132C3×C22⋊C4—81no—
133C3×C22⋊C4—81no—
134C3×C22⋊C4C2×C812yescomplemented
135C3×C42—21no—
136C3×C42—21no—
137C3×C42—81no—
138C2×C3⋊C8—24no—
139C2×C3⋊C8—42no—
140C2×C3⋊C8—82no—
141C2×C3⋊C8—82no—
142C2×C3⋊C8—82no—
143C25C2×Dic311yescharacteristic
144C22×D4—21no—
145C22×D4C2×Dic311yescharacteristic
146C23×C4C2×Dic311yescharacteristic
147C23×C4#6711yescharacteristic
148C23×C4C22×D311yescharacteristic
149C22×C8—62no—
150C22×C8—62no—
151C22×C8—62no—
152#120—22no—
153#120C3⋊C814yescomplemented
154C4×D4—22no—
155C4×D4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4#6711yescharacteristic
158C2×C22⋊C4—21no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4C2×Dic311yescharacteristic
167C2×C22⋊C4C2×Dic311yescharacteristic
168C2×C22⋊C4C2×Dic311yescharacteristic
169C2×C22⋊C4#6711yescharacteristic
170C2×C42—21no—
171C2×C42#6711yescharacteristic
172C4⋊C8—62no—
173C22⋊C8—34no—
174C22⋊C8—62no—
175C22⋊C8—62no—
176C22⋊C8—62no—
177C22⋊C8—62no—
178C4×C8—62no—
179#95—21no—
180#95—21no—
181#95—21no—
182C23×C3—21no—
183C23×C3—21no—
184C23×C3—22no—
185C23×C3—24no—
186C23×C3—41no—
187C23×C3—41no—
188C23×C3—41no—
189C23×C3—41no—
190C23×C3—41no—
191C23×C3—41no—
192C23×C3—41no—
193C23×C3—41no—
194C23×C3—41no—
195C23×C3—41no—
196C23×C3—42no—
197C23×C3—82no—
198C23×C3—82no—
199C23×C3—82no—
200C23×C3—82no—
201C23×C3—84no—
202C23×C3C22⋊C812yes—
203C23×C3C2×C22⋊C411yescharacteristic
204C23×C3C4×D412yes—
205C23×C3C22×C811yescharacteristic
206C23×C3C2×M1611yescharacteristic
207C23×C3#13112yes—
208C3×D4—24no—
209C3×D4—24no—
210C3×D4—84no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—22no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—42no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—81no—
232C2×C3×C4—82no—
233C2×C3×C4C22⋊C812yes—
234C2×C3×C4#12011yescharacteristic
235C2×C3×C4#12112yes—
236C2×C3×C4#12311yescharacteristic
237C3⋊C8—44no—
238C3⋊C8—82no—
239C24—21no—
240C24—21no—
241C24—22no—
242C24—22no—
243C24—81no—
244C24—81no—
245C24C2×C3⋊C812yescomplemented
246C24#21711yescharacteristic
247C24C22×Dic311yescharacteristic
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—24no—
251C2×D4—81no—
252C2×D4—81no—
253C2×D4C2×C3⋊C812yes—
254C2×D4#20812yes—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4#21711yescharacteristic
265C22×C4#21711yescharacteristic
266C22×C4#21711yescharacteristic
267C22×C4D3×D412yes—
268C22×C4#23712yes—
269C22×C4C2×#6711yescharacteristic
270C22×C4C2×#6711yescharacteristic
271C2×C8—64no—
272C2×C8—122no—
273C2×C8—242no—
274C2×C8—242no—
275C2×C8—242no—
276C4⋊C4—22no—
277C4⋊C4—81no—
278C22⋊C4—21no—
279C22⋊C4—22no—
280C22⋊C4—24no—
281C22⋊C4—22no—
282C22⋊C4—22no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—42no—
286C22⋊C4—81no—
287C22⋊C4—81no—
288C22⋊C4—81no—
289C22⋊C4C2×C3⋊C812yescomplemented
290C42—21no—
291C42—21no—
292C42—81no—
293C22×C3—21no—
294C22×C3—22no—
295C22×C3—21no—
296C22×C3—22no—
297C22×C3—22no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—42no—
301C22×C3—42no—
302C22×C3—42no—
303C22×C3—42no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—42no—
307C22×C3—81no—
308C22×C3—84no—
309C22×C3—81no—
310C22×C3—81no—
311C22×C3—84no—
312C22×C3—81no—
313C22×C3—81no—
314C22×C3#38611yescharacteristic
315C22×C3C2×C22⋊C811yescharacteristic
316C22×C3#40811yescharacteristic
317C22×C3C8×D412yes—
318C22×C3#43512yes—
319C3×C4—22no—
320C3×C4—21no—
321C3×C4—22no—
322C3×C4—21no—
323C3×C4—42no—
324C3×C4—41no—
325C3×C4—41no—
326C3×C4—81no—
327C3×C4—81no—
328C23—21no—
329C23—21no—
330C23—22no—
331C23—24no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—42no—
343C23—82no—
344C23—82no—
345C23—82no—
346C23—82no—
347C23—84no—
348C23#83012yes—
349C23C22×C3⋊C811yescharacteristic
350C23C2×#20811yescharacteristic
351C23D4×Dic312yes—
352C23#94812yes—
353C23C2×#21711yescharacteristic
354D4—24no—
355D4—24no—
356D4—84no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—81no—
378C2×C4—82no—
379C2×C4#83012yes—
380C2×C4#93511yescharacteristic
381C2×C4#93711yescharacteristic
382C2×C4#93912yes—
383C8—124no—
384C8—242no—
385C2×C3—21no—
386C2×C3—21no—
387C2×C3—22no—
388C2×C3—22no—
389C2×C3—42no—
390C2×C3—82no—
391C2×C3#186911yescharacteristic
392C2×C3#191311yescharacteristic
393C2×C3#191411yescharacteristic
394C22—21no—
395C22—22no—
396C22—21no—
397C22—22no—
398C22—22no—
399C22—41no—
400C22—41no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—42no—
405C22—41no—
406C22—41no—
407C22—42no—
408C22—81no—
409C22—84no—
410C22—81no—
411C22—81no—
412C22—84no—
413C22—81no—
414C22—81no—
415C22D4×C3⋊C812yes—
416C22#501112yes—
417C22C2×#83011yescharacteristic
418C22#522011yescharacteristic
419C22#523411yescharacteristic
420C4—21no—
421C4—22no—
422C4—21no—
423C4—22no—
424C4—42no—
425C4—41no—
426C4—41no—
427C4—81no—
428C4—81no—
429C3#910711yescharacteristic, complemented, Sylow
430C2—21no—
431C2—21no—
432C2—22no—
433C2—22no—
434C2—42no—
435C2—82no—
436C2#7161011yescharacteristic
437C2#7184811yescharacteristic
438C2#7185511yescharacteristic
subgroupG/H
centerC22#5220
commutatorC23×C3C22×C8
FrattiniC23×C4C22×D3
FittingC3×#2321C2
socleC22×C3#386
prime radicalC2×C3×#120C4
cosocleC23×C3×C4C23
hypercenter#2321D3
hypocenterC3#9107
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2321

Complements

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

NH ≅ G/Ntype
2. #82301C2semidirect
3. #71600C2semidirect
4. C2×#4525C2semidirect
5. #67617C2semidirect
38. C3×#120C8semidirect
67. #830D4semidirect
96. C24×C3C2×C8semidirect
134. C3×C22⋊C4C2×C8semidirect
153. #120C3⋊C8semidirect
245. C24C2×C3⋊C8semidirect
289. C22⋊C4C2×C3⋊C8semidirect
429. C3#9107semidirect

Maximal subgroups

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

subgroupindex
C3×#23212
#823012
#716002
C2×#45252
#676172
#91073