Finite Group Databasebeta

← #10690524

Subgroups — #10690524

Order 768.

810 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#121C211yesmaximal, characteristic
2C22×C4×C3⋊C8C211yesmaximal, characteristic, complemented
3C2×D4⋊(C3⋊C8)C212yesmaximal, complemented
4#71530C211yesmaximal, characteristic, complemented
5#67619C212yesmaximal, complemented
6#9072—31nomaximal, Sylow
7C2×C3×#121C411yescharacteristic
8C2×C3×C4×D4—21no—
9C2×C3×C4×D4C2211yescharacteristic
10C22×C3×C42C2211yescharacteristic
11C23×C3⋊C8—21no—
12C3×#390C411yescharacteristic
13C3×#389C411yescharacteristic
14C3×#388—21no—
15C3×#386—21no—
16C3×#380C411yescharacteristic
17C3×C4×C4⋊C4C2211yescharacteristic
18C3×C4×C22⋊C4—21no—
19C2×#830—21no—
20C2×C4⋊(C3⋊C8)C2212yes—
21C2×C4×C3⋊C8—21no—
22C2×C4×C3⋊C8—24no—
23C2×C4×C3⋊C8C2212yes—
24D4⋊(C3⋊C8)—24no—
25#4524—21no—
26C22×C4×C8—31no—
27C4×#121D311yescharacteristic
28#1797—31no—
29C2×D4⋊C8—32no—
30#1279—32no—
31C22×C3×D4—21no—
32C22×C3×D4C2×C411yescharacteristic
33C23×C3×C4—21no—
34C23×C3×C4C2×C411yescharacteristic
35C23×C3×C4D411yescharacteristic
36C23×C3×C4D411yescharacteristic
37C3×#121—22no—
38C3×#121C814yescomplemented
39C3×C4×D4—22no—
40C3×C4×D4—22no—
41C3×C4×D4—42no—
42C3×C4×D4—42no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4C2×C411yescharacteristic
45C2×C3×C4⋊C4C2×C411yescharacteristic
46C2×C3×C4⋊C4C2×C411yescharacteristic
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C42—22no—
53C2×C3×C42—22no—
54C2×C3×C42—21no—
55C2×C3×C42C2×C411yescharacteristic
56C2×C3×C42D411yescharacteristic
57C2×C3×C42D411yescharacteristic
58C2×C3×C42C2311yescharacteristic
59C22×C3⋊C8—24no—
60C22×C3⋊C8—24no—
61C22×C3⋊C8—24no—
62C22×C3⋊C8—22no—
63C22×C3⋊C8—22no—
64C3×#95—21no—
65C3×#95—21no—
66C3×#95—21no—
67#830—42no—
68C4⋊(C3⋊C8)—22no—
69C4⋊(C3⋊C8)—22no—
70C4×C3⋊C8—24no—
71C4×C3⋊C8—22no—
72C4×C3⋊C8D414yescomplemented
73C23×C8—61no—
74C2×#121Dic311yescharacteristic
75C2×C4×D4—21no—
76C2×C4×D4C2×D311yescharacteristic
77C22×C42C2×D311yescharacteristic
78C2×C4⋊C8—32no—
79C2×C22⋊C8—61no—
80C2×C4×C8—32no—
81C2×C4×C8—64no—
82C2×C4×C8—61no—
83#390Dic311yescharacteristic
84#389Dic311yescharacteristic
85#388—21no—
86#386—21no—
87#380Dic311yescharacteristic
88C4×C4⋊C4C2×D311yescharacteristic
89C4×C22⋊C4—21no—
90#336—61no—
91D4⋊C8—64no—
92C24×C3—21no—
93C24×C3C22⋊C411yescharacteristic
94C2×C3×D4—24no—
95C2×C3×D4—22no—
96C2×C3×D4—21no—
97C2×C3×D4—22no—
98C2×C3×D4—42no—
99C2×C3×D4—42no—
100C2×C3×D4C2×C812yescomplemented
101C22×C3×C4—22no—
102C22×C3×C4—22no—
103C22×C3×C4—22no—
104C22×C3×C4—21no—
105C22×C3×C4—22no—
106C22×C3×C4—21no—
107C22×C3×C4—22no—
108C22×C3×C4—22no—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4—42no—
113C22×C3×C4—41no—
114C22×C3×C4—41no—
115C22×C3×C4—42no—
116C22×C3×C4C22⋊C411yescharacteristic
117C22×C3×C4C22⋊C411yescharacteristic
118C22×C3×C4C22⋊C411yescharacteristic
119C22×C3×C4C2×C812yes—
120C22×C3×C4M1612yes—
121C22×C3×C4D812yes—
122C22×C3×C4SD1612yes—
123C22×C3×C4C22×C411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C3×C4⋊C4—21no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—42no—
130C3×C4⋊C4C2×C812yescomplemented
131C3×C22⋊C4—24no—
132C3×C22⋊C4—42no—
133C3×C22⋊C4—42no—
134C3×C22⋊C4—42no—
135C3×C22⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C42—22no—
138C3×C42—21no—
139C3×C42—22no—
140C3×C42—21no—
141C3×C42—41no—
142C3×C42—41no—
143C3×C42C2×D412yes—
144C3×C42#4112yes—
145C2×C3⋊C8—216no—
146C2×C3⋊C8—24no—
147C2×C3⋊C8—24no—
148C2×C3⋊C8—24no—
149C2×C3⋊C8—42no—
150C2×C3⋊C8—42no—
151C2×C3⋊C8—42no—
152C22×D4—21no—
153C22×D4C2×Dic311yescharacteristic
154C23×C4—21no—
155C23×C4C2×Dic311yescharacteristic
156C23×C4#6711yescharacteristic
157C23×C4#6711yescharacteristic
158C22×C8—64no—
159C22×C8—64no—
160C22×C8—64no—
161C22×C8—62no—
162C22×C8—62no—
163#121—22no—
164#121C3⋊C814yescomplemented
165C4×D4—22no—
166C4×D4—22no—
167C4×D4—42no—
168C4×D4—42no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4C2×Dic311yescharacteristic
171C2×C4⋊C4C2×Dic311yescharacteristic
172C2×C4⋊C4C2×Dic311yescharacteristic
173C2×C22⋊C4—21no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—21no—
181C2×C42C2×Dic311yescharacteristic
182C2×C42#6711yescharacteristic
183C2×C42#6711yescharacteristic
184C2×C42C22×D311yescharacteristic
185C4⋊C8—62no—
186C4⋊C8—62no—
187C22⋊C8—122no—
188C4×C8—34no—
189C4×C8—64no—
190C4×C8—62no—
191#95—21no—
192#95—21no—
193#95—21no—
194C23×C3—21no—
195C23×C3—21no—
196C23×C3—21no—
197C23×C3—21no—
198C23×C3—22no—
199C23×C3—41no—
200C23×C3—41no—
201C23×C3—41no—
202C23×C3—41no—
203C23×C3—41no—
204C23×C3—41no—
205C23×C3C22⋊C812yes—
206C23×C3#10212yes—
207C23×C3#10412yes—
208C23×C3C2×C22⋊C411yescharacteristic
209C3×D4—24no—
210C3×D4—48no—
211C3×D4—42no—
212C2×C3×C4—24no—
213C2×C3×C4—24no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—24no—
223C2×C3×C4—24no—
224C2×C3×C4—24no—
225C2×C3×C4—24no—
226C2×C3×C4—22no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—82no—
236C2×C3×C4—82no—
237C2×C3×C4C22⋊C812yes—
238C2×C3×C4#10212yes—
239C2×C3×C4#10412yes—
240C2×C3×C4C4×D412yes—
241C2×C3×C4#12011yescharacteristic
242C2×C3×C4#12112yes—
243C2×C3×C4#12311yescharacteristic
244C2×C3×C4C22×C811yescharacteristic
245C2×C3×C4C2×M1611yescharacteristic
246C2×C3×C4#13112yes—
247C2×C3×C4C2×D811yescharacteristic
248C2×C3×C4C2×SD1611yescharacteristic
249C2×C3×C4#13512yes—
250C3⋊C8—28no—
251C3⋊C8—82no—
252C24—21no—
253C24#21711yescharacteristic
254C2×D4—24no—
255C2×D4—21no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4C2×C3⋊C812yescomplemented
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4C2×C3⋊C812yes—
277C22×C4#20812yes—
278C22×C4#21312yes—
279C22×C4#21412yes—
280C22×C4#21711yescharacteristic
281C22×C4#21711yescharacteristic
282C22×C4#21711yescharacteristic
283C22×C4C22×Dic311yescharacteristic
284C22×C4C2×#6711yescharacteristic
285C22×C4C2×#6711yescharacteristic
286C2×C8—616no—
287C2×C8—64no—
288C2×C8—64no—
289C2×C8—64no—
290C2×C8—122no—
291C2×C8—122no—
292C2×C8—122no—
293C4⋊C4—21no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—42no—
297C4⋊C4C2×C3⋊C812yescomplemented
298C22⋊C4—24no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C42—22no—
305C42—21no—
306C42—21no—
307C42—22no—
308C42—41no—
309C42—41no—
310C42D3×D412yes—
311C42#23712yes—
312C22×C3—22no—
313C22×C3—22no—
314C22×C3—22no—
315C22×C3—22no—
316C22×C3—22no—
317C22×C3—22no—
318C22×C3—42no—
319C22×C3—42no—
320C22×C3—42no—
321C22×C3—81no—
322C22×C3—81no—
323C22×C3—81no—
324C22×C3—81no—
325C22×C3D4⋊C814yescomplemented
326C22×C3#38611yescharacteristic
327C22×C3C2×C22⋊C811yescharacteristic
328C22×C3#40811yescharacteristic
329C22×C3C2×#10211yescharacteristic
330C22×C3#41611yescharacteristic
331C22×C3C2×#10411yescharacteristic
332C22×C3C2×#10411yescharacteristic
333C3×C4—22no—
334C3×C4—22no—
335C3×C4—21no—
336C3×C4—22no—
337C3×C4—21no—
338C3×C4—42no—
339C3×C4—81no—
340C3×C4—81no—
341C3×C4C8×D412yes—
342C3×C4#43512yes—
343C3×C4#46512yes—
344C3×C4#46612yes—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—22no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23#83012yes—
357C23#83212yes—
358C23#83712yes—
359C23C2×#21711yescharacteristic
360D4—24no—
361D4—48no—
362D4—42no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—22no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4#83012yes—
389C2×C4#83212yes—
390C2×C4#83712yes—
391C2×C4C22×C3⋊C811yescharacteristic
392C2×C4C2×#20811yescharacteristic
393C2×C4C2×#21311yescharacteristic
394C2×C4C2×#21411yescharacteristic
395C2×C4D4×Dic312yes—
396C2×C4#93511yescharacteristic
397C2×C4#93711yescharacteristic
398C2×C4#93912yes—
399C2×C4#94812yes—
400C2×C4#95012yes—
401C8—68no—
402C8—242no—
403C2×C3—24no—
404C2×C3—82no—
405C2×C3C2×D4⋊C811yescharacteristic
406C2×C3#147611yescharacteristic
407C2×C3#149211yescharacteristic
408C2×C3#149411yescharacteristic
409C2×C3#186911yescharacteristic
410C2×C3#189311yescharacteristic
411C2×C3#189611yescharacteristic
412C22—22no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—42no—
419C22—42no—
420C22—42no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22D4⋊(C3⋊C8)14yescomplemented
426C22C2×#83011yescharacteristic
427C22C2×#83211yescharacteristic
428C22#522011yescharacteristic
429C22#523311yescharacteristic
430C22#523411yescharacteristic
431C22C2×#83711yescharacteristic
432C22C2×#83711yescharacteristic
433C4—22no—
434C4—21no—
435C4—22no—
436C4—21no—
437C4—22no—
438C4—42no—
439C4—81no—
440C4—81no—
441C4D4×C3⋊C812yes—
442C4#501112yes—
443C4#503812yes—
444C4#504112yes—
445C3#907211yescharacteristic, complemented, Sylow
446C2—24no—
447C2—82no—
448C2C2×D4⋊(C3⋊C8)11yescharacteristic
449C2#7161011yescharacteristic
450C2#7163811yescharacteristic
451C2#7168111yescharacteristic
452C2#7171111yescharacteristic
453C2#7175211yescharacteristic
454C2#7180211yescharacteristic
subgroupG/H
centerC22×C4C2×#67
commutatorC2×C3×C4C22×C8
FrattiniC2×C42C22×D3
FittingC3×C4×#121C2
socleC23×C3C2×C22⋊C4
prime radicalC2×C3×#121C4
cosocleC2×C3×C42C23
hypercenterC4×#121D3
hypocenterC3#9072
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
2. C22×C4×C3⋊C8C2semidirect
3. C2×D4⋊(C3⋊C8)C2semidirect
4. #71530C2semidirect
5. #67619C2semidirect
38. C3×#121C8semidirect
72. C4×C3⋊C8D4semidirect
100. C2×C3×D4C2×C8semidirect
130. C3×C4⋊C4C2×C8semidirect
164. #121C3⋊C8semidirect
260. C2×D4C2×C3⋊C8semidirect
297. C4⋊C4C2×C3⋊C8semidirect
325. C22×C3D4⋊C8semidirect
425. C22D4⋊(C3⋊C8)semidirect
445. C3#9072semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#1212
C22×C4×C3⋊C82
C2×D4⋊(C3⋊C8)2
#715302
#676192
#90723