Finite Group Databasebeta

← #10690213

Subgroups — #10690213

Order 768.

812 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4993C211yesmaximal, characteristic
2#74005C211yesmaximal, characteristic
3C3×#1824C211yesmaximal, characteristic, complemented
4#71554C211yesmaximal, characteristic, complemented
5C2×#4468C211yesmaximal, characteristic
6#67675C211yesmaximal, characteristic, complemented
7#67624C211yesmaximal, characteristic, complemented
8#9021—31nomaximal, Sylow
9C22×C3×C4⋊C4C2211yescharacteristic
10C22×(C2×D3)⋊C4—21no—
11C22×(C2×D3)⋊C4C2211yescharacteristic
12C22×Dic3⋊C4—21no—
13C2×D3×C22⋊C4—21no—
14C2×C3×C22⋊C8C2211yescharacteristic
15C3×#380C2211yescharacteristic
16#5212C2211yescharacteristic
17C2×C3⋊#95C2211yescharacteristic
18C2×#830C2211yescharacteristic
19#4993—22no—
20#4993C414yescomplemented
21#4989—22no—
22#4942—21no—
23#4668—22no—
24C3×#340—22no—
25#4529—22no—
26#4468—22no—
27C2×#399—31no—
28#2315—31no—
29#1824—31no—
30#1824D311yescharacteristic, complemented
31C2×#323—31no—
32#1294—31no—
33#1294—31no—
34C24×D3D411yescharacteristic
35C23×C3×C4—21no—
36C23×C3×C4C2311yescharacteristic
37C23×Dic3—21no—
38C23×Dic3D411yescharacteristic
39C22×C4×D3—22no—
40C22×C3×C8—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4C2×C412yescomplemented
47C2×C3×C4⋊C4D412yes—
48C2×C3×C22⋊C4—21no—
49C2×#217—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4—21no—
53C2×(C2×D3)⋊C4—21no—
54C2×(C2×D3)⋊C4—22no—
55C2×(C2×D3)⋊C4—24no—
56C2×(C2×D3)⋊C4—44no—
57C2×(C2×D3)⋊C4C2×C412yes—
58C2×(C2×D3)⋊C4D412yes—
59C2×C4⋊Dic3D412yes—
60C2×Dic3⋊C4—24no—
61C2×Dic3⋊C4—22no—
62C2×Dic3⋊C4—44no—
63C22×C3⋊C8—21no—
64D3×C22⋊C4—42no—
65D3×C22⋊C4—42no—
66C3×C22⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×#95—21no—
69C3⋊#95—22no—
70C3⋊#95—21no—
71C3⋊#95—21no—
72C3⋊#95C2×C412yescomplemented
73#830—21no—
74#830—21no—
75C22×C4⋊C4—61no—
76C22×C4⋊C4C2×D311yescharacteristic
77C22×C22⋊C4—31no—
78C22×C22⋊C4—61no—
79C22×C22⋊C4—61no—
80C2×C22⋊C8—31no—
81C2×C22⋊C8C2×D311yescharacteristic
82#399—34no—
83#399—62no—
84#385—62no—
85#385—62no—
86#380—31no—
87#380—61no—
88#380C2×D311yescharacteristic
89C2×#95—31no—
90#340—22no—
91#340—62no—
92#323—62no—
93C24×C3C2×D411yescharacteristic
94C23×D3—21no—
95C23×D3—21no—
96C23×D3—21no—
97C23×D3—21no—
98C23×D3—42no—
99C23×D3—42no—
100C23×D3C22⋊C412yes—
101C23×D3SD1612yes—
102C23×D3Q1612yes—
103C22×C3×C4—22no—
104C22×C3×C4—22no—
105C22×C3×C4—42no—
106C22×C3×C4—42no—
107C22×C3×C4—42no—
108C22×C3×C4—41no—
109C22×C3×C4—41no—
110C22×C3×C4C22×C411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4#4111yescharacteristic
115C22×C3×C4#4111yescharacteristic
116C22×Dic3—24no—
117C22×Dic3—22no—
118C22×Dic3—21no—
119C22×Dic3—21no—
120C22×Dic3—22no—
121C22×Dic3—42no—
122C22×Dic3—41no—
123C22×Dic3—41no—
124C22×Dic3—42no—
125C22×Dic3C22⋊C412yes—
126C2×C4×D3—44no—
127C2×C4×D3—82no—
128C2×C4×D3—82no—
129C2×C3×C8—42no—
130C2×C3×C8—41no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—22no—
133C3×C4⋊C4—42no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C22⋊C4—41no—
138#217—41no—
139#217—41no—
140(C2×D3)⋊C4—22no—
141(C2×D3)⋊C4—22no—
142(C2×D3)⋊C4—22no—
143(C2×D3)⋊C4—22no—
144(C2×D3)⋊C4—48no—
145(C2×D3)⋊C4—44no—
146C4⋊Dic3—42no—
147Dic3⋊C4—48no—
148Dic3⋊C4—82no—
149C2×C3⋊C8—42no—
150C2×C3⋊C8—41no—
151C25—31no—
152C23×C4—21no—
153C23×C4—31no—
154C23×C4—62no—
155C23×C4—61no—
156C23×C4C22×D311yescharacteristic
157C22×C8—21no—
158C22×C8—61no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—32no—
165C2×C4⋊C4—64no—
166C2×C4⋊C4—62no—
167C2×C4⋊C4—124no—
168C2×C4⋊C4C4×D312yescomplemented
169C2×C4⋊C4#6712yes—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—32no—
172C2×C22⋊C4—32no—
173C2×C22⋊C4—61no—
174C2×C22⋊C4—61no—
175C2×C22⋊C4—61no—
176C2×C22⋊C4—61no—
177C2×C22⋊C4—64no—
178C2×C22⋊C4—62no—
179C2×C22⋊C4—61no—
180C2×C22⋊C4—122no—
181C2×C22⋊C4—122no—
182C2×C22⋊C4—124no—
183C22⋊C8—21no—
184C22⋊C8—21no—
185C22⋊C8—61no—
186C22⋊C8—61no—
187#95—21no—
188#95—32no—
189#95—62no—
190#95—61no—
191#95—61no—
192C23×C3—22no—
193C23×C3—22no—
194C23×C3C2×C22⋊C411yescharacteristic
195C23×C3#12011yescharacteristic
196C23×C3#12311yescharacteristic
197C23×C3C2×SD1611yescharacteristic
198C23×C3C2×Q1611yescharacteristic
199C23×C3AGL(1,ℤ/8)11yescharacteristic
200C23×C3#13711yescharacteristic
201C22×D3—22no—
202C22×D3—22no—
203C22×D3—22no—
204C22×D3—22no—
205C22×D3—22no—
206C22×D3—22no—
207C22×D3—42no—
208C22×D3—42no—
209C22×D3—42no—
210C22×D3—42no—
211C22×D3—44no—
212C22×D3—44no—
213C22×D3—82no—
214C22×D3—82no—
215C22×D3—81no—
216C22×D3—81no—
217C22×D3#10314yes—
218C2×C3×C4—22no—
219C2×C3×C4—22no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—42no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4—42no—
230C2×C3×C4—82no—
231C2×C3×C4—82no—
232C2×C3×C4C4×D411yescharacteristic
233C2×C3×C4C4×D411yescharacteristic
234C2×C3×C4#12111yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×Dic3—24no—
237C2×Dic3—22no—
238C2×Dic3—44no—
239C2×Dic3—44no—
240C2×Dic3—42no—
241C2×Dic3—42no—
242C2×Dic3—42no—
243C2×Dic3—42no—
244C2×Dic3—42no—
245C2×Dic3—42no—
246C2×Dic3—42no—
247C2×Dic3—81no—
248C4×D3—88no—
249C3×C8—81no—
250C3⋊C8—81no—
251C24—32no—
252C24—32no—
253C24—32no—
254C24—61no—
255C24—61no—
256C24—61no—
257C24—61no—
258C24—122no—
259C24—122no—
260C24D3×D411yescharacteristic
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—32no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—64no—
270C22×C4—62no—
271C22×C4—61no—
272C22×C4—61no—
273C22×C4—62no—
274C22×C4—124no—
275C22×C4—122no—
276C22×C4—121no—
277C22×C4—121no—
278C22×C4—122no—
279C22×C4—242no—
280C22×C4—242no—
281C22×C4C2×C4×D311yescharacteristic
282C22×C4#23511yescharacteristic
283C22×C4D3×D411yescharacteristic
284C22×C4D3×D411yescharacteristic
285C22×C4#23711yescharacteristic
286C22×C4C2×#6711yescharacteristic
287C2×C8—42no—
288C2×C8—41no—
289C2×C8—122no—
290C2×C8—121no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—42no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—122no—
297C4⋊C4—128no—
298C4⋊C4—242no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—62no—
302C22⋊C4—62no—
303C22⋊C4—62no—
304C22⋊C4—62no—
305C22⋊C4—128no—
306C22⋊C4—124no—
307C22⋊C4—121no—
308C22⋊C4—121no—
309C22×C3—22no—
310C22×C3—22no—
311C22×C3—22no—
312C22×C3—22no—
313C22×C3—21no—
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3#38611yescharacteristic
318C22×C3C2×#10311yescharacteristic
319C22×C3#41711yescharacteristic
320C22×C3#45011yescharacteristic
321C22×C3#45111yescharacteristic
322C22×C3#45312yes—
323C22×C3#45812yes—
324C22×C3#48311yescharacteristic
325C22×C3#48411yescharacteristic
326C2×D3—24no—
327C2×D3—44no—
328C2×D3—44no—
329C2×D3—42no—
330C2×D3—42no—
331C2×D3—44no—
332C2×D3—82no—
333C2×D3—82no—
334C2×D3—82no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—82no—
340C3×C4—82no—
341Dic3—44no—
342Dic3—42no—
343Dic3—82no—
344C23—22no—
345C23—22no—
346C23—34no—
347C23—62no—
348C23—62no—
349C23—62no—
350C23—62no—
351C23—62no—
352C23—62no—
353C23—122no—
354C23—124no—
355C23—124no—
356C23—122no—
357C23—122no—
358C23—122no—
359C23—242no—
360C23—242no—
361C23—241no—
362C23—241no—
363C23D3×C22⋊C411yescharacteristic
364C23#88311yescharacteristic
365C23D3×SD1611yescharacteristic
366C23#91411yescharacteristic
367C23D3×Q1611yescharacteristic
368C23#91811yescharacteristic
369C23#93711yescharacteristic
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—64no—
383C2×C4—62no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—124no—
387C2×C4—124no—
388C2×C4—122no—
389C2×C4—122no—
390C2×C4—122no—
391C2×C4—122no—
392C2×C4—122no—
393C2×C4—122no—
394C2×C4—122no—
395C2×C4—248no—
396C2×C4—241no—
397C2×C4#67⋊C411yescharacteristic
398C2×C4#88411yescharacteristic
399C2×C4C4×#6711yescharacteristic
400C2×C4#93811yescharacteristic
401C8—81no—
402C8—241no—
403C2×C3—21no—
404C2×C3—21no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3#161312yes—
408C2×C3#161412yes—
409C2×C3#187211yescharacteristic
410C2×C3#188711yescharacteristic
411C2×C3#189011yescharacteristic
412D3—44no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—64no—
422C22—124no—
423C22—124no—
424C22—124no—
425C22—122no—
426C22—122no—
427C22—242no—
428C22—242no—
429C22—242no—
430C22D3×#10311yescharacteristic
431C22#480311yescharacteristic
432C22#480511yescharacteristic
433C22#481111yescharacteristic
434C22#495611yescharacteristic
435C22#505312yes—
436C22#516911yescharacteristic
437C22#518611yescharacteristic
438C22#519812yes—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—82no—
444C4—82no—
445C4—124no—
446C4—122no—
447C4—242no—
448C3#902111yescharacteristic, complemented, Sylow
449C2—21no—
450C2—21no—
451C2—21no—
452C2—21no—
453C2—124no—
454C2#6953811yescharacteristic
455C2#7026811yescharacteristic
456C2#7032311yescharacteristic
457C2#7116812yes—
458C2#7117612yes—
subgroupG/H
centerC23#937
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1824C2
socleC23×C3#120
prime radicalC24×D3D4
cosocleC23×C3×C4C23
hypercenter#1824D3
hypocenterC3#9021
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
3. C3×#1824C2semidirect
4. #71554C2semidirect
6. #67675C2semidirect
7. #67624C2semidirect
20. #4993C4semidirect
30. #1824D3semidirect
46. C2×C3×C4⋊C4C2×C4semidirect
72. C3⋊#95C2×C4semidirect
168. C2×C4⋊C4C4×D3semidirect
448. C3#9021semidirect

Maximal subgroups

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

subgroupindex
C2×#49932
#740052
C3×#18242
#715542
C2×#44682
#676752
#676242
#90213