Finite Group Databasebeta

← #10690758

Subgroups — #10690758

Order 768.

694 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×#2854C211yesmaximal, characteristic
2C2×#4925C211yesmaximal, characteristic, complemented
3C2×#4537C212yesmaximal
4#71534C211yesmaximal, characteristic, complemented
5#67624C212yesmaximal, complemented
6#9114—31nomaximal, Sylow
7C2×C3×#123—21no—
8C2×C3×#121C2211yescharacteristic
9C22×C3×C42C2211yescharacteristic
10C3×#390C411yescharacteristic, complemented
11C3×#390C411yescharacteristic, complemented
12C3×#389C411yescharacteristic, complemented
13C3×#389C411yescharacteristic, complemented
14C3×#388—21no—
15C3×#388—21no—
16C3×#386—21no—
17C3×#380C2211yescharacteristic
18C2×C3⋊#95—21no—
19C2×C3⋊#95—21no—
20C2×C3⋊#95C2212yes—
21C2×#830C2212yes—
22#4925—24no—
23C2×C4⋊(C3⋊C8)—21no—
24C2×C4×C3⋊C8—21no—
25#4537—24no—
26C2×#431—31no—
27#2854D311yescharacteristic
28C2×#327—32no—
29#1434—31no—
30#1294—32no—
31C22×C3×D4—21no—
32C22×C3×D4C2×C411yescharacteristic
33C23×C3×C4D411yescharacteristic
34C23×C3×C4D411yescharacteristic
35C23×C3×C4C2311yescharacteristic
36C23×Dic3—22no—
37C3×#123—42no—
38C3×#123—42no—
39C3×#121—22no—
40C3×#121—22no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4C2×C411yescharacteristic
44C2×C3×C4⋊C4C2×C411yescharacteristic
45C2×C3×C4⋊C4C2×C411yescharacteristic
46C2×C3×C22⋊C4—21no—
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×C42C2×C411yescharacteristic
55C2×C3×C42C2×C411yescharacteristic
56C2×C3×C42D411yescharacteristic
57C2×C3×C42D411yescharacteristic
58C22×C3⋊C8—22no—
59C3×#95—21no—
60C3×#95—21no—
61C3×#95—21no—
62C3⋊#95—24no—
63C3⋊#95—22no—
64C3⋊#95—22no—
65C3⋊#95—42no—
66#830—22no—
67#830D414yes—
68C4⋊(C3⋊C8)—24no—
69C4×C3⋊C8—24no—
70C2×#123—21no—
71C2×#121C2×D311yescharacteristic
72C22×C42C2×D311yescharacteristic
73#431—64no—
74C2×C4⋊C8—61no—
75C2×C22⋊C8—32no—
76C2×C4×C8—61no—
77#390Dic311yescharacteristic, complemented
78#390Dic311yescharacteristic, complemented
79#389Dic311yescharacteristic, complemented
80#389Dic311yescharacteristic, complemented
81#388—21no—
82#388—21no—
83#386—21no—
84#380C2×D311yescharacteristic
85C2×#95—32no—
86C2×#95—61no—
87C2×#95—61no—
88#327—64no—
89C24×C3—21no—
90C24×C3C2×D411yescharacteristic
91C2×C3×D4—22no—
92C2×C3×D4—41no—
93C2×C3×D4—41no—
94C2×C3×D4—42no—
95C2×C3×D4—41no—
96C2×C3×D4—41no—
97C2×C3×D4—42no—
98C22×C3×C4—22no—
99C22×C3×C4—22no—
100C22×C3×C4—22no—
101C22×C3×C4—22no—
102C22×C3×C4—21no—
103C22×C3×C4—21no—
104C22×C3×C4—21no—
105C22×C3×C4—41no—
106C22×C3×C4—41no—
107C22×C3×C4—41no—
108C22×C3×C4C22⋊C411yescharacteristic
109C22×C3×C4C22⋊C411yescharacteristic
110C22×C3×C4C22⋊C411yescharacteristic
111C22×C3×C4C22⋊C411yescharacteristic
112C22×C3×C4D812yes—
113C22×C3×C4SD1612yes—
114C22×C3×C4C22×C411yescharacteristic
115C22×C3×C4C2×D412yes—
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4#4112yes—
118C22×Dic3—24no—
119C22×Dic3—42no—
120C22×Dic3—42no—
121C22×Dic3—42no—
122C22×Dic3—42no—
123C22×Dic3—42no—
124C22×Dic3—42no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—41no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—42no—
129C3×C4⋊C4—42no—
130C3×C22⋊C4—42no—
131C3×C22⋊C4—41no—
132C3×C22⋊C4—41no—
133C3×C22⋊C4—42no—
134C3×C22⋊C4—42no—
135C3×C22⋊C4—42no—
136C3×C22⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C42—22no—
139C3×C42—22no—
140C3×C42—21no—
141C3×C42—21no—
142C3×C42—21no—
143C3×C42—21no—
144C2×C3⋊C8—24no—
145C2×C3⋊C8—42no—
146C2×C3⋊C8—42no—
147C22×D4—21no—
148C22×D4C2×Dic311yescharacteristic
149C23×C4—62no—
150C23×C4#6711yescharacteristic
151C23×C4#6711yescharacteristic
152C23×C4C22×D311yescharacteristic
153C22×C8—62no—
154#123—42no—
155#123—42no—
156#121—22no—
157#121—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4C2×Dic311yescharacteristic
161C2×C4⋊C4C2×Dic311yescharacteristic
162C2×C4⋊C4C2×Dic311yescharacteristic
163C2×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—21no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42C2×Dic311yescharacteristic
172C2×C42C2×Dic311yescharacteristic
173C2×C42#6711yescharacteristic
174C2×C42#6711yescharacteristic
175C4⋊C8—64no—
176C22⋊C8—34no—
177C22⋊C8—62no—
178C4×C8—64no—
179#95—21no—
180#95—21no—
181#95—21no—
182#95—64no—
183#95—62no—
184#95—62no—
185#95—122no—
186C23×C3—21no—
187C23×C3—21no—
188C23×C3—21no—
189C23×C3—21no—
190C23×C3—41no—
191C23×C3—41no—
192C23×C3—41no—
193C23×C3—41no—
194C23×C3—41no—
195C23×C3—41no—
196C23×C3—41no—
197C23×C3C2×C22⋊C411yescharacteristic
198C23×C3#12112yes—
199C23×C3C2×D811yescharacteristic
200C23×C3C2×SD1611yescharacteristic
201C23×C3#13512yes—
202C3×D4—82no—
203C3×D4—82no—
204C3×D4—82no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—24no—
216C2×C3×C4—24no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—44no—
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—41no—
232C2×C3×C4#10212yes—
233C2×C3×C4#10212yes—
234C2×C3×C4#10412yes—
235C2×C3×C4C4×D412yes—
236C2×C3×C4#12011yescharacteristic
237C2×C3×C4#12311yescharacteristic
238C2×Dic3—44no—
239C2×Dic3—44no—
240C2×Dic3—44no—
241C2×Dic3—82no—
242C2×Dic3—82no—
243C2×Dic3—82no—
244C2×Dic3—82no—
245C3⋊C8—44no—
246C24—21no—
247C24C2×#6711yescharacteristic
248C2×D4—22no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—42no—
254C2×D4—42no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—64no—
266C22×C4—122no—
267C22×C4—122no—
268C22×C4—122no—
269C22×C4—122no—
270C22×C4—122no—
271C22×C4—122no—
272C22×C4#21312yes—
273C22×C4#21412yes—
274C22×C4#21711yescharacteristic
275C22×C4#21711yescharacteristic
276C22×C4#21711yescharacteristic
277C22×C4#21711yescharacteristic
278C22×C4D3×D412yes—
279C22×C4#23712yes—
280C22×C4C22×Dic311yescharacteristic
281C22×C4C2×#6711yescharacteristic
282C2×C8—64no—
283C2×C8—122no—
284C2×C8—122no—
285C4⋊C4—22no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C42—22no—
299C42—22no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42—21no—
304C22×C3—22no—
305C22×C3—22no—
306C22×C3—22no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—22no—
312C22×C3—81no—
313C22×C3—81no—
314C22×C3—81no—
315C22×C3—81no—
316C22×C3—81no—
317C22×C3—81no—
318C22×C3—81no—
319C22×C3#38611yescharacteristic
320C22×C3#41011yescharacteristic
321C22×C3#41011yescharacteristic
322C22×C3C2×#10211yescharacteristic
323C22×C3#41611yescharacteristic
324C22×C3C2×#10411yescharacteristic
325C22×C3#42111yescharacteristic
326C22×C3#46512yes—
327C22×C3#46612yes—
328C3×C4—22no—
329C3×C4—22no—
330C3×C4—22no—
331C3×C4—21no—
332C3×C4—22no—
333C3×C4—21no—
334C3×C4—41no—
335C3×C4—81no—
336C3×C4—81no—
337C3×C4—81no—
338Dic3—84no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23C2×#21311yescharacteristic
351C23C2×#21411yescharacteristic
352C23#93912yes—
353C23#95012yes—
354C23C2×#21711yescharacteristic
355D4—82no—
356D4—82no—
357D4—82no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—124no—
386C2×C4—124no—
387C2×C4—124no—
388C2×C4—242no—
389C2×C4—242no—
390C2×C4—242no—
391C2×C4—242no—
392C2×C4#83212yes—
393C2×C4#83212yes—
394C2×C4#83712yes—
395C2×C4D4×Dic312yes—
396C2×C4#93511yescharacteristic
397C2×C4#93711yescharacteristic
398C8—124no—
399C2×C3—22no—
400C2×C3—22no—
401C2×C3—161no—
402C2×C3#150011yescharacteristic
403C2×C3#150211yescharacteristic
404C2×C3#151711yescharacteristic
405C2×C3#152111yescharacteristic
406C2×C3#187111yescharacteristic
407C2×C3#189311yescharacteristic
408C2×C3#189911yescharacteristic
409C22—22no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22#503812yes—
425C22#504112yes—
426C22C2×#83211yescharacteristic
427C22#522011yescharacteristic
428C22#523311yescharacteristic
429C22C2×#83711yescharacteristic
430C22#523611yescharacteristic
431C22#524011yescharacteristic
432C22#524011yescharacteristic
433C4—22no—
434C4—22no—
435C4—21no—
436C4—22no—
437C4—21no—
438C4—22no—
439C4—41no—
440C4—81no—
441C4—81no—
442C4—81no—
443C4—244no—
444C3#911411yescharacteristic, complemented, Sylow
445C2—22no—
446C2—22no—
447C2—161no—
448C2#7168111yescharacteristic
449C2#7170511yescharacteristic
450C2#7171311yescharacteristic
451C2#7175411yescharacteristic
452C2#7175711yescharacteristic
453C2#7181111yescharacteristic
454C2#7181411yescharacteristic
subgroupG/H
centerC23C2×#217
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2854C2
socleC23×C3C2×C22⋊C4
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenter#2854D3
hypocenterC3#9114
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2854

Complements

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

NH ≅ G/Ntype
2. C2×#4925C2semidirect
4. #71534C2semidirect
5. #67624C2semidirect
10. C3×#390C4semidirect
11. C3×#390C4semidirect
12. C3×#389C4semidirect
13. C3×#389C4semidirect
77. #390Dic3semidirect
78. #390Dic3semidirect
79. #389Dic3semidirect
80. #389Dic3semidirect
444. C3#9114semidirect

Maximal subgroups

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

subgroupindex
C3×#28542
C2×#49252
C2×#45372
#715342
#676242
#91143