Finite Group Databasebeta

← #10694316

Subgroups — #10694316

Order 768.

742 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
1#82077C211yesmaximal, characteristic
2C3×#1797C211yesmaximal, characteristic, complemented
3C4×C3⋊#95C211yesmaximal, characteristic, complemented
4C2×#4467C212yesmaximal, complemented
5C2×#4470C212yesmaximal
6#9713—31nomaximal, Sylow
7C22×C3×C42C2211yescharacteristic
8C2×#930C2211yescharacteristic
9C2×#924C2211yescharacteristic
10C22×C4×Dic3—21no—
11C2×#875C411yescharacteristic, complemented
12C2×C3×C4⋊C8—21no—
13C2×C3×C22⋊C8C2212yes—
14#5213—21no—
15#5213—21no—
16C2×C3⋊#95—21no—
17C2×C3⋊#95C2212yes—
18#4940—21no—
19#4940—21no—
20#4939C411yescharacteristic, complemented
21#4936C411yescharacteristic, complemented
22#4929C411yescharacteristic, complemented
23C3×#336—21no—
24#4470—24no—
25#4467—24no—
26#2850—31no—
27#1797D311yescharacteristic, complemented
28C2×#328—32no—
29C2×#327—32no—
30C4×#95—31no—
31C23×C3×C4D411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C23×C3×C4C2311yescharacteristic
34C22×#67—21no—
35C23×Dic3—22no—
36C22×D12C2×C411yescharacteristic
37C22×Dic6C2×C411yescharacteristic
38C22×C3×C8—22no—
39C2×C3×C42—42no—
40C2×C3×C42C2×C411yescharacteristic
41C2×C3×C42C2×C411yescharacteristic
42C2×C3×C42D411yescharacteristic
43C2×C3×C42D411yescharacteristic
44C2×#217—21no—
45#930—22no—
46#930—22no—
47C2×(C2×D3)⋊C4—21no—
48C2×(C2×D3)⋊C4—21no—
49C2×(C2×D3)⋊C4—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—21no—
52C2×C4⋊Dic3C2×C411yescharacteristic
53C2×C4⋊Dic3C2×C411yescharacteristic
54#924—22no—
55#924—22no—
56C2×Dic3⋊C4—21no—
57C2×C4×Dic3—24no—
58C2×C4×Dic3—42no—
59C2×C4×Dic3—42no—
60#875—42no—
61C3×C4⋊C8—42no—
62C3×C22⋊C8—22no—
63C3×C22⋊C8—22no—
64C3⋊#95—24no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95—21no—
69C3⋊#95—22no—
70C3⋊#95D414yes—
71C2×#124—31no—
72C2×C22⋊Q8—31no—
73C2×#121—31no—
74C22×C42—61no—
75C22×C42C2×D311yescharacteristic
76C2×C4⋊C8—21no—
77C2×C22⋊C8C2×D312yes—
78#391—31no—
79#390—31no—
80#388—61no—
81#388—61no—
82#386—61no—
83#386—61no—
84#382—31no—
85C2×#95—32no—
86C2×#95—61no—
87#336—21no—
88#328—64no—
89#327—64no—
90C24×C3C2×D411yescharacteristic
91C23×D3—21no—
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4—22no—
95C22×C3×C4—22no—
96C22×C3×C4—22no—
97C22×C3×C4—22no—
98C22×C3×C4—41no—
99C22×C3×C4—41no—
100C22×C3×C4—41no—
101C22×C3×C4C22⋊C411yescharacteristic
102C22×C3×C4C22⋊C411yescharacteristic
103C22×C3×C4C22⋊C411yescharacteristic
104C22×C3×C4C22⋊C411yescharacteristic
105C22×C3×C4C22×C411yescharacteristic
106C22×C3×C4C2×D412yes—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4#4112yes—
109C2×#67—42no—
110C2×#67—42no—
111C2×#67—41no—
112C2×#67—41no—
113C22×Dic3—24no—
114C22×Dic3—24no—
115C22×Dic3—24no—
116C22×Dic3—21no—
117C22×Dic3—21no—
118C22×Dic3—21no—
119C22×Dic3—42no—
120C22×Dic3—42no—
121C22×Dic3—42no—
122C22×Dic3—42no—
123C2×D12—22no—
124C2×D12—41no—
125C2×D12—41no—
126C2×Dic6—22no—
127C2×Dic6—41no—
128C2×Dic6—41no—
129C2×C3×C8—42no—
130C2×C3×C8—42no—
131C2×C3×C8—42no—
132C3×C42—22no—
133C3×C42—22no—
134C3×C42—41no—
135C3×C42—41no—
136#217—42no—
137(C2×D3)⋊C4—42no—
138(C2×D3)⋊C4—42no—
139(C2×D3)⋊C4—42no—
140(C2×D3)⋊C4—42no—
141(C2×D3)⋊C4—41no—
142(C2×D3)⋊C4—41no—
143C4⋊Dic3—22no—
144C4⋊Dic3—22no—
145Dic3⋊C4—41no—
146Dic3⋊C4—41no—
147C4×Dic3—48no—
148C22×Q8—31no—
149C22×D4—31no—
150C22×D4—61no—
151C23×C4—62no—
152C23×C4D1211yescharacteristic
153C23×C4D1211yescharacteristic
154C23×C4C22×D311yescharacteristic
155C22×C8—22no—
156#124—122no—
157C22⋊Q8—62no—
158C22⋊Q8—62no—
159#121—62no—
160#121—62no—
161C2×C4⋊C4—31no—
162C2×C4⋊C4—31no—
163C2×C4⋊C4—61no—
164C2×C22⋊C4—61no—
165C2×C22⋊C4—61no—
166C2×C22⋊C4—61no—
167C2×C22⋊C4—61no—
168C2×C22⋊C4—61no—
169C2×C22⋊C4—61no—
170C2×C42—42no—
171C2×C42—64no—
172C2×C42—122no—
173C2×C42—122no—
174C2×C42C4×D311yescharacteristic
175C2×C42C4×D311yescharacteristic
176C2×C42#6711yescharacteristic
177C2×C42#6711yescharacteristic
178C4⋊C8—42no—
179C22⋊C8—22no—
180C22⋊C8—22no—
181#95—34no—
182#95—64no—
183#95—61no—
184#95—61no—
185#95—61no—
186#95—61no—
187#95—62no—
188C23×C3—21no—
189C23×C3—21no—
190C23×C3—21no—
191C23×C3—21no—
192C23×C3C2×C22⋊C411yescharacteristic
193C23×C3#12112yes—
194C23×C3#13512yes—
195C23×C3AGL(1,ℤ/8)11yescharacteristic
196C23×C3#13711yescharacteristic
197C22×D3—41no—
198C22×D3—41no—
199C22×D3—41no—
200C22×D3—41no—
201C22×D3—41no—
202C22×D3—41no—
203C22×D3—41no—
204C2×C3×C4—24no—
205C2×C3×C4—24no—
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—21no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—44no—
222C2×C3×C4#9912yes—
223C2×C3×C4#10412yes—
224C2×C3×C4#10412yes—
225C2×C3×C4C4×D412yes—
226C2×C3×C4#12411yescharacteristic
227C2×C3×C4#12711yescharacteristic
228#67—82no—
229#67—82no—
230C2×Dic3—28no—
231C2×Dic3—44no—
232C2×Dic3—44no—
233C2×Dic3—44no—
234C2×Dic3—44no—
235C2×Dic3—44no—
236C2×Dic3—44no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246D12—82no—
247Dic6—82no—
248C3×C8—82no—
249C24—61no—
250C24C2×D1211yescharacteristic
251C2×Q8—62no—
252C2×Q8—121no—
253C2×Q8—121no—
254C2×D4—62no—
255C2×D4—121no—
256C2×D4—121no—
257C2×D4—122no—
258C2×D4—122no—
259C2×D4—121no—
260C2×D4—121no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—64no—
271C22×C4—64no—
272C22×C4—64no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4—122no—
277C22×C4—122no—
278C22×C4—122no—
279C22×C4—122no—
280C22×C4(C2×D3)⋊C411yescharacteristic
281C22×C4(C2×D3)⋊C411yescharacteristic
282C22×C4(C2×D3)⋊C411yescharacteristic
283C22×C4(C2×D3)⋊C411yescharacteristic
284C22×C4C2×C4×D311yescharacteristic
285C22×C4D3×D412yes—
286C22×C4#23912yes—
287C22×C4C2×#6711yescharacteristic
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C4⋊C4—62no—
292C4⋊C4—62no—
293C4⋊C4—121no—
294C4⋊C4—121no—
295C22⋊C4—122no—
296C22⋊C4—122no—
297C22⋊C4—122no—
298C22⋊C4—122no—
299C22⋊C4—122no—
300C22⋊C4—121no—
301C22⋊C4—121no—
302C42—22no—
303C42—22no—
304C42—41no—
305C42—41no—
306C42—128no—
307C22×C3—22no—
308C22×C3—22no—
309C22×C3—22no—
310C22×C3—22no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—21no—
315C22×C3#39011yescharacteristic
316C22×C3C2×#9911yescharacteristic
317C22×C3#41011yescharacteristic
318C22×C3#41611yescharacteristic
319C22×C3#41711yescharacteristic
320C22×C3C2×#10411yescharacteristic
321C22×C3C2×#10411yescharacteristic
322C22×C3#46312yes—
323C22×C3#46412yes—
324C2×D3—81no—
325C2×D3—81no—
326C2×D3—81no—
327C2×D3—81no—
328C2×D3—81no—
329C2×D3—81no—
330C2×D3—81no—
331C3×C4—21no—
332C3×C4—22no—
333C3×C4—21no—
334C3×C4—22no—
335C3×C4—22no—
336C3×C4—22no—
337C3×C4—41no—
338Dic3—48no—
339Dic3—81no—
340Dic3—81no—
341Dic3—81no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—121no—
347C23—121no—
348C23—121no—
349C23—121no—
350C23—121no—
351C23—121no—
352C23—121no—
353C23#89512yes—
354C23#90412yes—
355C23#90811yescharacteristic
356C23#90911yescharacteristic
357C23C2×(C2×D3)⋊C411yescharacteristic
358Q8—242no—
359D4—242no—
360D4—242no—
361D4—242no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—21no—
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—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—68no—
381C2×C4—124no—
382C2×C4—124no—
383C2×C4—124no—
384C2×C4—124no—
385C2×C4—124no—
386C2×C4—124no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4—121no—
391C2×C4—121no—
392C2×C4—121no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4—121no—
396C2×C4#80612yes—
397C2×C4#82512yes—
398C2×C4#82512yes—
399C2×C4#89312yes—
400C2×C4#94011yescharacteristic
401C2×C4#94711yescharacteristic
402C8—82no—
403C2×C3—22no—
404C2×C3—22no—
405C2×C3#150011yescharacteristic
406C2×C3#150111yescharacteristic
407C2×C3#150411yescharacteristic
408C2×C3#150911yescharacteristic
409C2×C3#196211yescharacteristic
410C2×C3#196411yescharacteristic
411C2×C3#198311yescharacteristic
412D3—161no—
413C22—22no—
414C22—21no—
415C22—21no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—21no—
420C22—21no—
421C22—241no—
422C22—241no—
423C22—241no—
424C22—241no—
425C22—241no—
426C22—241no—
427C22—241no—
428C22#484412yes—
429C22#484512yes—
430C22C2×#80611yescharacteristic
431C22#498811yescharacteristic
432C22#500611yescharacteristic
433C22#511311yescharacteristic
434C22#513311yescharacteristic
435C22C2×#82511yescharacteristic
436C22C2×#82511yescharacteristic
437C4—22no—
438C4—22no—
439C4—22no—
440C4—21no—
441C4—22no—
442C4—21no—
443C4—41no—
444C4—128no—
445C4—241no—
446C4—241no—
447C4—241no—
448C3#971311yescharacteristic, complemented, Sylow
449C2—22no—
450C2—22no—
451C2—481no—
452C2#6952811yescharacteristic
453C2#6994211yescharacteristic
454C2#6997611yescharacteristic
455C2#7045811yescharacteristic
456C2#7051211yescharacteristic
457C2#7078811yescharacteristic
458C2#7083311yescharacteristic
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1797C2
socleC23×C3C2×C22⋊C4
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1797D3
hypocenterC3#9713
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
2. C3×#1797C2semidirect
3. C4×C3⋊#95C2semidirect
4. C2×#4467C2semidirect
11. C2×#875C4semidirect
20. #4939C4semidirect
21. #4936C4semidirect
22. #4929C4semidirect
27. #1797D3semidirect
448. C3#9713semidirect

Maximal subgroups

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

subgroupindex
#820772
C3×#17972
C4×C3⋊#952
C2×#44672
C2×#44702
#97133