Finite Group Databasebeta

← #10694329

Subgroups — #10694329

Order 768.

650 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82077C211yesmaximal, characteristic
2C3×#1797C211yesmaximal, characteristic, complemented
3#71534C211yesmaximal, characteristic, complemented
4C2×#4467C212yesmaximal, complemented
5C2×#4470C212yesmaximal
6#9716—31nomaximal, Sylow
7C22×C3×C42C2211yescharacteristic
8C2×#930C2211yescharacteristic
9C2×#924C2211yescharacteristic
10C2×#875C411yescharacteristic, complemented
11C2×C3×C4⋊C8—21no—
12C2×C3×C22⋊C8C2212yes—
13#5213—21no—
14#5213—21no—
15C2×C3⋊#95—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—
30#1434—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—
57#875—42no—
58C3×C4⋊C8—42no—
59C3×C22⋊C8—22no—
60C3×C22⋊C8—22no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—22no—
66C3⋊#95—42no—
67C3⋊#95—42no—
68C3⋊#95D414yes—
69C2×#124—31no—
70C2×C22⋊Q8—31no—
71C2×#121—31no—
72C22×C42C2×D311yescharacteristic
73C2×C4⋊C8—21no—
74C2×C22⋊C8C2×D312yes—
75#391—31no—
76#390—31no—
77#388—61no—
78#388—61no—
79#386—61no—
80#386—61no—
81#382—31no—
82C2×#95—32no—
83C2×#95—61no—
84C2×#95—61no—
85#336—21no—
86#328—64no—
87#327—64no—
88C24×C3C2×D411yescharacteristic
89C23×D3—21no—
90C22×C3×C4—22no—
91C22×C3×C4—21no—
92C22×C3×C4—21no—
93C22×C3×C4—22no—
94C22×C3×C4—41no—
95C22×C3×C4—41no—
96C22×C3×C4—41no—
97C22×C3×C4—41no—
98C22×C3×C4—41no—
99C22×C3×C4C22⋊C411yescharacteristic
100C22×C3×C4C22⋊C411yescharacteristic
101C22×C3×C4C22⋊C411yescharacteristic
102C22×C3×C4C22⋊C411yescharacteristic
103C22×C3×C4C22×C411yescharacteristic
104C22×C3×C4C2×D412yes—
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4#4112yes—
107C2×#67—42no—
108C2×#67—42no—
109C2×#67—41no—
110C2×#67—41no—
111C22×Dic3—21no—
112C22×Dic3—21no—
113C22×Dic3—21no—
114C22×Dic3—24no—
115C22×Dic3—42no—
116C22×Dic3—42no—
117C22×Dic3—42no—
118C22×Dic3—42no—
119C22×Dic3—42no—
120C22×Dic3—42no—
121C2×D12—22no—
122C2×D12—41no—
123C2×D12—41no—
124C2×Dic6—22no—
125C2×Dic6—41no—
126C2×Dic6—41no—
127C2×C3×C8—42no—
128C2×C3×C8—42no—
129C2×C3×C8—42no—
130C3×C42—22no—
131C3×C42—22no—
132C3×C42—41no—
133C3×C42—41no—
134#217—42no—
135(C2×D3)⋊C4—42no—
136(C2×D3)⋊C4—42no—
137(C2×D3)⋊C4—42no—
138(C2×D3)⋊C4—42no—
139(C2×D3)⋊C4—41no—
140(C2×D3)⋊C4—41no—
141C4⋊Dic3—22no—
142C4⋊Dic3—22no—
143Dic3⋊C4—41no—
144Dic3⋊C4—41no—
145C22×Q8—31no—
146C22×D4—31no—
147C22×D4—61no—
148C23×C4—62no—
149C23×C4D1211yescharacteristic
150C23×C4D1211yescharacteristic
151C23×C4C22×D311yescharacteristic
152C22×C8—22no—
153#124—122no—
154C22⋊Q8—62no—
155C22⋊Q8—62no—
156#121—62no—
157#121—62no—
158C2×C4⋊C4—31no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—61no—
161C2×C22⋊C4—61no—
162C2×C22⋊C4—61no—
163C2×C22⋊C4—61no—
164C2×C22⋊C4—61no—
165C2×C22⋊C4—61no—
166C2×C22⋊C4—61no—
167C2×C42—42no—
168C2×C42C4×D311yescharacteristic
169C2×C42C4×D311yescharacteristic
170C2×C42#6711yescharacteristic
171C2×C42#6711yescharacteristic
172C4⋊C8—42no—
173C22⋊C8—22no—
174C22⋊C8—22no—
175#95—34no—
176#95—61no—
177#95—61no—
178#95—61no—
179#95—61no—
180#95—62no—
181#95—122no—
182#95—122no—
183C23×C3—21no—
184C23×C3—21no—
185C23×C3—21no—
186C23×C3—21no—
187C23×C3C2×C22⋊C411yescharacteristic
188C23×C3#12112yes—
189C23×C3#13512yes—
190C23×C3AGL(1,ℤ/8)11yescharacteristic
191C23×C3#13711yescharacteristic
192C22×D3—41no—
193C22×D3—41no—
194C22×D3—41no—
195C22×D3—41no—
196C22×D3—41no—
197C22×D3—41no—
198C22×D3—41no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
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—42no—
215C2×C3×C4—42no—
216C2×C3×C4—44no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4#9912yes—
220C2×C3×C4C4×D412yes—
221C2×C3×C4#12411yescharacteristic
222C2×C3×C4#12711yescharacteristic
223#67—82no—
224#67—82no—
225C2×Dic3—44no—
226C2×Dic3—44no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—44no—
237C2×Dic3—82no—
238C2×Dic3—82no—
239C2×Dic3—82no—
240C2×Dic3—82no—
241D12—82no—
242Dic6—82no—
243C3×C8—82no—
244C24—61no—
245C24C2×D1211yescharacteristic
246C2×Q8—62no—
247C2×Q8—121no—
248C2×Q8—121no—
249C2×D4—62no—
250C2×D4—121no—
251C2×D4—121no—
252C2×D4—122no—
253C2×D4—122no—
254C2×D4—121no—
255C2×D4—121no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—64no—
266C22×C4—61no—
267C22×C4—61no—
268C22×C4—61no—
269C22×C4—122no—
270C22×C4—122no—
271C22×C4—122no—
272C22×C4—122no—
273C22×C4—122no—
274C22×C4—122no—
275C22×C4(C2×D3)⋊C411yescharacteristic
276C22×C4(C2×D3)⋊C411yescharacteristic
277C22×C4(C2×D3)⋊C411yescharacteristic
278C22×C4(C2×D3)⋊C411yescharacteristic
279C22×C4C2×C4×D311yescharacteristic
280C22×C4D3×D412yes—
281C22×C4#23912yes—
282C22×C4C2×#6711yescharacteristic
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C4⋊C4—62no—
287C4⋊C4—62no—
288C4⋊C4—121no—
289C4⋊C4—121no—
290C22⋊C4—122no—
291C22⋊C4—122no—
292C22⋊C4—122no—
293C22⋊C4—122no—
294C22⋊C4—122no—
295C22⋊C4—121no—
296C22⋊C4—121no—
297C42—22no—
298C42—22no—
299C42—41no—
300C42—41no—
301C22×C3—22no—
302C22×C3—22no—
303C22×C3—21no—
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—22no—
308C22×C3—22no—
309C22×C3#39011yescharacteristic
310C22×C3C2×#9911yescharacteristic
311C22×C3#41011yescharacteristic
312C22×C3#41611yescharacteristic
313C22×C3#41711yescharacteristic
314C22×C3#42111yescharacteristic
315C22×C3#42111yescharacteristic
316C22×C3#46312yes—
317C22×C3#46412yes—
318C2×D3—81no—
319C2×D3—81no—
320C2×D3—81no—
321C2×D3—81no—
322C2×D3—81no—
323C2×D3—81no—
324C2×D3—81no—
325C3×C4—21no—
326C3×C4—22no—
327C3×C4—21no—
328C3×C4—22no—
329C3×C4—41no—
330C3×C4—41no—
331C3×C4—41no—
332Dic3—84no—
333Dic3—81no—
334Dic3—81no—
335Dic3—81no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—121no—
341C23—121no—
342C23—121no—
343C23—121no—
344C23—121no—
345C23—121no—
346C23—121no—
347C23#89512yes—
348C23#90412yes—
349C23#90811yescharacteristic
350C23#90911yescharacteristic
351C23C2×(C2×D3)⋊C411yescharacteristic
352Q8—242no—
353D4—242no—
354D4—242no—
355D4—242no—
356C2×C4—21no—
357C2×C4—21no—
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—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—124no—
377C2×C4—124no—
378C2×C4—121no—
379C2×C4—121no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4—121no—
386C2×C4—124no—
387C2×C4—121no—
388C2×C4—242no—
389C2×C4—242no—
390C2×C4—242no—
391C2×C4—242no—
392C2×C4#80612yes—
393C2×C4#89312yes—
394C2×C4#94011yescharacteristic
395C2×C4#94711yescharacteristic
396C8—82no—
397C2×C3—22no—
398C2×C3—22no—
399C2×C3#151211yescharacteristic
400C2×C3#151311yescharacteristic
401C2×C3#151611yescharacteristic
402C2×C3#152111yescharacteristic
403C2×C3#196311yescharacteristic
404C2×C3#196411yescharacteristic
405C2×C3#198311yescharacteristic
406D3—161no—
407C22—22no—
408C22—21no—
409C22—21no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—21no—
414C22—21no—
415C22—241no—
416C22—241no—
417C22—241no—
418C22—241no—
419C22—241no—
420C22—241no—
421C22—241no—
422C22#484412yes—
423C22#484512yes—
424C22C2×#80611yescharacteristic
425C22#498811yescharacteristic
426C22#500611yescharacteristic
427C22#511311yescharacteristic
428C22#513311yescharacteristic
429C22#513911yescharacteristic
430C22#513911yescharacteristic
431C4—22no—
432C4—21no—
433C4—22no—
434C4—21no—
435C4—41no—
436C4—41no—
437C4—41no—
438C4—244no—
439C4—241no—
440C4—241no—
441C4—241no—
442C3#971611yescharacteristic, complemented, Sylow
443C2—22no—
444C2—22no—
445C2—481no—
446C2#6952811yescharacteristic
447C2#6994311yescharacteristic
448C2#6997911yescharacteristic
449C2#7045811yescharacteristic
450C2#7051311yescharacteristic
451C2#7079111yescharacteristic
452C2#7083411yescharacteristic
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#9716
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. #71534C2semidirect
4. C2×#4467C2semidirect
10. C2×#875C4semidirect
20. #4939C4semidirect
21. #4936C4semidirect
22. #4929C4semidirect
27. #1797D3semidirect
442. C3#9716semidirect

Maximal subgroups

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

subgroupindex
#820772
C3×#17972
#715342
C2×#44672
C2×#44702
#97163