Finite Group Databasebeta

← #10690962

Subgroups — #10690962

Order 768.

1024 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#380C211yesmaximal, characteristic, complemented
2#84032C211yesmaximal, characteristic
3#72126C211yesmaximal, characteristic, complemented
4C2×#4451C212yesmaximal, complemented
5#67612C212yesmaximal, complemented
6#9155—31nomaximal, Sylow
7C24×C3×C4C2211yescharacteristic
8C22×C3×C4⋊C4—21no—
9C22×C3×C22⋊C4—21no—
10C2×#953—21no—
11C2×#930C412yescomplemented
12C2×#930C2211yescharacteristic
13C3×#380—24no—
14C2×C3×#95C2212yes—
15#5213—21no—
16#5213—22no—
17#5212C412yescomplemented
18#5212C2211yescharacteristic
19C2×#830—22no—
20C2×#830C2212yes—
21#4451—24no—
22#2847—31no—
23C2×#380D311yescharacteristic, complemented
24#1779—31no—
25C2×#327—32no—
26#1294—32no—
27C25×C3D411yescharacteristic
28C23×C3×C4—22no—
29C23×C3×C4—22no—
30C23×C3×C4—22no—
31C23×C3×C4—44no—
32C23×C3×C4C2×C412yes—
33C23×C3×C4D412yes—
34C23×C3×C4D411yescharacteristic
35C23×C3×C4C2311yescharacteristic
36C22×#67—22no—
37C22×#67—21no—
38C22×D12C2×C411yescharacteristic
39C2×C3×C4⋊C4—24no—
40C2×C3×C4⋊C4—44no—
41C2×C3×C22⋊C4—24no—
42C2×C3×C22⋊C4—42no—
43C2×C3×C22⋊C4—42no—
44#953—42no—
45#953—42no—
46C2×#217—22no—
47C2×#217—21no—
48#930—22no—
49#930—22no—
50#930—44no—
51C2×(C2×D3)⋊C4—22no—
52C2×(C2×D3)⋊C4—21no—
53C2×C4⋊Dic3C2×C411yescharacteristic
54C2×C4⋊Dic3C2×C412yes—
55C22×C3⋊C8—22no—
56C3×#95—22no—
57C3×#95D414yescomplemented
58C3⋊#95—22no—
59C3⋊#95—21no—
60#830—22no—
61#830—22no—
62#830—44no—
63C24×C4C2×D311yescharacteristic
64C2×#121—31no—
65C2×#121—32no—
66C2×#120—61no—
67C22×C4⋊C4—21no—
68C22×C22⋊C4—21no—
69C2×C22⋊C8—32no—
70C2×C22⋊C8—62no—
71#386—62no—
72#386—61no—
73#380—24no—
74#380—32no—
75#380—31no—
76C2×#95C2×D312yes—
77#327—64no—
78C24×C3—21no—
79C24×C3—21no—
80C24×C3—21no—
81C24×C3—21no—
82C24×C3—42no—
83C24×C3—42no—
84C24×C3C22⋊C412yes—
85C24×C3D812yes—
86C24×C3SD1612yes—
87C24×C3C2×D411yescharacteristic
88C23×D3—21no—
89C22×C3×C4—21no—
90C22×C3×C4—21no—
91C22×C3×C4—24no—
92C22×C3×C4—24no—
93C22×C3×C4—24no—
94C22×C3×C4—24no—
95C22×C3×C4—24no—
96C22×C3×C4—24no—
97C22×C3×C4—42no—
98C22×C3×C4—42no—
99C22×C3×C4—42no—
100C22×C3×C4—48no—
101C22×C3×C4—48no—
102C22×C3×C4—44no—
103C22×C3×C4—44no—
104C22×C3×C4—42no—
105C22×C3×C4—42no—
106C22×C3×C4C22⋊C412yes—
107C22×C3×C4C22×C411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D412yes—
110C22×C3×C4#4112yes—
111C2×#67—44no—
112C2×#67—44no—
113C2×#67—44no—
114C2×#67—41no—
115C2×#67—41no—
116C2×#67—41no—
117C2×#67—41no—
118C2×#67—41no—
119C2×#67—41no—
120C22×Dic3—21no—
121C22×Dic3—22no—
122C2×D12—22no—
123C2×D12—42no—
124C3×C4⋊C4—48no—
125C3×C22⋊C4—48no—
126#217—44no—
127#217—41no—
128#217—41no—
129(C2×D3)⋊C4—44no—
130(C2×D3)⋊C4—42no—
131C4⋊Dic3—22no—
132C4⋊Dic3—42no—
133C2×C3⋊C8—44no—
134C2×C3⋊C8—42no—
135C25#6711yescharacteristic
136C22×D4—31no—
137C22×D4—62no—
138C22×D4—61no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—44no—
143C23×C4C4×D312yes—
144C23×C4D1212yes—
145C23×C4#6711yescharacteristic
146C23×C4C22×D311yescharacteristic
147C22×C8—62no—
148#121—62no—
149#121—62no—
150#121—124no—
151#120—122no—
152#120—122no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—31no—
155C2×C4⋊C4—32no—
156C2×C4⋊C4—44no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—42no—
159C2×C22⋊C4—42no—
160C2×C22⋊C4—62no—
161C2×C22⋊C4—61no—
162C2×C22⋊C4—62no—
163C2×C22⋊C4—61no—
164C22⋊C8—62no—
165C22⋊C8—62no—
166C22⋊C8—124no—
167#95—22no—
168#95—62no—
169#95—61no—
170#95#6714yescomplemented
171C23×C3—22no—
172C23×C3—22no—
173C23×C3—22no—
174C23×C3—22no—
175C23×C3—22no—
176C23×C3—22no—
177C23×C3—22no—
178C23×C3—22no—
179C23×C3—22no—
180C23×C3—22no—
181C23×C3—22no—
182C23×C3—22no—
183C23×C3—44no—
184C23×C3—44no—
185C23×C3—42no—
186C23×C3—42no—
187C23×C3—42no—
188C23×C3—42no—
189C23×C3—44no—
190C23×C3—44no—
191C23×C3#10214yes—
192C23×C3C2×C22⋊C411yescharacteristic
193C23×C3#12011yescharacteristic
194C23×C3#12311yescharacteristic
195C23×C3C2×D811yescharacteristic
196C23×C3C2×SD1611yescharacteristic
197C23×C3AGL(1,ℤ/8)11yescharacteristic
198C23×C3#13711yescharacteristic
199C22×D3—41no—
200C22×D3—42no—
201C22×D3—41no—
202C22×D3—41no—
203C22×D3—42no—
204C2×C3×C4—24no—
205C2×C3×C4—24no—
206C2×C3×C4—22no—
207C2×C3×C4—22no—
208C2×C3×C4—22no—
209C2×C3×C4—22no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4—48no—
213C2×C3×C4—48no—
214C2×C3×C4—44no—
215C2×C3×C4—44no—
216C2×C3×C4—44no—
217C2×C3×C4—84no—
218C2×C3×C4—84no—
219C2×C3×C4#9914yes—
220C2×C3×C4C4×D412yes—
221C2×C3×C4#12112yes—
222#67—88no—
223#67—81no—
224#67—81no—
225#67—81no—
226#67—81no—
227C2×Dic3—41no—
228C2×Dic3—42no—
229C2×Dic3—41no—
230C2×Dic3—42no—
231C2×Dic3—41no—
232C2×Dic3—42no—
233D12—82no—
234C3⋊C8—82no—
235C24—21no—
236C24—21no—
237C24—21no—
238C24—21no—
239C24—42no—
240C24—42no—
241C24—61no—
242C24(C2×D3)⋊C412yes—
243C24#21312yes—
244C24#21512yes—
245C24C2×#6711yescharacteristic
246C2×D4—62no—
247C2×D4—122no—
248C2×D4—124no—
249C2×D4—124no—
250C2×D4—124no—
251C2×D4—121no—
252C2×D4—121no—
253C2×D4—121no—
254C2×D4—121no—
255C2×D4—121no—
256C2×D4—121no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—48no—
269C22×C4—48no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—62no—
275C22×C4—61no—
276C22×C4(C2×D3)⋊C412yes—
277C22×C4C2×C4×D311yescharacteristic
278C22×C4C2×D1211yescharacteristic
279C22×C4#23512yes—
280C22×C4C2×#6712yes—
281C2×C8—124no—
282C2×C8—122no—
283C4⋊C4—48no—
284C4⋊C4—62no—
285C4⋊C4—122no—
286C22⋊C4—48no—
287C22⋊C4—124no—
288C22⋊C4—121no—
289C22⋊C4—121no—
290C22⋊C4—124no—
291C22⋊C4—122no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—22no—
297C22×C3—22no—
298C22×C3—22no—
299C22×C3—22no—
300C22×C3—24no—
301C22×C3—24no—
302C22×C3—24no—
303C22×C3—44no—
304C22×C3—44no—
305C22×C3—44no—
306C22×C3—44no—
307C22×C3—42no—
308C22×C3—42no—
309C22×C3—42no—
310C22×C3—42no—
311C22×C3#38611yescharacteristic
312C22×C3C2×#9912yes—
313C22×C3C2×#10211yescharacteristic
314C22×C3#41711yescharacteristic
315C22×C3#42112yes—
316C22×C3#44711yescharacteristic
317C22×C3#44811yescharacteristic
318C22×C3#48011yescharacteristic
319C22×C3#48111yescharacteristic
320C2×D3—81no—
321C2×D3—81no—
322C2×D3—81no—
323C2×D3—82no—
324C2×D3—82no—
325C3×C4—22no—
326C3×C4—22no—
327C3×C4—24no—
328C3×C4—84no—
329Dic3—81no—
330Dic3—82no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—44no—
344C23—44no—
345C23—42no—
346C23—42no—
347C23—44no—
348C23—44no—
349C23—42no—
350C23—42no—
351C23—122no—
352C23—121no—
353C23—121no—
354C23—121no—
355C23—122no—
356C23#80914yes—
357C23C2×(C2×D3)⋊C411yescharacteristic
358C23#92911yescharacteristic
359C23C2×#21311yescharacteristic
360C23#93211yescharacteristic
361C23C2×#21511yescharacteristic
362C23#94211yescharacteristic
363C23#95311yescharacteristic
364D4—242no—
365D4—248no—
366D4—241no—
367D4—241no—
368D4—241no—
369D4—241no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—48no—
379C2×C4—48no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—84no—
384C2×C4—84no—
385C2×C4—122no—
386C2×C4—121no—
387C2×C4—122no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4—122no—
391C2×C4#80614yes—
392C2×C4C4×#6712yes—
393C2×C4#93012yes—
394C8—242no—
395C2×C3—21no—
396C2×C3—21no—
397C2×C3—21no—
398C2×C3—21no—
399C2×C3—44no—
400C2×C3#151312yes—
401C2×C3#151912yes—
402C2×C3#187411yescharacteristic
403C2×C3#189711yescharacteristic
404C2×C3#189811yescharacteristic
405D3—161no—
406C22—21no—
407C22—21no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—21no—
413C22—21no—
414C22—24no—
415C22—24no—
416C22—24no—
417C22—44no—
418C22—44no—
419C22—44no—
420C22—44no—
421C22—42no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—241no—
426C22—242no—
427C22—242no—
428C22—241no—
429C22—241no—
430C22C2×#80612yes—
431C22C2×#80911yescharacteristic
432C22#496611yescharacteristic
433C22#496711yescharacteristic
434C22#496811yescharacteristic
435C22#500512yes—
436C22#521311yescharacteristic
437C22#521711yescharacteristic
438C22#522711yescharacteristic
439C4—22no—
440C4—24no—
441C4—22no—
442C4—84no—
443C4—242no—
444C4—241no—
445C3#915511yescharacteristic, complemented, Sylow
446C2—21no—
447C2—21no—
448C2—21no—
449C2—21no—
450C2—44no—
451C2—481no—
452C2#6950812yes—
453C2#6950912yes—
454C2#7154311yescharacteristic
455C2#7156011yescharacteristic
456C2#7159111yescharacteristic
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#380C2
socleC23×C3C2×C22⋊C4
prime radical#84032C2
cosocleC23×C3×C4C23
hypercenterC2×#380D3
hypocenterC3#9155
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
1. C2×C3×#380C2semidirect
3. #72126C2semidirect
4. C2×#4451C2semidirect
5. #67612C2semidirect
11. C2×#930C4semidirect
17. #5212C4semidirect
23. C2×#380D3semidirect
57. C3×#95D4semidirect
170. #95#67semidirect
445. C3#9155semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3802
#840322
#721262
C2×#44512
#676122
#91553