Finite Group Databasebeta

← #10711425

Subgroups — #10711425

Order 768.

992 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 992 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#386C211yesmaximal, characteristic
2#72130C211yesmaximal, characteristic
3#72129C211yesmaximal, characteristic
4#67637C214yesmaximal, complemented
5#12852—31nomaximal, Sylow
6C23×C3×D4C2211yescharacteristic
7C24×C3×C4C411yescharacteristic, complemented
8C22×C3×C22⋊C4—21no—
9C22×C3×C22⋊C4C411yescharacteristic, complemented
10C3×#386—21no—
11C3×#386—21no—
12C3×#386C412yescomplemented
13C3×#386C2212yes—
14C2×C3×#95—21no—
15#5247—21no—
16C2×#834—21no—
17C2×#834C2212yes—
18C2×#833—21no—
19C2×#833C2212yes—
20#5207—21no—
21C2×#386D311yescharacteristic
22#1783—31no—
23#1781—31no—
24#1390—34no—
25C25×C3C2×C411yescharacteristic
26C25×C3D411yescharacteristic
27C22×C3×D4—24no—
28C22×C3×D4—41no—
29C22×C3×D4—41no—
30C22×C3×D4—42no—
31C22×C3×D4C2×C411yescharacteristic
32C22×C3×D4D411yescharacteristic
33C22×C3×D4D411yescharacteristic
34C22×C3×D4C2311yescharacteristic
35C23×C3×C4—21no—
36C23×C3×C4—21no—
37C23×C3×C4—21no—
38C23×C3×C4—44no—
39C23×C3×C4—41no—
40C23×C3×C4—41no—
41C23×C3×C4C2×C412yes—
42C23×C3×C4D411yescharacteristic
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—22no—
45C2×C3×C22⋊C4—22no—
46C2×C3×C22⋊C4—44no—
47C2×C3×C22⋊C4—42no—
48C2×C3×C22⋊C4C2×C412yes—
49C2×#217—22no—
50C2×#217—41no—
51C2×#217—41no—
52C2×#208—22no—
53C3×#95—22no—
54C3×#95—22no—
55#834—24no—
56#834—42no—
57#833—24no—
58#833—42no—
59#830—41no—
60#830—41no—
61C23×D4C2×D311yescharacteristic
62C24×C4Dic311yescharacteristic, complemented
63C22×C22⋊C4—21no—
64C22×C22⋊C4Dic311yescharacteristic, complemented
65C2×#100—32no—
66C2×#100—61no—
67C2×#99—32no—
68C2×#99—61no—
69#407—61no—
70#386—21no—
71#386—21no—
72#386Dic312yescomplemented
73#386C2×D312yes—
74#379—61no—
75C2×#95—21no—
76C24×C3—22no—
77C24×C3—21no—
78C24×C3—21no—
79C24×C3—22no—
80C24×C3—22no—
81C24×C3—21no—
82C24×C3—21no—
83C24×C3—41no—
84C24×C3—42no—
85C24×C3—42no—
86C24×C3—42no—
87C24×C3—42no—
88C24×C3C22⋊C411yescharacteristic
89C24×C3C22⋊C411yescharacteristic
90C24×C3C22⋊C411yescharacteristic
91C24×C3C22×C411yescharacteristic
92C24×C3C2×D411yescharacteristic
93C2×C3×D4—22no—
94C2×C3×D4—22no—
95C2×C3×D4—22no—
96C2×C3×D4—22no—
97C2×C3×D4—22no—
98C2×C3×D4—41no—
99C2×C3×D4—41no—
100C2×C3×D4—41no—
101C2×C3×D4—41no—
102C2×C3×D4—41no—
103C2×C3×D4—41no—
104C2×C3×D4—48no—
105C2×C3×D4—48no—
106C2×C3×D4#4114yes—
107C22×C3×C4—24no—
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4—22no—
111C22×C3×C4—22no—
112C22×C3×C4—22no—
113C22×C3×C4—41no—
114C22×C3×C4—44no—
115C22×C3×C4—44no—
116C22×C3×C4—42no—
117C22×C3×C4—42no—
118C22×C3×C4—48no—
119C22×C3×C4—44no—
120C22×C3×C4—41no—
121C22×C3×C4—41no—
122C22×C3×C4—42no—
123C22×C3×C4—44no—
124C22×C3×C4—42no—
125C22×C3×C4—42no—
126C22×C3×C4—42no—
127C22×C3×C4C22⋊C411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×Dic3—42no—
130C3×C22⋊C4—44no—
131C3×C22⋊C4—44no—
132C3×C22⋊C4—42no—
133C3×C22⋊C4—42no—
134#217—42no—
135#217—42no—
136#217—82no—
137#217—82no—
138#208—42no—
139#208—42no—
140C2×C3⋊C8—42no—
141C25C2×Dic311yescharacteristic
142C25#6711yescharacteristic
143C22×D4—24no—
144C22×D4—41no—
145C22×D4—41no—
146C22×D4—42no—
147C22×D4C2×Dic311yescharacteristic
148C22×D4#6711yescharacteristic
149C22×D4#6711yescharacteristic
150C22×D4C22×D311yescharacteristic
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—44no—
155C23×C4—41no—
156C23×C4—41no—
157C23×C4C2×Dic312yes—
158C23×C4#6711yescharacteristic
159C2×M16—62no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—44no—
164C2×C22⋊C4—42no—
165C2×C22⋊C4—62no—
166C2×C22⋊C4—121no—
167C2×C22⋊C4—121no—
168C2×C22⋊C4C2×Dic312yes—
169#100—64no—
170#100—122no—
171#99—64no—
172#99—122no—
173C22⋊C8—121no—
174C22⋊C8—121no—
175#95—22no—
176#95—22no—
177C23×C3—22no—
178C23×C3—22no—
179C23×C3—21no—
180C23×C3—21no—
181C23×C3—24no—
182C23×C3—24no—
183C23×C3—21no—
184C23×C3—21no—
185C23×C3—21no—
186C23×C3—21no—
187C23×C3—21no—
188C23×C3—21no—
189C23×C3—21no—
190C23×C3—21no—
191C23×C3—21no—
192C23×C3—42no—
193C23×C3—42no—
194C23×C3—41no—
195C23×C3—41no—
196C23×C3—41no—
197C23×C3—41no—
198C23×C3—42no—
199C23×C3—48no—
200C23×C3—48no—
201C23×C3—42no—
202C23×C3—41no—
203C23×C3—41no—
204C23×C3—41no—
205C23×C3—41no—
206C23×C3—48no—
207C23×C3—42no—
208C23×C3—42no—
209C23×C3—42no—
210C23×C3—42no—
211C23×C3—81no—
212C23×C3—81no—
213C23×C3—84no—
214C23×C3#9911yescharacteristic
215C23×C3#9911yescharacteristic
216C23×C3#9912yes—
217C23×C3C2×C22⋊C411yescharacteristic
218C23×C3#11712yes—
219C23×C3#12312yes—
220C3×D4—44no—
221C3×D4—44no—
222C3×D4—44no—
223C3×D4—44no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—24no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—42no—
233C2×C3×C4—44no—
234C2×C3×C4—48no—
235C2×C3×C4—44no—
236C2×C3×C4—44no—
237C2×C3×C4—42no—
238C2×C3×C4—44no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—84no—
243C2×C3×C4#12312yes—
244C2×Dic3—82no—
245C2×Dic3—82no—
246C2×Dic3—82no—
247C3⋊C8—82no—
248C24—22no—
249C24—21no—
250C24—21no—
251C24—22no—
252C24—22no—
253C24—21no—
254C24—21no—
255C24—41no—
256C24—42no—
257C24—42no—
258C24—42no—
259C24—42no—
260C24#21711yescharacteristic
261C24#21711yescharacteristic
262C24#21711yescharacteristic
263C24C22×Dic311yescharacteristic
264C24C2×#6711yescharacteristic
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—48no—
277C2×D4—48no—
278C2×D4#23514yes—
279C22×C4—24no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—41no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—48no—
291C22×C4—44no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—44no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—122no—
300C22×C4#21711yescharacteristic
301C22×C4C2×#6711yescharacteristic
302M16—122no—
303M16—122no—
304C2×C8—122no—
305C22⋊C4—44no—
306C22⋊C4—44no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—122no—
310C22⋊C4—122no—
311C22⋊C4—242no—
312C22⋊C4—242no—
313C22×C3—24no—
314C22×C3—24no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—22no—
322C22×C3—22no—
323C22×C3—22no—
324C22×C3—42no—
325C22×C3—42no—
326C22×C3—42no—
327C22×C3—42no—
328C22×C3—42no—
329C22×C3—48no—
330C22×C3—48no—
331C22×C3—42no—
332C22×C3—41no—
333C22×C3—41no—
334C22×C3—41no—
335C22×C3—41no—
336C22×C3—41no—
337C22×C3—41no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3—41no—
341C22×C3—81no—
342C22×C3—84no—
343C22×C3—84no—
344C22×C3—81no—
345C22×C3AΣL(1,𝔽16)12yes—
346C22×C3#35212yes—
347C22×C3#38111yescharacteristic
348C22×C3C2×#9911yescharacteristic
349C22×C3C2×#9911yescharacteristic
350C3×C4—22no—
351C3×C4—41no—
352C3×C4—44no—
353C3×C4—82no—
354Dic3—162no—
355C23—21no—
356C23—21no—
357C23—22no—
358C23—22no—
359C23—24no—
360C23—24no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—42no—
371C23—42no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—42no—
377C23—48no—
378C23—48no—
379C23—42no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—48no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—81no—
390C23—81no—
391C23—84no—
392C23#83411yescharacteristic
393C23#83411yescharacteristic
394C23#83412yes—
395C23#92612yes—
396C23#92912yes—
397C23C2×#21711yescharacteristic
398D4—44no—
399D4—44no—
400D4—44no—
401D4—44no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—24no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—48no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—84no—
421C2×C4—242no—
422C2×C4—242no—
423C2×C4—242no—
424C2×C4#92912yes—
425C8—242no—
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—22no—
429C2×C3—22no—
430C2×C3—42no—
431C2×C3—42no—
432C2×C3—42no—
433C2×C3—81no—
434C2×C3#181911yescharacteristic
435C2×C3C2×AΣL(1,𝔽16)11yescharacteristic
436C2×C3C2×#35211yescharacteristic
437C22—24no—
438C22—24no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—42no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—48no—
454C22—48no—
455C22—42no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—81no—
466C22—84no—
467C22—84no—
468C22—81no—
469C22#453612yes—
470C22#453912yes—
471C22#521111yescharacteristic
472C22C2×#83411yescharacteristic
473C22C2×#83411yescharacteristic
474C4—22no—
475C4—41no—
476C4—44no—
477C4—82no—
478C4—482no—
479C3#1285211yescharacteristic, complemented, Sylow
480C2—21no—
481C2—21no—
482C2—22no—
483C2—22no—
484C2—42no—
485C2—42no—
486C2—42no—
487C2—81no—
488C2#7154211yescharacteristic
489C2C2×#453611yescharacteristic
490C2C2×#453911yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC24×C3C22×C4
FrattiniC22×D4C22×D3
FittingC2×C3×#386C2
socleC22×C3C2×#99
prime radicalC23×C3×D4C22
cosocleC22×C3×D4C23
hypercenterC2×#386D3
hypocenterC3#12852
LCSG ▷ C24×C3 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ C2×#386

Complements

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

NH ≅ G/Ntype
4. #67637C2semidirect
7. C24×C3×C4C4semidirect
9. C22×C3×C22⋊C4C4semidirect
12. C3×#386C4semidirect
62. C24×C4Dic3semidirect
64. C22×C22⋊C4Dic3semidirect
72. #386Dic3semidirect
479. C3#12852semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3862
#721302
#721292
#676372
#128523