Finite Group Databasebeta

← #11712714

Subgroups — #11712714

Order 768.

829 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 829 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C22×A4⋊Q8C211yesmaximal, characteristic
2A4×#117C211yesmaximal, characteristic
3C2×#5409C211yesmaximal, characteristic
4#73052C214yesmaximal, complemented
5#20911—31nomaximal, Sylow
6C22×C4×A4C2211yescharacteristic
7C22×SL(2,ℤ/4)—21no—
8C2×A4⋊Q8—22no—
9C2×A4⋊Q8—21no—
10C2×A4⋊Q8C414yescomplemented
11C2×A4⋊Q8C2212yes—
12A4×C4⋊C4C2212yes—
13A4×C22⋊C4—21no—
14C42×A4—21no—
15#5409—22no—
16C2×A4⋊C8C2212yes—
17#5004—41nomaximal
18C22×C22⋊Q8—31no—
19C22×#117D311yescharacteristic
20C2×#419—31no—
21C2×#419—31no—
22C2×#407—31no—
23#1873—34no—
24#1872—34no—
25#1736—31no—
26#1736—31no—
27C23×A4D411yescharacteristic
28C22×Dic6—41no—
29C2×C4×A4—22no—
30C2×C4×A4C2×C412yes—
31C2×C4×A4D412yes—
32C2×C4×A4D411yescharacteristic
33C2×C4×A4C2311yescharacteristic
34C2×SL(2,ℤ/4)—24no—
35C2×SL(2,ℤ/4)—22no—
36A4⋊Q8—24no—
37A4⋊Q8—22no—
38A4⋊Q8C2×C414yes—
39C3×#117—41no—
40C2×#208—41no—
41A4⋊C8—22no—
42#810—44no—
43C23×Q8—31no—
44C24×C4C2×D311yescharacteristic
45C22×M16—31no—
46C22×M16—31no—
47C2×C22⋊Q8—32no—
48C2×C22⋊Q8—34no—
49C2×C22⋊Q8—34no—
50C2×C22⋊Q8—32no—
51C2×C22⋊Q8—61no—
52C2×C22⋊Q8—62no—
53C2×C22⋊Q8—62no—
54C2×C22⋊Q8—61no—
55C2×#117—31no—
56C2×#117—31no—
57C2×#117—32no—
58C2×#117—61no—
59C2×#117—61no—
60C22×C4⋊C4—31no—
61C22×C4⋊C4—61no—
62C22×C4⋊C4C2×D312yes—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—61no—
65C22×C42—21no—
66#419—62no—
67#419—62no—
68#419—61no—
69#419—62no—
70#419—61no—
71C2×#103—34no—
72C2×#103—34no—
73#407—62no—
74#407—62no—
75C2×C22⋊C8—32no—
76C2×C22⋊C8—32no—
77#340—34no—
78#340—34no—
79C22×A4—22no—
80C22×A4C22⋊C412yes—
81C22×A4C2×D411yescharacteristic
82C22×C3×C4—41no—
83C22×Dic3—81no—
84C2×Dic6—44no—
85C2×Dic6—42no—
86C2×Dic6—82no—
87C2×Dic6—81no—
88C4×A4—42no—
89C4×A4C22⋊C412yes—
90C4×A4C22×C411yescharacteristic
91C4×A4C2×D411yescharacteristic
92SL(2,ℤ/4)—24no—
93C3×C4⋊C4—42no—
94C3×C22⋊C4—81no—
95C3×C42—81no—
96#208—82no—
97C2×C3⋊C8—42no—
98C25#6711yescharacteristic
99C22×Q8—34no—
100C22×Q8—32no—
101C22×Q8—62no—
102C22×Q8—61no—
103C22×Q8—62no—
104C22×Q8—62no—
105C22×Q8—61no—
106C22×Q8—61no—
107C22×Q8—61no—
108C22×Q8—61no—
109C23×C4—22no—
110C23×C4—31no—
111C23×C4—31no—
112C23×C4—31no—
113C23×C4—31no—
114C23×C4—32no—
115C23×C4—32no—
116C23×C4—61no—
117C23×C4—61no—
118C23×C4C4×D312yes—
119C23×C4D1212yes—
120C23×C4#6711yescharacteristic
121C23×C4C22×D311yescharacteristic
122C2×M16—62no—
123C2×M16—62no—
124C2×M16—62no—
125C2×M16—62no—
126C2×M16—61no—
127C2×M16—61no—
128C2×M16—61no—
129C2×M16—61no—
130C22×C8—32no—
131C22×C8—32no—
132C22⋊Q8—38no—
133C22⋊Q8—34no—
134C22⋊Q8—34no—
135C22⋊Q8—64no—
136C22⋊Q8—62no—
137C22⋊Q8—64no—
138C22⋊Q8—64no—
139C22⋊Q8—64no—
140C22⋊Q8—64no—
141C22⋊Q8—62no—
142#117—32no—
143#117—32no—
144#117—31no—
145#117—32no—
146#117—61no—
147#117—61no—
148#117—61no—
149#117—64no—
150#117S411yescharacteristic
151C2×C4⋊C4—32no—
152C2×C4⋊C4—34no—
153C2×C4⋊C4—32no—
154C2×C4⋊C4—32no—
155C2×C4⋊C4—62no—
156C2×C4⋊C4—62no—
157C2×C4⋊C4—61no—
158C2×C4⋊C4—62no—
159C2×C4⋊C4—62no—
160C2×C4⋊C4—64no—
161C2×C4⋊C4—64no—
162C2×C22⋊C4—61no—
163C2×C22⋊C4—62no—
164C2×C22⋊C4—64no—
165C2×C22⋊C4—64no—
166C2×C22⋊C4—62no—
167C2×C22⋊C4—62no—
168C2×C22⋊C4—61no—
169C2×C42—61no—
170C2×C42—62no—
171C2×C42—61no—
172#103—68no—
173#103—64no—
174#103—64no—
175C22⋊C8—64no—
176C22⋊C8—62no—
177C22⋊C8—62no—
178C23×C3—41no—
179C2×A4—22no—
180C2×A4C2×C22⋊C411yescharacteristic
181C2×A4#13712yes—
182C2×C3×C4—42no—
183C2×C3×C4—42no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—82no—
187C2×Dic3—84no—
188C2×Dic3—82no—
189Dic6—44no—
190Dic6—82no—
191Dic6—84no—
192C3⋊C8—82no—
193C24—22no—
194C24—31no—
195C24—31no—
196C24—32no—
197C24—62no—
198C24(C2×D3)⋊C412yes—
199C24C2×#6711yescharacteristic
200C2×Q8—34no—
201C2×Q8—62no—
202C2×Q8—64no—
203C2×Q8—62no—
204C2×Q8—64no—
205C2×Q8—62no—
206C2×Q8—64no—
207C2×Q8—62no—
208C2×Q8—64no—
209C2×Q8—64no—
210C2×Q8—62no—
211C2×Q8—122no—
212C2×Q8—121no—
213C2×Q8—122no—
214C2×Q8—122no—
215C2×Q8—122no—
216C2×Q8—122no—
217C2×Q8—121no—
218C22×C4—31no—
219C22×C4—32no—
220C22×C4—32no—
221C22×C4—32no—
222C22×C4—32no—
223C22×C4—31no—
224C22×C4—31no—
225C22×C4—31no—
226C22×C4—31no—
227C22×C4—32no—
228C22×C4—31no—
229C22×C4—31no—
230C22×C4—31no—
231C22×C4—32no—
232C22×C4—32no—
233C22×C4—32no—
234C22×C4—31no—
235C22×C4—32no—
236C22×C4—32no—
237C22×C4—32no—
238C22×C4—42no—
239C22×C4—62no—
240C22×C4—61no—
241C22×C4—61no—
242C22×C4—62no—
243C22×C4—61no—
244C22×C4—62no—
245C22×C4—62no—
246C22×C4—64no—
247C22×C4—64no—
248C22×C4—62no—
249C22×C4—62no—
250C22×C4—122no—
251C22×C4—121no—
252C22×C4—124no—
253C22×C4—121no—
254C22×C4—122no—
255C22×C4(C2×D3)⋊C412yes—
256C22×C4C2×C4×D311yescharacteristic
257C22×C4C2×D1211yescharacteristic
258C22×C4C2×S411yescharacteristic
259M16—122no—
260M16—122no—
261M16—122no—
262M16—122no—
263C2×C8—62no—
264C2×C8—62no—
265C2×C8—62no—
266C2×C8—62no—
267C2×C8—62no—
268C2×C8—62no—
269C4⋊C4—34no—
270C4⋊C4—34no—
271C4⋊C4—32no—
272C4⋊C4—62no—
273C4⋊C4—62no—
274C4⋊C4—64no—
275C4⋊C4—68no—
276C4⋊C4—64no—
277C4⋊C4—64no—
278C4⋊C4C2×S412yes—
279C22⋊C4—21no—
280C22⋊C4—62no—
281C22⋊C4—62no—
282C22⋊C4—64no—
283C22⋊C4—68no—
284C22⋊C4—64no—
285C22⋊C4—61no—
286C42—21no—
287C42—62no—
288C42—62no—
289C42—61no—
290C22×C3—42no—
291C22×C3—41no—
292C22×C3—82no—
293A4#41911yescharacteristic, complemented
294C3×C4—41no—
295C3×C4—41no—
296C3×C4—42no—
297C3×C4—162no—
298Dic3—84no—
299C23—22no—
300C23—31no—
301C23—31no—
302C23—32no—
303C23—32no—
304C23—32no—
305C23—32no—
306C23—31no—
307C23—62no—
308C23—62no—
309C23—61no—
310C23—62no—
311C23—64no—
312C23—62no—
313C23—62no—
314C23—122no—
315C23C2×(C2×D3)⋊C411yescharacteristic
316C23#95112yes—
317C23GL(2,ℤ/4)11yescharacteristic
318Q8—64no—
319Q8—64no—
320Q8—122no—
321Q8—124no—
322Q8—122no—
323Q8—124no—
324C2×C4—22no—
325C2×C4—32no—
326C2×C4—32no—
327C2×C4—31no—
328C2×C4—31no—
329C2×C4—31no—
330C2×C4—31no—
331C2×C4—32no—
332C2×C4—31no—
333C2×C4—31no—
334C2×C4—32no—
335C2×C4—32no—
336C2×C4—32no—
337C2×C4—32no—
338C2×C4—32no—
339C2×C4—32no—
340C2×C4—31no—
341C2×C4—31no—
342C2×C4—61no—
343C2×C4—62no—
344C2×C4—62no—
345C2×C4—61no—
346C2×C4—64no—
347C2×C4—64no—
348C2×C4—62no—
349C2×C4—122no—
350C2×C4—122no—
351C2×C4—124no—
352C2×C4—122no—
353C2×C4—124no—
354C2×C4—128no—
355C2×C4—124no—
356C2×C4—122no—
357C2×C4—122no—
358C2×C4—122no—
359C2×C4C4×S412yes—
360C2×C4C4⋊S412yes—
361C2×C4GL(2,ℤ/4)11yescharacteristic
362C2×C4C22×S411yescharacteristic
363C8—122no—
364C8—122no—
365C2×C3—42no—
366C2×C3—41no—
367C2×C3—82no—
368C22—22no—
369C22—31no—
370C22—32no—
371C22—32no—
372C22—32no—
373C22—31no—
374C22—31no—
375C22—32no—
376C22—62no—
377C22—62no—
378C22—64no—
379C22—61no—
380C22—62no—
381C22—62no—
382C22—62no—
383C22—122no—
384C22#500411yescharacteristic, complemented
385C22(C2×S4)⋊C412yes—
386C22C2×GL(2,ℤ/4)11yescharacteristic
387C4—31no—
388C4—31no—
389C4—32no—
390C4—42no—
391C4—124no—
392C4—122no—
393C4—124no—
394C4(C2×S4)⋊C412yes—
395C4C2×C4×S411yescharacteristic
396C4C2×C4⋊S411yescharacteristic
397C3—41noSylow
398C2—22no—
399C2—31no—
400C2—31no—
401C2—32no—
402C2—62no—
403C2C2×(C2×S4)⋊C411yescharacteristic
404C2#8551212yes—
subgroupG/H
centerC22C2×GL(2,ℤ/4)
commutatorC4×A4C22×C4
FrattiniC2×C4C22×S4
FittingC22×#117D3
socleC24C2×#67
prime radicalC23×A4D4
cosocleC2×C4×A4C23
hypercenter#117S4
hypocenterA4#419
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
4. #73052C2semidirect
10. C2×A4⋊Q8C4semidirect
293. A4#419semidirect
384. C22#5004semidirect

Maximal subgroups

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

subgroupindex
C22×A4⋊Q82
A4×#1172
C2×#54092
#730522
#209113
#50044