Finite Group Databasebeta

← #11714017

Subgroups — #11714017

Order 768.

480 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 59 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
1C4×Q8×A4C211yesmaximal, characteristic
2#85284C211yesmaximal, characteristic
3#73147C211yesmaximal, characteristic
4#73147C211yesmaximal, characteristic
5#73051C211yesmaximal, characteristic
6#73050C211yesmaximal, characteristic
7C4⋊(A4⋊C8)C211yesmaximal, characteristic
8#32960—31nomaximal, Sylow
9C2×Q8×A4C2211yescharacteristic
10A4×C4⋊C4—21no—
11A4×C4⋊C4C2211yescharacteristic
12C42×A4—21no—
13C42×A4C2211yescharacteristic
14C4⋊SL(2,ℤ/4)C2211yescharacteristic
15C4⋊SL(2,ℤ/4)C2211yescharacteristic
16SL(2,ℤ/4)⋊C4—21no—
17SL(2,ℤ/4)⋊C4—21no—
18C2×A4⋊C8C2211yescharacteristic
19C2×A4⋊C8C2211yescharacteristic
20#5022—41nomaximal
21C22×C4×Q8D311yescharacteristic
22C2×#479—31no—
23C2×#479—31no—
24#2851—31no—
25#1865—31no—
26#1865—31no—
27#1863—31no—
28#1863—31no—
29#1853—31no—
30#1852—31no—
31#1805—31no—
32#1805—31no—
33#1797—31no—
34#1787—31no—
35#1787—31no—
36Q8×A4—21no—
37Q8×A4Q812yes—
38C2×C4×A4—21no—
39C2×C4×A4—21no—
40C2×C4×A4D411yescharacteristic
41C2×C4×A4D411yescharacteristic
42C2×C4×A4C2311yescharacteristic
43C2×SL(2,ℤ/4)—21no—
44C2×SL(2,ℤ/4)—21no—
45C3×C4×Q8—41no—
46#870—41no—
47A4⋊C8—21no—
48A4⋊C8—21no—
49#835—41no—
50#835—41no—
51#808—41no—
52#807—41no—
53C4⋊(C3⋊C8)—41no—
54C23×Q8C2×D311yescharacteristic
55C2×#125—31no—
56C2×C4×Q8—31no—
57C2×C4×Q8—31no—
58C2×C4×Q8—31no—
59C2×C4×Q8—31no—
60C2×C4×Q8—61no—
61C2×C4×Q8—61no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4C2×D311yescharacteristic
64C22×C42—21no—
65C22×C42C2×D311yescharacteristic
66#479—62no—
67#479—62no—
68#479—61no—
69#479—61no—
70#479—61no—
71#479—61no—
72C2×#107—31no—
73C2×#107—31no—
74C2×#106—31no—
75C2×#106—31no—
76C2×C4⋊C8—31no—
77C2×C4⋊C8—31no—
78C2×#103—31no—
79C2×#103—31no—
80C2×#103—31no—
81C2×#103—31no—
82C2×#103—31no—
83C2×#103—31no—
84C2×#103—31no—
85C2×#103—31no—
86C2×C22⋊C8—31no—
87C2×C22⋊C8—31no—
88#389—31no—
89#389—31no—
90#387—61no—
91#387—61no—
92#385—61no—
93#385—61no—
94#382—31no—
95#380—31no—
96#380—31no—
97#340—31no—
98#340—31no—
99#340—31no—
100#340—31no—
101#336—31no—
102#336—31no—
103C22×A4C2×D411yescharacteristic
104C2×C3×Q8—41no—
105C4×A4—22no—
106C4×A4—21no—
107C4×A4—21no—
108C4×A4—41no—
109C4×A4C2×Q811yescharacteristic
110C4×A4#4111yescharacteristic
111SL(2,ℤ/4)—41no—
112SL(2,ℤ/4)—41no—
113C3×C4⋊C4—41no—
114C3×C4⋊C4—81no—
115C3×C42—41no—
116C3×C42—81no—
117C4⋊Dic3—41no—
118C4⋊Dic3—41no—
119Dic3⋊C4—81no—
120Dic3⋊C4—81no—
121C2×C3⋊C8—41no—
122C2×C3⋊C8—41no—
123C22×Q8—21no—
124C22×Q8—31no—
125C22×Q8—31no—
126C22×Q8—32no—
127C22×Q8—61no—
128C22×Q8—61no—
129C22×Q8Dic612yes—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4#6711yescharacteristic
133C23×C4#6711yescharacteristic
134C23×C4C22×D311yescharacteristic
135C22×C8—31no—
136C22×C8—31no—
137C22×C8—31no—
138C22×C8—31no—
139#125—61no—
140#125—61no—
141#125—61no—
142#125—61no—
143C4×Q8—32no—
144C4×Q8—32no—
145C4×Q8—31no—
146C4×Q8—31no—
147C4×Q8—31no—
148C4×Q8—61no—
149C4×Q8—61no—
150C4×Q8—61no—
151C4×Q8—62no—
152C4×Q8—62no—
153C4×Q8S411yescharacteristic
154C2×C4⋊C4—31no—
155C2×C4⋊C4—31no—
156C2×C4⋊C4—31no—
157C2×C4⋊C4—31no—
158C2×C4⋊C4—31no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—61no—
161C2×C4⋊C4—61no—
162C2×C4⋊C4—61no—
163C2×C4⋊C4—61no—
164C2×C4⋊C4—61no—
165C2×C4⋊C4—61no—
166C2×C4⋊C4—61no—
167C2×C4⋊C4—61no—
168C2×C4⋊C4—61no—
169C2×C22⋊C4—61no—
170C2×C22⋊C4—61no—
171C2×C42—31no—
172C2×C42—31no—
173C2×C42—31no—
174C2×C42—31no—
175C2×C42—61no—
176C2×C42—61no—
177C2×C42—61no—
178C2×C42—61no—
179#107—62no—
180#107—61no—
181#107—61no—
182#106—62no—
183#106—61no—
184#106—61no—
185C4⋊C8—61no—
186C4⋊C8—61no—
187C4⋊C8—61no—
188C4⋊C8—61no—
189#103—62no—
190#103—62no—
191#103—62no—
192#103—62no—
193#103—62no—
194#103—62no—
195#103—61no—
196#103—61no—
197#103—61no—
198#103—61no—
199C22⋊C8—61no—
200C22⋊C8—61no—
201C22⋊C8—61no—
202C22⋊C8—61no—
203#95—61no—
204#95—61no—
205#95—61no—
206#95—61no—
207#95—61no—
208#95—61no—
209C2×A4C22⋊Q811yescharacteristic
210C2×A4#13511yescharacteristic
211C2×A4#13711yescharacteristic
212C3×Q8—42no—
213C3×Q8—81no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—81no—
218C2×C3×C4—81no—
219C2×Dic3—81no—
220C2×Dic3—81no—
221C3⋊C8—81no—
222C3⋊C8—81no—
223C24C2×#6711yescharacteristic
224C2×Q8—32no—
225C2×Q8—32no—
226C2×Q8—32no—
227C2×Q8—32no—
228C2×Q8—32no—
229C2×Q8—31no—
230C2×Q8—61no—
231C2×Q8—61no—
232C2×Q8—61no—
233C2×Q8—62no—
234C2×Q8—61no—
235C2×Q8—61no—
236C2×Q8—62no—
237C2×Q8—64no—
238C2×Q8C2×S411yescharacteristic
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—31no—
243C22×C4—31no—
244C22×C4—31no—
245C22×C4—31no—
246C22×C4—31no—
247C22×C4—31no—
248C22×C4—31no—
249C22×C4—31no—
250C22×C4—41no—
251C22×C4—61no—
252C22×C4—61no—
253C22×C4—61no—
254C22×C4—61no—
255C22×C4—62no—
256C22×C4—61no—
257C22×C4—61no—
258C22×C4—61no—
259C22×C4—61no—
260C22×C4—61no—
261C22×C4—61no—
262C22×C4—121no—
263C22×C4C2×Dic611yescharacteristic
264C22×C4#23511yescharacteristic
265C2×C8—61no—
266C2×C8—61no—
267C2×C8—61no—
268C2×C8—61no—
269C2×C8—61no—
270C2×C8—61no—
271C2×C8—61no—
272C2×C8—61no—
273C2×C8—61no—
274C2×C8—61no—
275C2×C8—61no—
276C2×C8—61no—
277C4⋊C4—21no—
278C4⋊C4—32no—
279C4⋊C4—31no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4—61no—
283C4⋊C4—62no—
284C4⋊C4—61no—
285C4⋊C4—61no—
286C4⋊C4—61no—
287C4⋊C4—61no—
288C4⋊C4—61no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—122no—
294C4⋊C4—121no—
295C4⋊C4—121no—
296C4⋊C4—122no—
297C4⋊C4—121no—
298C4⋊C4—121no—
299C4⋊C4C2×S411yescharacteristic
300C22⋊C4—121no—
301C22⋊C4—121no—
302C22⋊C4—121no—
303C22⋊C4—121no—
304C42—21no—
305C42—31no—
306C42—31no—
307C42—31no—
308C42—61no—
309C42—61no—
310C42—61no—
311C42—62no—
312C42—61no—
313C42C2×S411yescharacteristic
314C22×C3—41no—
315A4#47911yescharacteristic, complemented
316C3×C4—41no—
317C3×C4—41no—
318C3×C4—81no—
319C3×C4—81no—
320C3×C4—82no—
321C3×C4—161no—
322Dic3—161no—
323Dic3—161no—
324C23—31no—
325C23—31no—
326C23—31no—
327C23—31no—
328C23#92411yescharacteristic
329C23#94211yescharacteristic
330C23#95011yescharacteristic
331Q8—21no—
332Q8—34no—
333Q8—32no—
334Q8—62no—
335Q8—62no—
336Q8—62no—
337Q8—61no—
338Q8A4⋊Q812yes—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—31no—
342C2×C4—31no—
343C2×C4—31no—
344C2×C4—31no—
345C2×C4—31no—
346C2×C4—31no—
347C2×C4—31no—
348C2×C4—32no—
349C2×C4—31no—
350C2×C4—31no—
351C2×C4—61no—
352C2×C4—61no—
353C2×C4—62no—
354C2×C4—61no—
355C2×C4—61no—
356C2×C4—62no—
357C2×C4—61no—
358C2×C4—62no—
359C2×C4—61no—
360C2×C4—61no—
361C2×C4—61no—
362C2×C4—61no—
363C2×C4—62no—
364C2×C4—61no—
365C2×C4—121no—
366C2×C4—121no—
367C2×C4—121no—
368C2×C4—121no—
369C2×C4—121no—
370C2×C4—121no—
371C2×C4—121no—
372C2×C4—121no—
373C2×C4—121no—
374C2×C4—121no—
375C2×C4—121no—
376C2×C4—121no—
377C2×C4—121no—
378C2×C4—121no—
379C2×C4—121no—
380C2×C4—121no—
381C2×C4GL(2,ℤ/4)11yescharacteristic
382C2×C4GL(2,ℤ/4)11yescharacteristic
383C2×C4C22×S411yescharacteristic
384C8—121no—
385C8—121no—
386C8—121no—
387C8—121no—
388C2×C3—41no—
389C2×C3—41no—
390C2×C3—41no—
391C22—31no—
392C22—31no—
393C22—31no—
394C22—31no—
395C22—31no—
396C22—31no—
397C22—31no—
398C22—31no—
399C22—31no—
400C22—61no—
401C22#502211yescharacteristic, complemented
402C22C2×GL(2,ℤ/4)11yescharacteristic
403C4—21no—
404C4—22no—
405C4—21no—
406C4—31no—
407C4—31no—
408C4—41no—
409C4—61no—
410C4—62no—
411C4—61no—
412C4—121no—
413C4—241no—
414C4—241no—
415C4—241no—
416C4—241no—
417C4C2×A4⋊Q811yescharacteristic
418C4#591211yescharacteristic
419C3—41noSylow
420C2—31no—
421C2—31no—
422C2—31no—
423C2—31no—
424C2#8548011yescharacteristic
425C2#8549911yescharacteristic
426C2#8551411yescharacteristic
subgroupG/H
centerC22C2×GL(2,ℤ/4)
commutator, cosocleC2×C4×A4C23
FrattiniC2×C4C22×S4
FittingC22×C4×Q8D3
socleC24C2×#67
prime radicalC22×A4C2×D4
hypercenterC4×Q8S4
hypocenterA4#479
LCSG ▷ C2×C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
315. A4#479semidirect
401. C22#5022semidirect

Maximal subgroups

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

subgroupindex
C4×Q8×A42
#852842
#731472
#730512
#730502
C4⋊(A4⋊C8)2
#329603
#50224