Finite Group Databasebeta

← #11711776

Subgroups — #11711776

Order 768.

773 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 773 conjugacy classes of subgroups — too many to draw. Only its 83 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×A4×C4⋊C4C211yesmaximal, characteristic
2C2×C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic
3A4⋊#95C211yesmaximal, characteristic
4A4⋊#95C212yesmaximal
5A4⋊#95C212yesmaximal
6#14822—31nomaximal, Sylow
7C22×C4×A4C2211yescharacteristic
8C22×C4×A4C2211yescharacteristic
9C22×C4×A4C2211yescharacteristic
10C22×SL(2,ℤ/4)C2212yes—
11C22×SL(2,ℤ/4)C2212yes—
12A4×C4⋊C4—22no—
13C4⋊SL(2,ℤ/4)—22no—
14#4986—41nomaximal
15C23×C4⋊C4D311yescharacteristic
16C2×#400—31no—
17C2×#400—31no—
18C2×#380—31no—
19#1442—32no—
20#1442—32no—
21#1441—31no—
22#1441—31no—
23C22⋊#95—31no—
24C22⋊#95—32no—
25C22⋊#95—32no—
26C23×A4C2311yescharacteristic
27C2×C4×A4—21no—
28C2×C4×A4—21no—
29C2×C4×A4—22no—
30C2×C4×A4—21no—
31C2×C4×A4—22no—
32C2×C4×A4D412yes—
33C2×C4×A4Q812yes—
34C2×SL(2,ℤ/4)—22no—
35C2×SL(2,ℤ/4)—22no—
36C2×SL(2,ℤ/4)—22no—
37C2×SL(2,ℤ/4)—22no—
38C2×SL(2,ℤ/4)—22no—
39C2×SL(2,ℤ/4)—22no—
40C2×C3×C4⋊C4—41no—
41C2×C4⋊Dic3—41no—
42C3⋊#95—41no—
43C3⋊#95—42no—
44C3⋊#95—42no—
45C24×C4C2×D311yescharacteristic
46C24×C4C2×D311yescharacteristic
47C24×C4C2×D311yescharacteristic
48C22×C4⋊C4—22no—
49C22×C4⋊C4—31no—
50C22×C4⋊C4—31no—
51C22×C4⋊C4—31no—
52C22×C4⋊C4—31no—
53C22×C4⋊C4—31no—
54C22×C4⋊C4—31no—
55C22×C4⋊C4—62no—
56C22×C22⋊C4—32no—
57C22×C22⋊C4—32no—
58#400—64no—
59#400—61no—
60#400—61no—
61#400—61no—
62#400—61no—
63#380—62no—
64#380—62no—
65C2×#95—31no—
66C2×#95—31no—
67C2×#95—32no—
68C2×#95—32no—
69C2×#95—32no—
70C2×#95—32no—
71C2×#95—32no—
72C2×#95—32no—
73C2×#95—32no—
74C2×#95—32no—
75C2×#95—32no—
76C2×#95—32no—
77C22×A4C2×D411yescharacteristic
78C22×A4C2×Q811yescharacteristic
79C22×A4#4111yescharacteristic
80C22×A4#4112yes—
81C22×A4#4112yes—
82C22×C3×C4—41no—
83C22×C3×C4—41no—
84C22×C3×C4—41no—
85C22×Dic3—42no—
86C22×Dic3—42no—
87C4×A4—22no—
88C4×A4—22no—
89C4×A4—41no—
90SL(2,ℤ/4)—42no—
91SL(2,ℤ/4)—42no—
92C3×C4⋊C4—82no—
93C4⋊Dic3—82no—
94C25C22×D311yescharacteristic
95C23×C4—21no—
96C23×C4—22no—
97C23×C4—21no—
98C23×C4—22no—
99C23×C4—21no—
100C23×C4—31no—
101C23×C4—31no—
102C23×C4—31no—
103C23×C4—31no—
104C23×C4—31no—
105C23×C4—32no—
106C23×C4—32no—
107C23×C4—31no—
108C23×C4—32no—
109C23×C4—32no—
110C23×C4—32no—
111C23×C4—32no—
112C23×C4—61no—
113C23×C4—61no—
114C23×C4—61no—
115C23×C4—62no—
116C23×C4—62no—
117C23×C4Dic612yes—
118C23×C4#6712yes—
119C2×C4⋊C4—34no—
120C2×C4⋊C4—31no—
121C2×C4⋊C4—31no—
122C2×C4⋊C4—31no—
123C2×C4⋊C4—62no—
124C2×C4⋊C4—62no—
125C2×C4⋊C4—61no—
126C2×C4⋊C4—62no—
127C2×C4⋊C4—62no—
128C2×C4⋊C4—62no—
129C2×C4⋊C4—62no—
130C2×C4⋊C4—62no—
131C2×C4⋊C4—62no—
132C2×C4⋊C4—62no—
133C2×C4⋊C4—61no—
134C2×C4⋊C4—62no—
135C2×C4⋊C4—61no—
136C2×C4⋊C4—122no—
137C2×C4⋊C4S411yescharacteristic
138C2×C22⋊C4—62no—
139C2×C22⋊C4—62no—
140C2×C22⋊C4—62no—
141C2×C22⋊C4—62no—
142C2×C22⋊C4—62no—
143C2×C22⋊C4—62no—
144C2×C22⋊C4—62no—
145C2×C22⋊C4—62no—
146C2×C22⋊C4—62no—
147C2×C22⋊C4—62no—
148C2×C22⋊C4—62no—
149C2×C22⋊C4—62no—
150#95—64no—
151#95—64no—
152#95—64no—
153#95—64no—
154#95—62no—
155#95—61no—
156#95—62no—
157#95—64no—
158#95—62no—
159#95—61no—
160#95—62no—
161#95—62no—
162#95—122no—
163#95—122no—
164#95—122no—
165C23×C3—41no—
166C2×A4C22⋊Q811yescharacteristic
167C2×A4#12311yescharacteristic
168C2×A4#12411yescharacteristic
169C2×A4#12512yes—
170C2×A4#12612yes—
171C2×C3×C4—42no—
172C2×C3×C4—42no—
173C2×C3×C4—81no—
174C2×C3×C4—81no—
175C2×C3×C4—82no—
176C2×C3×C4—81no—
177C2×C3×C4—82no—
178C2×Dic3—82no—
179C2×Dic3—82no—
180C2×Dic3—82no—
181C2×Dic3—82no—
182C2×Dic3—82no—
183C2×Dic3—82no—
184C24—31no—
185C24—32no—
186C24—31no—
187C24—31no—
188C24—32no—
189C24—31no—
190C24C2×Dic611yescharacteristic
191C24#23511yescharacteristic
192C24#23712yes—
193C24#23912yes—
194C24C2×#6711yescharacteristic
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—32no—
198C22×C4—32no—
199C22×C4—32no—
200C22×C4—31no—
201C22×C4—31no—
202C22×C4—32no—
203C22×C4—32no—
204C22×C4—32no—
205C22×C4—32no—
206C22×C4—32no—
207C22×C4—32no—
208C22×C4—31no—
209C22×C4—41no—
210C22×C4—61no—
211C22×C4—61no—
212C22×C4—62no—
213C22×C4—61no—
214C22×C4—62no—
215C22×C4—61no—
216C22×C4—62no—
217C22×C4—61no—
218C22×C4—61no—
219C22×C4—62no—
220C22×C4—61no—
221C22×C4—62no—
222C22×C4—61no—
223C22×C4—64no—
224C22×C4—61no—
225C22×C4—62no—
226C22×C4—62no—
227C22×C4—61no—
228C22×C4—62no—
229C22×C4—62no—
230C22×C4—62no—
231C22×C4—64no—
232C22×C4—62no—
233C22×C4—62no—
234C22×C4—61no—
235C22×C4—62no—
236C22×C4—62no—
237C22×C4—62no—
238C22×C4—62no—
239C22×C4—62no—
240C22×C4—62no—
241C22×C4—62no—
242C22×C4—62no—
243C22×C4—62no—
244C22×C4—62no—
245C22×C4—62no—
246C22×C4—62no—
247C22×C4—62no—
248C22×C4—62no—
249C22×C4—62no—
250C22×C4—62no—
251C22×C4—62no—
252C22×C4—62no—
253C22×C4—62no—
254C22×C4—62no—
255C22×C4—62no—
256C22×C4—62no—
257C22×C4—62no—
258C22×C4—62no—
259C22×C4—62no—
260C22×C4—62no—
261C22×C4—62no—
262C22×C4—62no—
263C22×C4—122no—
264C22×C4—122no—
265C22×C4—122no—
266C22×C4—121no—
267C22×C4C2×S411yescharacteristic
268C22×C4C2×S411yescharacteristic
269C22×C4C2×S411yescharacteristic
270C4⋊C4—22no—
271C4⋊C4—62no—
272C4⋊C4—62no—
273C4⋊C4—64no—
274C4⋊C4—62no—
275C4⋊C4—122no—
276C4⋊C4—124no—
277C4⋊C4—122no—
278C22⋊C4—122no—
279C22⋊C4—122no—
280C22⋊C4—122no—
281C22⋊C4—122no—
282C22⋊C4—122no—
283C22⋊C4—122no—
284C22⋊C4—122no—
285C22⋊C4—122no—
286C22×C3—41no—
287C22×C3—41no—
288C22×C3—42no—
289C22×C3—42no—
290C22×C3—41no—
291A4#40011yescharacteristic, complemented
292C3×C4—82no—
293C3×C4—82no—
294C3×C4—161no—
295Dic3—162no—
296Dic3—162no—
297C23—31no—
298C23—31no—
299C23—31no—
300C23—31no—
301C23—32no—
302C23—32no—
303C23—32no—
304C23—32no—
305C23—31no—
306C23—31no—
307C23—32no—
308C23—31no—
309C23—32no—
310C23—32no—
311C23—32no—
312C23—32no—
313C23—31no—
314C23—31no—
315C23—31no—
316C23—31no—
317C23—32no—
318C23—32no—
319C23—32no—
320C23—61no—
321C23—61no—
322C23—61no—
323C23—62no—
324C23—62no—
325C23#89012yes—
326C23#89812yes—
327C23#92411yescharacteristic
328C23#93511yescharacteristic
329C23#94711yescharacteristic
330C23C22×S411yescharacteristic
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—34no—
337C2×C4—32no—
338C2×C4—32no—
339C2×C4—62no—
340C2×C4—62no—
341C2×C4—62no—
342C2×C4—62no—
343C2×C4—62no—
344C2×C4—62no—
345C2×C4—64no—
346C2×C4—62no—
347C2×C4—62no—
348C2×C4—61no—
349C2×C4—62no—
350C2×C4—61no—
351C2×C4—62no—
352C2×C4—62no—
353C2×C4—62no—
354C2×C4—64no—
355C2×C4—62no—
356C2×C4—61no—
357C2×C4—121no—
358C2×C4—121no—
359C2×C4—122no—
360C2×C4—122no—
361C2×C4—121no—
362C2×C4—121no—
363C2×C4—122no—
364C2×C4—122no—
365C2×C4—122no—
366C2×C4—122no—
367C2×C4—122no—
368C2×C4—122no—
369C2×C4—122no—
370C2×C4—122no—
371C2×C4—122no—
372C2×C4—122no—
373C2×C4—122no—
374C2×C4—122no—
375C2×C4—122no—
376C2×C4—122no—
377C2×C4—122no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4—122no—
381C2×C4—122no—
382C2×C4—122no—
383C2×C4—122no—
384C2×C4—122no—
385C2×C4—122no—
386C2×C4—122no—
387C2×C4—122no—
388C2×C4—122no—
389C2×C4—122no—
390C2×C4—122no—
391C2×C4A4⋊Q812yes—
392C2×C4GL(2,ℤ/4)12yes—
393C2×C3—41no—
394C2×C3—42no—
395C2×C3—42no—
396C2×C3—41no—
397C2×C3—41no—
398C22—32no—
399C22—32no—
400C22—31no—
401C22—31no—
402C22—31no—
403C22—31no—
404C22—32no—
405C22—31no—
406C22—32no—
407C22—32no—
408C22—32no—
409C22—32no—
410C22—32no—
411C22—32no—
412C22—32no—
413C22—32no—
414C22—31no—
415C22—31no—
416C22—31no—
417C22—31no—
418C22—31no—
419C22—32no—
420C22—31no—
421C22—62no—
422C22—61no—
423C22—61no—
424C22—62no—
425C22—61no—
426C22#498611yescharacteristic, complemented
427C22C2×A4⋊Q811yescharacteristic
428C22#591211yescharacteristic
429C22#591412yes—
430C22#591912yes—
431C22C2×GL(2,ℤ/4)11yescharacteristic
432C4—22no—
433C4—22no—
434C4—41no—
435C4—62no—
436C4—62no—
437C4—121no—
438C4—242no—
439C4—242no—
440C4—242no—
441C4—242no—
442C3—41noSylow
443C2—31no—
444C2—32no—
445C2—31no—
446C2—32no—
447C2—31no—
448C2—31no—
449C2#8530512yes—
450C2#8531312yes—
451C2#8548011yescharacteristic
452C2#8549111yescharacteristic
453C2#8550811yescharacteristic
subgroupG/H
center, FrattiniC23C22×S4
commutator, prime radical, cosocleC23×A4C23
FittingC23×C4⋊C4D3
socleC25C22×D3
hypercenterC2×C4⋊C4S4
hypocenterA4#400
LCSG ▷ C23×A4 ▷ A4
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
291. A4#400semidirect
426. C22#4986semidirect

Maximal subgroups

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

subgroupindex
C2×A4×C4⋊C42
C2×C4⋊SL(2,ℤ/4)2
A4⋊#952
#148223
#49864