Finite Group Databasebeta

← #11711775

Subgroups — #11711775

Order 768.

861 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 861 conjugacy classes of subgroups — too many to draw. Only its 87 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
3C2×SL(2,ℤ/4)⋊C4C212yesmaximal
4A4⋊#95C212yesmaximal
5A4⋊#95C211yesmaximal, characteristic
6#14821—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—
14SL(2,ℤ/4)⋊C4—24no—
15#4981—41nomaximal
16C23×C4⋊C4D311yescharacteristic
17C2×#398—31no—
18C2×#398—31no—
19C2×#385—32no—
20C2×#380—31no—
21#1441—31no—
22#1441—31no—
23#1441—32no—
24#1441—32no—
25C22⋊#95—31no—
26C22⋊#95—32no—
27C23×A4C2311yescharacteristic
28C2×C4×A4—21no—
29C2×C4×A4—21no—
30C2×C4×A4—22no—
31C2×C4×A4—21no—
32C2×C4×A4—22no—
33C2×C4×A4D412yes—
34C2×C4×A4D412yes—
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×SL(2,ℤ/4)Q814yes—
41C2×C3×C4⋊C4—41no—
42C2×C4⋊Dic3—41no—
43C2×Dic3⋊C4—42no—
44C3⋊#95—41no—
45C3⋊#95—42no—
46C24×C4C2×D311yescharacteristic
47C24×C4C2×D311yescharacteristic
48C24×C4C2×D311yescharacteristic
49C22×C4⋊C4—22no—
50C22×C4⋊C4—31no—
51C22×C4⋊C4—31no—
52C22×C4⋊C4—31no—
53C22×C4⋊C4—31no—
54C22×C4⋊C4—31no—
55C22×C4⋊C4—32no—
56C22×C4⋊C4—32no—
57C22×C4⋊C4—31no—
58C22×C4⋊C4—62no—
59C22×C22⋊C4—32no—
60C22×C22⋊C4—32no—
61#398—64no—
62#398—61no—
63#398—61no—
64#398—61no—
65#398—61no—
66#385—64no—
67#385—64no—
68#380—62no—
69#380—62no—
70C2×#95—31no—
71C2×#95—32no—
72C2×#95—31no—
73C2×#95—32no—
74C2×#95—32no—
75C2×#95—32no—
76C2×#95—32no—
77C2×#95—32no—
78C2×#95—32no—
79C2×#95—32no—
80C22×A4C2×D411yescharacteristic
81C22×A4C2×D411yescharacteristic
82C22×A4C2×Q812yes—
83C22×A4#4111yescharacteristic
84C22×A4#4112yes—
85C22×C3×C4—41no—
86C22×C3×C4—41no—
87C22×C3×C4—41no—
88C22×Dic3—42no—
89C22×Dic3—42no—
90C4×A4—22no—
91C4×A4—22no—
92C4×A4—41no—
93SL(2,ℤ/4)—24no—
94SL(2,ℤ/4)—42no—
95C3×C4⋊C4—82no—
96C4⋊Dic3—82no—
97Dic3⋊C4—84no—
98C25C22×D311yescharacteristic
99C23×C4—21no—
100C23×C4—22no—
101C23×C4—21no—
102C23×C4—22no—
103C23×C4—21no—
104C23×C4—31no—
105C23×C4—31no—
106C23×C4—31no—
107C23×C4—31no—
108C23×C4—31no—
109C23×C4—32no—
110C23×C4—32no—
111C23×C4—32no—
112C23×C4—32no—
113C23×C4—31no—
114C23×C4—32no—
115C23×C4—32no—
116C23×C4—61no—
117C23×C4—61no—
118C23×C4—61no—
119C23×C4—62no—
120C23×C4—62no—
121C23×C4D1212yes—
122C23×C4#6712yes—
123C2×C4⋊C4—34no—
124C2×C4⋊C4—31no—
125C2×C4⋊C4—31no—
126C2×C4⋊C4—31no—
127C2×C4⋊C4—62no—
128C2×C4⋊C4—62no—
129C2×C4⋊C4—61no—
130C2×C4⋊C4—62no—
131C2×C4⋊C4—62no—
132C2×C4⋊C4—62no—
133C2×C4⋊C4—62no—
134C2×C4⋊C4—62no—
135C2×C4⋊C4—64no—
136C2×C4⋊C4—64no—
137C2×C4⋊C4—62no—
138C2×C4⋊C4—62no—
139C2×C4⋊C4—64no—
140C2×C4⋊C4—64no—
141C2×C4⋊C4—64no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4—62no—
144C2×C4⋊C4—62no—
145C2×C4⋊C4—62no—
146C2×C4⋊C4—61no—
147C2×C4⋊C4—122no—
148C2×C4⋊C4S411yescharacteristic
149C2×C22⋊C4—34no—
150C2×C22⋊C4—34no—
151C2×C22⋊C4—62no—
152C2×C22⋊C4—62no—
153C2×C22⋊C4—62no—
154C2×C22⋊C4—62no—
155C2×C22⋊C4—62no—
156C2×C22⋊C4—62no—
157C2×C22⋊C4—62no—
158C2×C22⋊C4—62no—
159C2×C22⋊C4—62no—
160C2×C22⋊C4—62no—
161#95—64no—
162#95—64no—
163#95—64no—
164#95—61no—
165#95—62no—
166#95—64no—
167#95—62no—
168#95—61no—
169#95—62no—
170#95—122no—
171#95—122no—
172#95—122no—
173C23×C3—41no—
174C2×A4#12111yescharacteristic
175C2×A4C22⋊Q812yes—
176C2×A4#12311yescharacteristic
177C2×A4#12512yes—
178C2×A4C4⋊Q811yescharacteristic
179C2×C3×C4—42no—
180C2×C3×C4—42no—
181C2×C3×C4—81no—
182C2×C3×C4—81no—
183C2×C3×C4—82no—
184C2×C3×C4—81no—
185C2×C3×C4—82no—
186C2×Dic3—44no—
187C2×Dic3—82no—
188C2×Dic3—82no—
189C2×Dic3—82no—
190C2×Dic3—82no—
191C2×Dic3—82no—
192C24—31no—
193C24—32no—
194C24—31no—
195C24—31no—
196C24—31no—
197C24—32no—
198C24C2×D1211yescharacteristic
199C24#23511yescharacteristic
200C24#23712yes—
201C24D3×Q812yes—
202C24C2×#6711yescharacteristic
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—32no—
206C22×C4—32no—
207C22×C4—32no—
208C22×C4—31no—
209C22×C4—31no—
210C22×C4—32no—
211C22×C4—32no—
212C22×C4—32no—
213C22×C4—32no—
214C22×C4—32no—
215C22×C4—32no—
216C22×C4—34no—
217C22×C4—34no—
218C22×C4—31no—
219C22×C4—41no—
220C22×C4—61no—
221C22×C4—61no—
222C22×C4—62no—
223C22×C4—61no—
224C22×C4—62no—
225C22×C4—61no—
226C22×C4—62no—
227C22×C4—61no—
228C22×C4—61no—
229C22×C4—62no—
230C22×C4—61no—
231C22×C4—62no—
232C22×C4—61no—
233C22×C4—64no—
234C22×C4—61no—
235C22×C4—62no—
236C22×C4—62no—
237C22×C4—61no—
238C22×C4—62no—
239C22×C4—62no—
240C22×C4—62no—
241C22×C4—64no—
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—61no—
257C22×C4—62no—
258C22×C4—62no—
259C22×C4—62no—
260C22×C4—62no—
261C22×C4—62no—
262C22×C4—62no—
263C22×C4—62no—
264C22×C4—62no—
265C22×C4—62no—
266C22×C4—62no—
267C22×C4—62no—
268C22×C4—62no—
269C22×C4—62no—
270C22×C4—62no—
271C22×C4—122no—
272C22×C4—122no—
273C22×C4—122no—
274C22×C4—121no—
275C22×C4C2×S411yescharacteristic
276C22×C4C2×S411yescharacteristic
277C22×C4C2×S411yescharacteristic
278C4⋊C4—22no—
279C4⋊C4—62no—
280C4⋊C4—62no—
281C4⋊C4—64no—
282C4⋊C4—62no—
283C4⋊C4—128no—
284C4⋊C4—124no—
285C4⋊C4—122no—
286C4⋊C4—124no—
287C4⋊C4—124no—
288C4⋊C4—122no—
289C22⋊C4—64no—
290C22⋊C4—64no—
291C22⋊C4—64no—
292C22⋊C4—64no—
293C22⋊C4—122no—
294C22⋊C4—122no—
295C22⋊C4—122no—
296C22⋊C4—122no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—42no—
300C22×C3—42no—
301C22×C3—41no—
302A4#39811yescharacteristic, complemented
303C3×C4—82no—
304C3×C4—82no—
305C3×C4—161no—
306Dic3—84no—
307Dic3—162no—
308C23—31no—
309C23—31no—
310C23—31no—
311C23—31no—
312C23—32no—
313C23—32no—
314C23—32no—
315C23—32no—
316C23—31no—
317C23—31no—
318C23—32no—
319C23—32no—
320C23—32no—
321C23—31no—
322C23—31no—
323C23—31no—
324C23—32no—
325C23—31no—
326C23—32no—
327C23—32no—
328C23—32no—
329C23—32no—
330C23—31no—
331C23—61no—
332C23—61no—
333C23—62no—
334C23—62no—
335C23—61no—
336C23#88912yes—
337C23#89712yes—
338C23#93011yescharacteristic
339C23#93511yescharacteristic
340C23#94411yescharacteristic
341C23C22×S411yescharacteristic
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—34no—
348C2×C4—32no—
349C2×C4—32no—
350C2×C4—62no—
351C2×C4—62no—
352C2×C4—62no—
353C2×C4—62no—
354C2×C4—62no—
355C2×C4—62no—
356C2×C4—64no—
357C2×C4—62no—
358C2×C4—62no—
359C2×C4—61no—
360C2×C4—62no—
361C2×C4—61no—
362C2×C4—62no—
363C2×C4—62no—
364C2×C4—62no—
365C2×C4—64no—
366C2×C4—64no—
367C2×C4—64no—
368C2×C4—64no—
369C2×C4—64no—
370C2×C4—64no—
371C2×C4—62no—
372C2×C4—64no—
373C2×C4—61no—
374C2×C4—121no—
375C2×C4—121no—
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—121no—
387C2×C4—121no—
388C2×C4—122no—
389C2×C4—122no—
390C2×C4—122no—
391C2×C4—122no—
392C2×C4—122no—
393C2×C4—122no—
394C2×C4—122no—
395C2×C4—122no—
396C2×C4—122no—
397C2×C4—122no—
398C2×C4—122no—
399C2×C4—122no—
400C2×C4—122no—
401C2×C4—122no—
402C2×C4C4⋊S412yes—
403C2×C4GL(2,ℤ/4)12yes—
404C2×C3—42no—
405C2×C3—42no—
406C2×C3—41no—
407C2×C3—41no—
408C2×C3—41no—
409C22—32no—
410C22—32no—
411C22—31no—
412C22—31no—
413C22—31no—
414C22—32no—
415C22—32no—
416C22—32no—
417C22—32no—
418C22—32no—
419C22—32no—
420C22—32no—
421C22—32no—
422C22—31no—
423C22—31no—
424C22—31no—
425C22—31no—
426C22—31no—
427C22—32no—
428C22—31no—
429C22—32no—
430C22—31no—
431C22—31no—
432C22—61no—
433C22—62no—
434C22—62no—
435C22—61no—
436C22—61no—
437C22#498111yescharacteristic, complemented
438C22C2×C4⋊S411yescharacteristic
439C22#591211yescharacteristic
440C22#591412yes—
441C22Q8×S412yes—
442C22C2×GL(2,ℤ/4)11yescharacteristic
443C4—22no—
444C4—22no—
445C4—41no—
446C4—62no—
447C4—62no—
448C4—124no—
449C4—124no—
450C4—121no—
451C4—242no—
452C4—242no—
453C3—41noSylow
454C2—32no—
455C2—31no—
456C2—31no—
457C2—32no—
458C2—31no—
459C2—31no—
460C2#8530412yes—
461C2#8531212yes—
462C2#8548611yescharacteristic
463C2#8549111yescharacteristic
464C2#8550211yescharacteristic
subgroupG/H
center, FrattiniC23C22×S4
commutator, prime radical, cosocleC23×A4C23
FittingC23×C4⋊C4D3
socleC25C22×D3
hypercenterC2×C4⋊C4S4
hypocenterA4#398
LCSG ▷ C23×A4 ▷ A4
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
302. A4#398semidirect
437. C22#4981semidirect

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
C2×SL(2,ℤ/4)⋊C42
A4⋊#952
#148213
#49814