Finite Group Databasebeta

← #11711780

Subgroups — #11711780

Order 768.

773 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 773 conjugacy classes of subgroups — too many to draw. Only its 79 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×SL(2,ℤ/4)⋊C4C211yesmaximal, characteristic
3A4⋊#95C211yesmaximal, characteristic
4A4⋊#95C212yesmaximal
5A4⋊#95C212yesmaximal
6#14843—31nomaximal, Sylow
7C22×C4×A4C2212yes—
8C22×C4×A4C2211yescharacteristic
9C22×SL(2,ℤ/4)C2212yes—
10C22×SL(2,ℤ/4)C2211yescharacteristic
11C22×SL(2,ℤ/4)C2211yescharacteristic
12A4×C4⋊C4—22no—
13SL(2,ℤ/4)⋊C4—22no—
14#4985—41nomaximal
15C23×C4⋊C4D311yescharacteristic
16C2×#400—31no—
17C2×#400—31no—
18C2×#385—31no—
19#1445—31no—
20#1445—31no—
21#1442—32no—
22#1441—32no—
23C22⋊#95—32no—
24C22⋊#95—31no—
25C22⋊#95—32no—
26C23×A4C2311yescharacteristic
27C2×C4×A4—22no—
28C2×C4×A4—22no—
29C2×C4×A4—22no—
30C2×C4×A4—22no—
31C2×C4×A4D412yes—
32C2×SL(2,ℤ/4)—21no—
33C2×SL(2,ℤ/4)—21no—
34C2×SL(2,ℤ/4)—21no—
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)Q812yes—
40C2×C3×C4⋊C4—41no—
41C2×Dic3⋊C4—41no—
42C3⋊#95—42no—
43C3⋊#95—41no—
44C3⋊#95—42no—
45C24×C4C2×D312yes—
46C24×C4C2×D311yescharacteristic
47C22×C4⋊C4—22no—
48C22×C4⋊C4—31no—
49C22×C4⋊C4—31no—
50C22×C4⋊C4—32no—
51C22×C4⋊C4—31no—
52C22×C4⋊C4—31no—
53C22×C4⋊C4—62no—
54C22×C22⋊C4—31no—
55C22×C22⋊C4—32no—
56C22×C22⋊C4—31no—
57#400—64no—
58#400—61no—
59#400—62no—
60#400—61no—
61#385—62no—
62#385—62no—
63C2×#95—32no—
64C2×#95—32no—
65C2×#95—32no—
66C2×#95—32no—
67C2×#95—31no—
68C2×#95—31no—
69C2×#95—32no—
70C2×#95—31no—
71C2×#95—31no—
72C2×#95—32no—
73C2×#95—32no—
74C2×#95—32no—
75C2×#95—32no—
76C22×A4C2×D411yescharacteristic
77C22×A4C2×Q811yescharacteristic
78C22×A4#4112yes—
79C22×A4#4112yes—
80C22×A4#4111yescharacteristic
81C22×C3×C4—41no—
82C22×C3×C4—42no—
83C22×Dic3—41no—
84C22×Dic3—42no—
85C22×Dic3—41no—
86C4×A4—22no—
87C4×A4—42no—
88SL(2,ℤ/4)—22no—
89SL(2,ℤ/4)—41no—
90SL(2,ℤ/4)—42no—
91C3×C4⋊C4—82no—
92Dic3⋊C4—82no—
93C25C22×D311yescharacteristic
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—31no—
99C23×C4—31no—
100C23×C4—32no—
101C23×C4—32no—
102C23×C4—31no—
103C23×C4—31no—
104C23×C4—32no—
105C23×C4—31no—
106C23×C4—32no—
107C23×C4—31no—
108C23×C4—32no—
109C23×C4—62no—
110C23×C4—62no—
111C23×C4—62no—
112C23×C4—62no—
113C23×C4#6712yes—
114C2×C4⋊C4—34no—
115C2×C4⋊C4—32no—
116C2×C4⋊C4—31no—
117C2×C4⋊C4—62no—
118C2×C4⋊C4—62no—
119C2×C4⋊C4—61no—
120C2×C4⋊C4—64no—
121C2×C4⋊C4—64no—
122C2×C4⋊C4—62no—
123C2×C4⋊C4—64no—
124C2×C4⋊C4—62no—
125C2×C4⋊C4—62no—
126C2×C4⋊C4—61no—
127C2×C4⋊C4—62no—
128C2×C4⋊C4—61no—
129C2×C4⋊C4S411yescharacteristic
130C2×C22⋊C4—32no—
131C2×C22⋊C4—32no—
132C2×C22⋊C4—61no—
133C2×C22⋊C4—61no—
134C2×C22⋊C4—61no—
135C2×C22⋊C4—61no—
136C2×C22⋊C4—62no—
137C2×C22⋊C4—62no—
138C2×C22⋊C4—62no—
139C2×C22⋊C4—62no—
140C2×C22⋊C4—61no—
141C2×C22⋊C4—62no—
142C2×C22⋊C4—62no—
143C2×C22⋊C4—62no—
144C2×C22⋊C4—62no—
145C2×C22⋊C4—61no—
146#95—64no—
147#95—62no—
148#95—62no—
149#95—61no—
150#95—62no—
151#95—64no—
152#95—64no—
153#95—62no—
154#95—62no—
155#95—62no—
156#95—61no—
157#95—62no—
158#95—122no—
159#95—122no—
160#95—122no—
161#95—122no—
162C23×C3—41no—
163C2×A4C22⋊Q811yescharacteristic
164C2×A4#12311yescharacteristic
165C2×A4#12411yescharacteristic
166C2×A4#12512yes—
167C2×A4#12612yes—
168C2×C3×C4—42no—
169C2×C3×C4—82no—
170C2×C3×C4—82no—
171C2×C3×C4—82no—
172C2×C3×C4—82no—
173C2×Dic3—42no—
174C2×Dic3—81no—
175C2×Dic3—81no—
176C2×Dic3—81no—
177C2×Dic3—82no—
178C2×Dic3—82no—
179C2×Dic3—82no—
180C2×Dic3—82no—
181C24—31no—
182C24—31no—
183C24—31no—
184C24—31no—
185C24—32no—
186C24—32no—
187C24#23512yes—
188C24#23712yes—
189C24D3×Q811yescharacteristic
190C24#23911yescharacteristic
191C24C2×#6711yescharacteristic
192C22×C4—22no—
193C22×C4—32no—
194C22×C4—34no—
195C22×C4—31no—
196C22×C4—32no—
197C22×C4—32no—
198C22×C4—34no—
199C22×C4—32no—
200C22×C4—32no—
201C22×C4—32no—
202C22×C4—42no—
203C22×C4—62no—
204C22×C4—62no—
205C22×C4—62no—
206C22×C4—62no—
207C22×C4—62no—
208C22×C4—62no—
209C22×C4—62no—
210C22×C4—62no—
211C22×C4—61no—
212C22×C4—62no—
213C22×C4—64no—
214C22×C4—64no—
215C22×C4—62no—
216C22×C4—62no—
217C22×C4—62no—
218C22×C4—62no—
219C22×C4—62no—
220C22×C4—61no—
221C22×C4—61no—
222C22×C4—61no—
223C22×C4—62no—
224C22×C4—61no—
225C22×C4—61no—
226C22×C4—62no—
227C22×C4—61no—
228C22×C4—61no—
229C22×C4—61no—
230C22×C4—62no—
231C22×C4—62no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4—62no—
235C22×C4—62no—
236C22×C4—62no—
237C22×C4—62no—
238C22×C4—62no—
239C22×C4—61no—
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—61no—
251C22×C4—61no—
252C22×C4—62no—
253C22×C4—62no—
254C22×C4—61no—
255C22×C4—124no—
256C22×C4—122no—
257C22×C4—122no—
258C22×C4—122no—
259C22×C4C2×S412yes—
260C22×C4C2×S411yescharacteristic
261C4⋊C4—22no—
262C4⋊C4—64no—
263C4⋊C4—64no—
264C4⋊C4—62no—
265C4⋊C4—124no—
266C4⋊C4—122no—
267C4⋊C4—122no—
268C22⋊C4—62no—
269C22⋊C4—62no—
270C22⋊C4—62no—
271C22⋊C4—62no—
272C22⋊C4—122no—
273C22⋊C4—121no—
274C22⋊C4—122no—
275C22⋊C4—122no—
276C22⋊C4—122no—
277C22⋊C4—122no—
278C22⋊C4—121no—
279C22×C3—42no—
280C22×C3—42no—
281C22×C3—41no—
282C22×C3—41no—
283C22×C3—41no—
284A4#40011yescharacteristic, complemented
285C3×C4—82no—
286C3×C4—162no—
287Dic3—82no—
288Dic3—162no—
289Dic3—161no—
290C23—32no—
291C23—32no—
292C23—32no—
293C23—32no—
294C23—31no—
295C23—31no—
296C23—31no—
297C23—31no—
298C23—31no—
299C23—31no—
300C23—31no—
301C23—31no—
302C23—31no—
303C23—32no—
304C23—31no—
305C23—32no—
306C23—32no—
307C23—32no—
308C23—32no—
309C23—31no—
310C23—32no—
311C23—32no—
312C23—32no—
313C23—61no—
314C23—61no—
315C23—61no—
316C23—62no—
317C23—62no—
318C23#88912yes—
319C23#89812yes—
320C23#92911yescharacteristic
321C23#93611yescharacteristic
322C23#94611yescharacteristic
323C23C22×S411yescharacteristic
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—34no—
329C2×C4—32no—
330C2×C4—62no—
331C2×C4—62no—
332C2×C4—64no—
333C2×C4—64no—
334C2×C4—62no—
335C2×C4—62no—
336C2×C4—62no—
337C2×C4—62no—
338C2×C4—64no—
339C2×C4—62no—
340C2×C4—64no—
341C2×C4—64no—
342C2×C4—62no—
343C2×C4—62no—
344C2×C4—62no—
345C2×C4—62no—
346C2×C4—122no—
347C2×C4—122no—
348C2×C4—122no—
349C2×C4—122no—
350C2×C4—121no—
351C2×C4—122no—
352C2×C4—121no—
353C2×C4—121no—
354C2×C4—121no—
355C2×C4—121no—
356C2×C4—121no—
357C2×C4—122no—
358C2×C4—122no—
359C2×C4—122no—
360C2×C4—121no—
361C2×C4—122no—
362C2×C4—121no—
363C2×C4—121no—
364C2×C4—122no—
365C2×C4—121no—
366C2×C4—122no—
367C2×C4—122no—
368C2×C4—122no—
369C2×C4—122no—
370C2×C4—122no—
371C2×C4—122no—
372C2×C4—122no—
373C2×C4—121no—
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—121no—
383C2×C4GL(2,ℤ/4)12yes—
384C2×C3—41no—
385C2×C3—42no—
386C2×C3—42no—
387C2×C3—41no—
388C2×C3—41no—
389C22—32no—
390C22—32no—
391C22—31no—
392C22—31no—
393C22—31no—
394C22—31no—
395C22—31no—
396C22—31no—
397C22—31no—
398C22—32no—
399C22—32no—
400C22—32no—
401C22—32no—
402C22—32no—
403C22—32no—
404C22—32no—
405C22—32no—
406C22—31no—
407C22—31no—
408C22—32no—
409C22—32no—
410C22—31no—
411C22—31no—
412C22—61no—
413C22—61no—
414C22—61no—
415C22—62no—
416C22—62no—
417C22#498511yescharacteristic, complemented
418C22#591212yes—
419C22#591412yes—
420C22Q8×S411yescharacteristic
421C22#591911yescharacteristic
422C22C2×GL(2,ℤ/4)11yescharacteristic
423C4—22no—
424C4—42no—
425C4—62no—
426C4—122no—
427C4—122no—
428C4—122no—
429C4—242no—
430C4—241no—
431C4—242no—
432C4—241no—
433C3—41noSylow
434C2—31no—
435C2—32no—
436C2—31no—
437C2—31no—
438C2—32no—
439C2—31no—
440C2#8530412yes—
441C2#8531312yes—
442C2#8548511yescharacteristic
443C2#8549211yescharacteristic
444C2#8550611yescharacteristic
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
284. A4#400semidirect
417. C22#4985semidirect

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×SL(2,ℤ/4)⋊C42
A4⋊#952
#148433
#49854