Finite Group Databasebeta

← #11714033

Subgroups — #11714033

Order 768.

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

normal subgroups only

This group has 636 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
1C2×A4×C4⋊C4C211yesmaximal, characteristic, complemented
2#85486C211yesmaximal, characteristic
3#73141C211yesmaximal, characteristic, complemented
4#73053C212yesmaximal, complemented
5#73051C212yesmaximal, complemented
6#32979—31nomaximal, Sylow
7C22×C4×A4—21no—
8C22×C4×A4C2211yescharacteristic
9C2×GL(2,ℤ/4)—21no—
10C2×C4⋊S4C2211yescharacteristic
11A4×C4⋊C4—21no—
12A4×C4⋊C4C2212yes—
13(C2×S4)⋊C4—21no—
14C4⋊SL(2,ℤ/4)C2211yescharacteristic
15C2×A4⋊C8C2212yes—
16#4968—41nomaximal
17C23×C4⋊C4D311yescharacteristic, complemented
18C2×#481—31no—
19C2×#481—31no—
20#2847—31no—
21#1874—32no—
22#1874—32no—
23#1853—32no—
24#1853—32no—
25#1824—31no—
26#1824—31no—
27#1779—31no—
28C23×A4D411yescharacteristic
29C22×S4—21no—
30C2×C4×A4—22no—
31C2×C4×A4—22no—
32C2×C4×A4—42no—
33C2×C4×A4D411yescharacteristic
34C2×C4×A4C2311yescharacteristic
35GL(2,ℤ/4)—42no—
36C2×SL(2,ℤ/4)—21no—
37C4⋊S4—22no—
38C2×C3×C4⋊C4—41no—
39#930—41no—
40A4⋊C8—22no—
41#830—41no—
42#809—42no—
43#808—42no—
44C24×C4—21no—
45C24×C4C2×D311yescharacteristic
46C2×#121—31no—
47C2×#121—31no—
48C2×#121—31no—
49C2×#120—61no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—31no—
52C22×C4⋊C4—31no—
53C22×C4⋊C4—32no—
54C22×C4⋊C4—61no—
55C22×C4⋊C4—61no—
56C22×C4⋊C4C2×D312yes—
57#481—62no—
58#481—62no—
59#481—62no—
60#481—61no—
61#481—61no—
62C2×#106—32no—
63C2×#106—32no—
64C2×#102—32no—
65C2×#102—32no—
66C2×C22⋊C8—31no—
67C2×C22⋊C8—31no—
68C2×C22⋊C8—32no—
69C2×C22⋊C8—32no—
70#386—61no—
71#386—61no—
72#386—61no—
73#380—31no—
74#380—31no—
75#380—31no—
76#340—32no—
77#340—32no—
78#340—32no—
79#340—32no—
80C22×A4—21no—
81C22×A4—21no—
82C22×A4SD1612yes—
83C22×A4C2×D411yescharacteristic
84C2×S4—41no—
85C2×S4—41no—
86C2×S4—41no—
87C22×C3×C4—41no—
88C22×C3×C4—81no—
89C2×#67—81no—
90C2×D12—41no—
91C4×A4—21no—
92C4×A4—42no—
93C4×A4#4112yes—
94SL(2,ℤ/4)—41no—
95C3×C4⋊C4—42no—
96C3×C4⋊C4—81no—
97(C2×D3)⋊C4—81no—
98C4⋊Dic3—41no—
99C2×C3⋊C8—42no—
100C25#6711yescharacteristic
101C22×D4—31no—
102C22×D4—61no—
103C22×D4—61no—
104C22×D4—61no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—31no—
108C23×C4—31no—
109C23×C4—31no—
110C23×C4—31no—
111C23×C4—42no—
112C23×C4—61no—
113C23×C4—62no—
114C23×C4—62no—
115C23×C4—61no—
116C23×C4—122no—
117C23×C4#6711yescharacteristic
118C23×C4C22×D311yescharacteristic
119C22×C8—32no—
120C22×C8—32no—
121#121—62no—
122#121—64no—
123#121—61no—
124#121—61no—
125#121—61no—
126#121—62no—
127#121—61no—
128#120—122no—
129#120—122no—
130C2×C4⋊C4—32no—
131C2×C4⋊C4—32no—
132C2×C4⋊C4—32no—
133C2×C4⋊C4—31no—
134C2×C4⋊C4—32no—
135C2×C4⋊C4—32no—
136C2×C4⋊C4—31no—
137C2×C4⋊C4—31no—
138C2×C4⋊C4—31no—
139C2×C4⋊C4—61no—
140C2×C4⋊C4—61no—
141C2×C4⋊C4—61no—
142C2×C4⋊C4—62no—
143C2×C4⋊C4—64no—
144C2×C4⋊C4—61no—
145C2×C4⋊C4—62no—
146C2×C4⋊C4—61no—
147C2×C4⋊C4S411yescharacteristic, complemented
148C2×C22⋊C4—61no—
149C2×C22⋊C4—61no—
150C2×C22⋊C4—61no—
151C2×C22⋊C4—61no—
152C2×C22⋊C4—61no—
153C2×C22⋊C4—61no—
154#106—64no—
155#106—62no—
156#106—62no—
157#102—64no—
158#102—62no—
159#102—62no—
160C22⋊C8—62no—
161C22⋊C8—64no—
162C22⋊C8—61no—
163C22⋊C8—61no—
164C22⋊C8—61no—
165C22⋊C8—62no—
166C22⋊C8—61no—
167#95—61no—
168#95—61no—
169#95—61no—
170C23×C3—41no—
171C22×D3—81no—
172C2×A4—22no—
173C2×A4#12311yescharacteristic
174C2×A4C2×SD1611yescharacteristic
175C2×A4AGL(1,ℤ/8)11yescharacteristic
176S4—81no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—82no—
180C2×C3×C4—82no—
181C2×C3×C4—162no—
182#67—162no—
183C2×Dic3—81no—
184D12—82no—
185C3⋊C8—82no—
186C24—21no—
187C24—21no—
188C24—31no—
189C24—31no—
190C24—32no—
191C24—61no—
192C24—61no—
193C24—61no—
194C24#21512yes—
195C24C2×#6711yescharacteristic
196C2×D4—61no—
197C2×D4—62no—
198C2×D4—62no—
199C2×D4—61no—
200C2×D4—122no—
201C2×D4—121no—
202C2×D4—122no—
203C2×D4—121no—
204C2×D4—122no—
205C2×D4—121no—
206C2×D4—121no—
207C2×D4—122no—
208C2×D4—122no—
209C2×D4—121no—
210C2×D4—121no—
211C2×D4—121no—
212C2×D4—121no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—31no—
216C22×C4—31no—
217C22×C4—31no—
218C22×C4—32no—
219C22×C4—34no—
220C22×C4—31no—
221C22×C4—31no—
222C22×C4—31no—
223C22×C4—31no—
224C22×C4—42no—
225C22×C4—62no—
226C22×C4—62no—
227C22×C4—62no—
228C22×C4—62no—
229C22×C4—61no—
230C22×C4—61no—
231C22×C4—62no—
232C22×C4—62no—
233C22×C4—62no—
234C22×C4—62no—
235C22×C4—62no—
236C22×C4—62no—
237C22×C4—61no—
238C22×C4—62no—
239C22×C4—61no—
240C22×C4—61no—
241C22×C4—61no—
242C22×C4—122no—
243C22×C4—122no—
244C22×C4—124no—
245C22×C4—124no—
246C22×C4—122no—
247C22×C4—122no—
248C22×C4—122no—
249C22×C4#23512yes—
250C22×C4C2×S411yescharacteristic
251C2×C8—62no—
252C2×C8—62no—
253C2×C8—62no—
254C2×C8—62no—
255C2×C8—62no—
256C2×C8—62no—
257C4⋊C4—21no—
258C4⋊C4—34no—
259C4⋊C4—32no—
260C4⋊C4—62no—
261C4⋊C4—62no—
262C4⋊C4—62no—
263C4⋊C4—61no—
264C4⋊C4—62no—
265C4⋊C4—62no—
266C4⋊C4—61no—
267C4⋊C4—61no—
268C4⋊C4C2×S412yes—
269C22⋊C4—122no—
270C22⋊C4—122no—
271C22⋊C4—121no—
272C22⋊C4—122no—
273C22⋊C4—121no—
274C22⋊C4—122no—
275C22⋊C4—121no—
276C22⋊C4—121no—
277C22×C3—42no—
278C22×C3—41no—
279C22×C3—81no—
280C22×C3—81no—
281C2×D3—161no—
282C2×D3—161no—
283C2×D3—161no—
284A4#48111yescharacteristic, complemented
285C3×C4—42no—
286C3×C4—81no—
287C3×C4—162no—
288Dic3—161no—
289C23—22no—
290C23—31no—
291C23—32no—
292C23—31no—
293C23—31no—
294C23—31no—
295C23—32no—
296C23—32no—
297C23—32no—
298C23—32no—
299C23—31no—
300C23—61no—
301C23—61no—
302C23—61no—
303C23—61no—
304C23—62no—
305C23—61no—
306C23—61no—
307C23—61no—
308C23—62no—
309C23—62no—
310C23—64no—
311C23—121no—
312C23—121no—
313C23—121no—
314C23—121no—
315C23—121no—
316C23—121no—
317C23—121no—
318C23#92911yescharacteristic
319C23#93211yescharacteristic
320C23C2×#21511yescharacteristic
321C23GL(2,ℤ/4)11yescharacteristic
322D4—122no—
323D4—122no—
324D4—241no—
325D4—241no—
326D4—241no—
327D4—244no—
328D4—242no—
329D4—242no—
330D4—241no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—31no—
334C2×C4—32no—
335C2×C4—32no—
336C2×C4—32no—
337C2×C4—32no—
338C2×C4—31no—
339C2×C4—42no—
340C2×C4—62no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—62no—
345C2×C4—64no—
346C2×C4—62no—
347C2×C4—61no—
348C2×C4—62no—
349C2×C4—122no—
350C2×C4—122no—
351C2×C4—124no—
352C2×C4—122no—
353C2×C4—124no—
354C2×C4—122no—
355C2×C4—122no—
356C2×C4—121no—
357C2×C4—121no—
358C2×C4—121no—
359C2×C4—121no—
360C2×C4—121no—
361C2×C4—121no—
362C2×C4—121no—
363C2×C4—121no—
364C2×C4—121no—
365C2×C4GL(2,ℤ/4)11yescharacteristic
366C2×C4C22×S411yescharacteristic
367C8—122no—
368C8—122no—
369C2×C3—41no—
370C2×C3—41no—
371C2×C3—41no—
372C2×C3—82no—
373D3—321no—
374C22—21no—
375C22—21no—
376C22—31no—
377C22—31no—
378C22—31no—
379C22—31no—
380C22—31no—
381C22—31no—
382C22—31no—
383C22—31no—
384C22—31no—
385C22—34no—
386C22—32no—
387C22—62no—
388C22—62no—
389C22—62no—
390C22—62no—
391C22—61no—
392C22—62no—
393C22—62no—
394C22—62no—
395C22—61no—
396C22—61no—
397C22—241no—
398C22—241no—
399C22—241no—
400C22—241no—
401C22—241no—
402C22—241no—
403C22—241no—
404C22#496811yescharacteristic, complemented
405C22#541712yes—
406C22C2×GL(2,ℤ/4)11yescharacteristic
407C4—21no—
408C4—32no—
409C4—42no—
410C4—61no—
411C4—122no—
412C4—241no—
413C4—241no—
414C4—241no—
415C4#591212yes—
416C3—41noSylow
417C2—22no—
418C2—31no—
419C2—31no—
420C2—31no—
421C2—31no—
422C2—62no—
423C2—481no—
424C2#8548511yescharacteristic
425C2#8548911yescharacteristic
426C2C2×#541711yescharacteristic
subgroupG/H
centerC22C2×GL(2,ℤ/4)
commutator, cosocleC2×C4×A4C23
FrattiniC2×C4C22×S4
FittingC23×C4⋊C4D3
socleC24C2×#67
prime radical#85486C2
hypercenterC2×C4⋊C4S4
hypocenterA4#481
LCSG ▷ C2×C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×A4×C4⋊C4C2semidirect
3. #73141C2semidirect
4. #73053C2semidirect
5. #73051C2semidirect
17. C23×C4⋊C4D3semidirect
147. C2×C4⋊C4S4semidirect
284. A4#481semidirect
404. C22#4968semidirect

Maximal subgroups

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

subgroupindex
C2×A4×C4⋊C42
#854862
#731412
#730532
#730512
#329793
#49684