Finite Group Databasebeta

← #72253045

Subgroups — #72253045

Order 2304.

1138 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 1138 conjugacy classes of subgroups — too many to draw. Only its 91 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×(#41⋊C3)⋊A4C211yesmaximal, characteristic
2C4×Ω+(4,𝔽3)C212yesmaximal, complemented
3#11710605C312yesmaximal, complemented
4#11710604C312yesmaximal, complemented
5(#41⋊C3)⋊A4C412yescomplemented
6C2×Ω+(4,𝔽3)C2211yescharacteristic
7C3×#520⋊C3—42nomaximal
8C2×#5950C2×C312yes—
9C2×#5945C2×C312yes—
10C4×#997C2×C314yescomplemented
11C4×2−1+4⋊C3—34no—
12C4×#994C2×C314yescomplemented
13#3428⋊C3—32no—
14C2×#520⋊C3—32no—
15C2×C3×#41⋊C3—42no—
16Ω+(4,𝔽3)C2×C411yescharacteristic, complemented
17C3×C4×SL(2,𝔽3)—44no—
18#62654C3211yescharacteristic, complemented, Sylow
19#5950C3×C414yescomplemented
20C2×#997C22×C312yes—
21#5945—34no—
22#5945C3×C414yescomplemented
23C2×2−1+4⋊C3—32no—
24C2×#994C22×C312yes—
25C22×#41⋊C3—32no—
26C3×#520—42no—
27#520⋊C3—32no—
28#520⋊C3—34no—
29#520⋊C3—44no—
30#520⋊C3A412yescomplemented
31#519⋊C3—34no—
32C4×#41⋊C3—34no—
33C2×C4×SL(2,𝔽3)—34no—
34A4×C4⋊C4—122no—
35C3×#41⋊C3—44no—
36C2×C3×SL(2,𝔽3)—42no—
37C32×C4⋊C4—161no—
38C2×#585C2×C3211yescharacteristic
39C4×2−1+4—34no—
40C4×H(5,𝔽2)—92no—
41C4×H(5,𝔽2)C2×C3212yes—
42#3428—32no—
43#3428—91no—
44C2×#520—32no—
45C2×#520—91no—
46C2×C3×#41—42no—
47#997C2×C3×C412yescomplemented
482−1+4⋊C3—34no—
492−1+4⋊C3—34no—
50#994—32no—
51#994C2×C3×C412yescomplemented
52C2×#41⋊C3—34no—
53C2×#41⋊C3—34no—
54C2×#41⋊C3—32no—
55C2×#41⋊C3—44no—
56C2×#41⋊C3C2×A412yes—
57C22×SL(2,𝔽3)—32no—
58C2×C4×A4—122no—
59C2×C4×A4—124no—
60C3×C4×Q8—44no—
61C3×C4×D4—124no—
62C3×#117—122no—
63C2×C3×C4⋊C4—122no—
64C4×SL(2,𝔽3)—34no—
65C4×SL(2,𝔽3)—48no—
66C4×SL(2,𝔽3)C2×A414yes—
67C2×C32×C4—161no—
68C2×C32×C4—162no—
69C3×SL(2,𝔽3)—42no—
70#585—34no—
71#585—92no—
72#585C32×C412yescomplemented
73C2×2−1+4—32no—
74C2×H(5,𝔽2)—91no—
75C2×H(5,𝔽2)C22×C3211yescharacteristic
76C22×#41—32no—
77C22×#41—91no—
78#520—32no—
79#520—34no—
80#520—92no—
81#520—98no—
82#520—94no—
83#520—94no—
84#520—92no—
85#520C3×A412yes—
86#519—34no—
87#519—92no—
88#518—94no—
89#518—94no—
90#518—92no—
91C4×#41—34no—
92C4×#41—98no—
93C4×#41—94no—
94C2×C4×Q8—34no—
95C2×C4×Q8—92no—
96C2×C4×D4—94no—
97C2×C4×D4—92no—
98C2×C4×D4—94no—
99C2×#117—92no—
100C2×#117—91no—
101C2×#117—92no—
102C22×C4⋊C4—32no—
103C22×C4⋊C4—91no—
104C22×A4—122no—
105C3×#41—44no—
106C3×#41—124no—
107C2×C3×Q8—42no—
108C2×C3×D4—122no—
109C22×C3×C4—122no—
110C22×C3×C4—124no—
111#41⋊C3—32no—
112#41⋊C3—32no—
113#41⋊C3—48no—
114#41⋊C3C4×A414yescomplemented
115C2×SL(2,𝔽3)—34no—
116C2×SL(2,𝔽3)—44no—
117C2×SL(2,𝔽3)C22×A412yes—
118C4×A4—124no—
119C4×A4—244no—
120C3×C4⋊C4—42no—
121C3×C4⋊C4—122no—
122C3×C4⋊C4—124no—
123C3×C4⋊C4—162no—
124C3×C22⋊C4—124no—
125C3×C42—124no—
126C22×C32—161no—
127C32×C4—162no—
128C32×C4—322no—
1292−1+4—34no—
1302−1+4—34no—
1312−1+4—94no—
1322−1+4—94no—
133H(5,𝔽2)—32no—
134H(5,𝔽2)—94no—
135H(5,𝔽2)—98no—
136H(5,𝔽2)—91no—
137H(5,𝔽2)—92no—
138H(5,𝔽2)—92no—
139H(5,𝔽2)C2×C32×C411yescharacteristic
140C2×#41—34no—
141C2×#41—34no—
142C2×#41—32no—
143C2×#41—92no—
144C2×#41—94no—
145C2×#41—92no—
146C2×#41—92no—
147C2×#41—98no—
148C2×#41—94no—
149C2×#41—92no—
150C2×#41—94no—
151C2×#41—94no—
152C2×#41C2×C3×A412yes—
153C22×Q8—32no—
154C22×Q8—91no—
155C22×D4—92no—
156C22×D4—92no—
157C22×D4—91no—
158C23×C4—32no—
159C23×C4—34no—
160C23×C4—91no—
161C23×C4—92no—
162C4×Q8—34no—
163C4×Q8—94no—
164C4×Q8—94no—
165C4×Q8—98no—
166C4×Q8C2×C3×A414yes—
167C4×D4—34no—
168C4×D4—94no—
169C4×D4—98no—
170C4×D4—94no—
171C4×D4—98no—
172C4×D4—916no—
173C4×D4—94no—
174C4×D4—98no—
175#117—32no—
176#117—94no—
177#117—92no—
178#117—94no—
179#117—98no—
180#117—94no—
181#117—94no—
182C2×C4⋊C4—32no—
183C2×C4⋊C4—91no—
184C2×C4⋊C4—94no—
185C2×C4⋊C4—91no—
186C2×C4⋊C4—92no—
187C2×C4⋊C4—92no—
188C2×C4⋊C4—94no—
189C2×C4⋊C4—92no—
190C2×C22⋊C4—92no—
191C2×C22⋊C4—94no—
192C2×C22⋊C4—94no—
193C2×C42—92no—
194C2×C42—94no—
195C2×C42—94no—
196C23×C3—122no—
197C2×A4—122no—
198C2×A4—122no—
199C2×A4—122no—
200C3×Q8—42no—
201C3×Q8—122no—
202C3×D4—124no—
203C3×D4—124no—
204C2×C3×C4—42no—
205C2×C3×C4—44no—
206C2×C3×C4—124no—
207C2×C3×C4—124no—
208C2×C3×C4—122no—
209C2×C3×C4—124no—
210C2×C3×C4—162no—
211C2×C3×C4—164no—
212C2×C3×C4—484no—
213SL(2,𝔽3)—44no—
214SL(2,𝔽3)C2×C4×A412yes—
215C2×C32—161no—
216C2×C32—161no—
217C2×C32—161no—
218C24—32no—
219C24—91no—
220#41—34no—
221#41—32no—
222#41—32no—
223#41—98no—
224#41—94no—
225#41—94no—
226#41—94no—
227#41—94no—
228#41—916no—
229#41—98no—
230#41—98no—
231#41—94no—
232#41—94no—
233#41—92no—
234#41—92no—
235#41C3×C4×A414yes—
236C2×Q8—34no—
237C2×Q8—92no—
238C2×Q8—92no—
239C2×Q8—94no—
240C2×Q8—92no—
241C2×Q8—94no—
242C2×Q8C22×C3×A412yes—
243C2×D4—32no—
244C2×D4—92no—
245C2×D4—94no—
246C2×D4—94no—
247C2×D4—98no—
248C2×D4—94no—
249C2×D4—92no—
250C2×D4—94no—
251C2×D4—98no—
252C2×D4—94no—
253C2×D4—94no—
254C2×D4—91no—
255C2×D4—91no—
256C22×C4—32no—
257C22×C4—34no—
258C22×C4—34no—
259C22×C4—91no—
260C22×C4—92no—
261C22×C4—94no—
262C22×C4—92no—
263C22×C4—92no—
264C22×C4—92no—
265C22×C4—91no—
266C22×C4—92no—
267C22×C4—94no—
268C22×C4—94no—
269C22×C4—92no—
270C22×C4—94no—
271C22×C4—244no—
272C22×C4—722no—
273C4⋊C4—34no—
274C4⋊C4—32no—
275C4⋊C4—92no—
276C4⋊C4—94no—
277C4⋊C4—94no—
278C4⋊C4—91no—
279C4⋊C4—92no—
280C4⋊C4A4211yescharacteristic
281C22⋊C4—34no—
282C22⋊C4—94no—
283C22⋊C4—98no—
284C42—34no—
285C42—94no—
286C42—98no—
287C22×C3—42no—
288C22×C3—124no—
289C22×C3—162no—
290C22×C3—242no—
291C22×C3—242no—
292A4—122no—
293C3×C4—44no—
294C3×C4—84no—
295C3×C4—122no—
296C3×C4—122no—
297C3×C4—164no—
298C3×C4—244no—
299C3×C4—324no—
300C32—161noSylow
301C23—32no—
302C23—32no—
303C23—92no—
304C23—92no—
305C23—91no—
306C23—92no—
307C23—122no—
308C23—122no—
309C23—361no—
310C23—361no—
311Q8—32no—
312Q8—94no—
313Q8—94no—
314Q8C2×C3×C4×A412yes—
315D4—34no—
316D4—34no—
317D4—92no—
318D4—92no—
319D4—94no—
320D4—98no—
321D4—98no—
322C2×C4—32no—
323C2×C4—34no—
324C2×C4—34no—
325C2×C4—34no—
326C2×C4—91no—
327C2×C4—92no—
328C2×C4—91no—
329C2×C4—94no—
330C2×C4—94no—
331C2×C4—94no—
332C2×C4—91no—
333C2×C4—92no—
334C2×C4—92no—
335C2×C4—94no—
336C2×C4—92no—
337C2×C4—124no—
338C2×C4—362no—
339C2×C4—362no—
340C2×C4—364no—
341C2×C4—364no—
342C2×C4C2×A4212yes—
343C2×C4C2×A4211yescharacteristic
344C2×C3—42no—
345C2×C3—42no—
346C2×C3—42no—
347C2×C3—162no—
348C2×C3—162no—
349C2×C3—162no—
350C2×C3—244no—
351C22—34no—
352C22—62no—
353C22—62no—
354C22—91no—
355C22—91no—
356C22—122no—
357C22—181no—
358C22—181no—
359C22—362no—
360C22—362no—
361C22—362no—
362C22C22×A4211yescharacteristic
363C4—22no—
364C4—32no—
365C4—32no—
366C4—64no—
367C4—92no—
368C4—182no—
369C4C4×A4212yes—
370C3—42no—
371C3—162no—
372C2—64no—
373C2—181no—
374C2—181no—
375C2C2×C4×A4211yescharacteristic
376C2(2−1+4⋊C3)⋊A411yescharacteristic
377C2#1192367311yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×A42
commutator, hypocenterH(5,𝔽2)C2×C32×C4
Fitting#62654C32
prime radicalC2×(#41⋊C3)⋊A4C2
cosocleC2×H(5,𝔽2)C22×C32
hypercenterC4⋊C4A42
LCSG ▷ H(5,𝔽2)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C4×Ω+(4,𝔽3)C2semidirect
3. #11710605C3semidirect
4. #11710604C3semidirect
5. (#41⋊C3)⋊A4C4semidirect
10. C4×#997C2×C3semidirect
12. C4×#994C2×C3semidirect
16. Ω+(4,𝔽3)C2×C4semidirect
18. #62654C32semidirect
19. #5950C3×C4semidirect
22. #5945C3×C4semidirect
30. #520⋊C3A4semidirect
47. #997C2×C3×C4semidirect
51. #994C2×C3×C4semidirect
72. #585C32×C4semidirect
114. #41⋊C3C4×A4semidirect

Maximal subgroups

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

subgroupindex
C2×(#41⋊C3)⋊A42
C4×Ω+(4,𝔽3)2
#117106053
#117106043
C3×#520⋊C34