Finite Group Databasebeta

← #82536⋊C3 #11921277

Subgroups — #82536⋊C3 #11921277

Order 1152.

399 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 399 conjugacy classes of subgroups — too many to draw. Only its 53 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
1#5964⋊C3C211yesmaximal, characteristic
2C3×C4×#41⋊C3C211yesmaximal, characteristic, complemented
3#5819⋊C3C211yesmaximal, characteristic, complemented
4#82536C311yesmaximal, characteristic, complemented
5#2901⋊C3—31nomaximal
6C2×C3×#41⋊C3C2211yescharacteristic
7#1010⋊C3C411yescharacteristic, complemented
8D3×#41⋊C3—21no—
9C3⋊#994C411yescharacteristic, complemented
10#1007⋊C3—21no—
11C3×C4×SL(2,𝔽3)—21no—
12C3⋊(#131⋊C3)—21no—
13C3×#805—41nomaximal
14#5964C2×C311yescharacteristic
15#5945—31no—
16C3×C4×#41C2×C311yescharacteristic, complemented
17#5819C2×C311yescharacteristic, complemented
18#568⋊C3—31no—
19C4×#41⋊C3—21no—
20C4×#41⋊C3D311yescharacteristic, complemented
21#5005—31no—
22#5005—31no—
23C2×#805—31no—
24C2×#805—31no—
25C3×#235—41no—
26C3×#41⋊C3C2×C411yescharacteristic
27C3×#41⋊C3D411yescharacteristic
28C2×C3×SL(2,𝔽3)D411yescharacteristic
29D3×SL(2,𝔽3)—21no—
30C3⋊(#41⋊C3)—21no—
31C32×C42—41no—
32C3×#208—41no—
33#2901—31noSylow
34C2×C3×#41C22×C311yescharacteristic
35#1010—31no—
36#1010C3×C411yescharacteristic, complemented
37#1009—31no—
38#1009C3×C411yescharacteristic, complemented
39D3×#41—21no—
40D3×#41—61no—
41#1007—21no—
42#1004—61no—
43C2×#235—31no—
44C2×#235—31no—
452−1+4⋊C3—31no—
462−1+4⋊C3—61no—
47#994—31no—
48C2×#41⋊C3—21no—
49C2×#41⋊C3—61no—
50C2×#41⋊C3C2×D311yescharacteristic
51C3×C4×Q8—21no—
52C3×C4×D4—61no—
53C3×#117—31no—
54C2×C3×C42—31no—
55#948—21no—
56#948—61no—
57C2×#208—31no—
58C2×#208—31no—
59#131⋊C3—61no—
60C4×SL(2,𝔽3)—22no—
61C4×SL(2,𝔽3)—21no—
62C3×#104—121no—
63#805—32no—
64#805—31no—
65#805—31no—
66#805—31no—
67#805A411yescharacteristic
68C2×C32×C4—41no—
69C2×C32×C4—81no—
70C3×#67—81no—
71C3×D12—41no—
72C3×C4×D3—81no—
73C3×Dic6—41no—
74C3×SL(2,𝔽3)C22⋊C411yescharacteristic
75C3×C3⋊C8—81no—
76#585—31no—
77#568—31no—
78C4×#41C3×D311yescharacteristic, complemented
79#421—91no—
80#421—91no—
81C2×#104—91no—
82C2×#104—91no—
83C3×#41—31no—
84C3×#41—31no—
85C3×#41—121no—
86C3×#41C2×C3×C411yescharacteristic
87C3×#41C3×D411yescharacteristic
88C2×C3×Q8C3×D411yescharacteristic
89C2×C3×D4—31no—
90C22×C3×C4—31no—
91C22×C3×C4—61no—
92C2×#67—61no—
93C2×#67—61no—
94#239—21no—
95#239—61no—
96D3×Q8—21no—
97D3×Q8—61no—
98#237—62no—
99#237—61no—
100#237—61no—
101D3×D4—61no—
102D3×D4—62no—
103D3×D4—61no—
104#235—32no—
105#235—31no—
106#235—31no—
107#235—31no—
108#235—61no—
109#235—61no—
110#235—61no—
111#235—61no—
112#235C2×A411yescharacteristic
113C2×D12—31no—
114C2×D12—31no—
115C2×C4×D3—61no—
116C2×C4×D3—61no—
117C2×Dic6—31no—
118C2×Dic6—31no—
119#41⋊C3—21no—
120#41⋊C3—21no—
121#41⋊C3—61no—
122#41⋊C3C4×D311yescharacteristic
123#41⋊C3D1211yescharacteristic
124C2×SL(2,𝔽3)—21no—
125C2×SL(2,𝔽3)—61no—
126C2×SL(2,𝔽3)#6711yescharacteristic
127C3×M16—121no—
128C3×C4⋊C4—61no—
129C3×C22⋊C4—61no—
130C3×C42—31no—
131C3×C42—41no—
132C3×C42—61no—
133C3×C42—81no—
134C3×C42C2×A411yescharacteristic
135#208—31no—
136#208—61no—
137#208—61no—
138#208C2×A411yescharacteristic
139C2×C3⋊C8—61no—
140C2×C3⋊C8—61no—
141C22×C32—41no—
142C2×C3×D3—81no—
143C32×C4—41no—
144C32×C4—41no—
145C32×C4—82no—
146C3×Dic3—81no—
1472−1+4—31no—
1482−1+4—61no—
1492−1+4—91no—
150H(5,𝔽2)—31no—
151H(5,𝔽2)—91no—
152H(5,𝔽2)—181no—
153C2×#41—61no—
154C2×#41—91no—
155C2×#41—91no—
156C2×#41—181no—
157C2×#41C2×C3×D311yescharacteristic
158#131—61no—
159#131—181no—
160C2×M16—91no—
161C2×M16—91no—
162C4×Q8—21no—
163C4×D4—61no—
164#117—31no—
165C2×C42—31no—
166#104—31no—
167#104—91no—
168#104—91no—
169#104—92no—
170#104—91no—
171C23×C3—31no—
172C22×D3—61no—
173C22×D3—61no—
174C3×Q8—31no—
175C3×Q8—121no—
176C3×Q8C3×C22⋊C411yescharacteristic
177C3×D4—31no—
178C3×D4—31no—
179C3×D4—32no—
180C3×D4—121no—
181C3×D4—241no—
182C2×C3×C4—21no—
183C2×C3×C4—31no—
184C2×C3×C4—31no—
185C2×C3×C4—31no—
186C2×C3×C4—31no—
187C2×C3×C4—31no—
188C2×C3×C4—41no—
189C2×C3×C4—61no—
190C2×C3×C4—81no—
191C2×C3×C4—82no—
192C2×C3×C4—81no—
193C2×C3×C4—122no—
194C2×C3×C4—241no—
195C2×C3×C4C22×A411yescharacteristic
196#67—21no—
197#67—61no—
198#67—61no—
199#67—62no—
200#67—61no—
201C2×Dic3—61no—
202C2×Dic3—61no—
203D12—31no—
204D12—61no—
205D12—61no—
206D12C4×A411yescharacteristic
207C4×D3—21no—
208C4×D3—61no—
209C4×D3—61no—
210C4×D3—62no—
211C4×D3—61no—
212Dic6—31no—
213Dic6—61no—
214Dic6—61no—
215Dic6C4×A411yescharacteristic
216SL(2,𝔽3)—21no—
217SL(2,𝔽3)(C2×D3)⋊C411yescharacteristic
218C3×C8—241no—
219C3⋊C8—21no—
220C3⋊C8—61no—
221C2×C32—41no—
222C2×C32—81no—
223C3×D3—161no—
224#41—31no—
225#41—31no—
226#41—31no—
227#41—61no—
228#41—91no—
229#41—91no—
230#41—92no—
231#41—91no—
232#41—181no—
233#41—181no—
234#41—181no—
235#41—181no—
236#41—182no—
237#41—181no—
238#41—181no—
239#41—181no—
240#41C3×C4×D311yescharacteristic
241#41C3×D1211yescharacteristic
242C2×Q8—61no—
243C2×Q8—91no—
244C2×Q8—91no—
245C2×Q8—181no—
246C2×Q8C3×#6711yescharacteristic
247C2×D4—31no—
248C2×D4—91no—
249C2×D4—91no—
250C2×D4—181no—
251C2×D4—182no—
252C2×D4—181no—
253C2×D4—181no—
254C2×D4—181no—
255C22×C4—31no—
256C22×C4—61no—
257C22×C4—181no—
258C22×C4—181no—
259M16—31no—
260M16—91no—
261M16—181no—
262M16—181no—
263C2×C8—181no—
264C2×C8—181no—
265C4⋊C4—61no—
266C22⋊C4—61no—
267C42—31no—
268C42—61no—
269C42D3×A411yescharacteristic
270C22×C3—31no—
271C22×C3—31no—
272C22×C3—41no—
273C22×C3—81no—
274C22×C3—121no—
275C22×C3—241no—
276C22×C3D4×A411yescharacteristic
277C2×D3—21no—
278C2×D3—61no—
279C2×D3—241no—
280C2×D3—241no—
281C3×C4—22no—
282C3×C4—31no—
283C3×C4—31no—
284C3×C4—41no—
285C3×C4—41no—
286C3×C4—82no—
287C3×C4—84no—
288C3×C4—81no—
289C3×C4—81no—
290C3×C4—121no—
291C3×C4—241no—
292C3×C4C2×C4×A411yescharacteristic
293C3×C4D4×A411yescharacteristic
294Dic3—21no—
295Dic3—61no—
296C32—41noSylow
297C23—31no—
298C23—181no—
299C23—181no—
300Q8—31no—
301Q8—31no—
302Q8—91no—
303Q8—181no—
304Q8—181no—
305Q8C3×(C2×D3)⋊C411yescharacteristic
306D4—31no—
307D4—31no—
308D4—32no—
309D4—31no—
310D4—61no—
311D4—91no—
312D4—181no—
313D4—181no—
314D4—181no—
315D4—181no—
316D4—182no—
317D4—181no—
318C2×C4—21no—
319C2×C4—31no—
320C2×C4—31no—
321C2×C4—31no—
322C2×C4—31no—
323C2×C4—31no—
324C2×C4—61no—
325C2×C4—61no—
326C2×C4—122no—
327C2×C4—181no—
328C2×C4—181no—
329C2×C4—182no—
330C2×C4—181no—
331C2×C4—181no—
332C2×C4—181no—
333C2×C4C2×D3×A411yescharacteristic
334C8—61no—
335C8—181no—
336C2×C3—21no—
337C2×C3—41no—
338C2×C3—61no—
339C2×C3—61no—
340C2×C3—81no—
341C2×C3—82no—
342C2×C3—81no—
343C2×C3—481no—
344C2×C3A4×C22⋊C411yescharacteristic
345D3—41no—
346D3—121no—
347C22—31no—
348C22—31no—
349C22—61no—
350C22—121no—
351C22—181no—
352C22—721no—
353C22—721no—
354C22A4×#6711yescharacteristic
355C4—22no—
356C4—31no—
357C4—31no—
358C4—61no—
359C4—121no—
360C4—181no—
361C4C4×D3×A411yescharacteristic
362C4A4×D1211yescharacteristic
363C3—41no—
364C3—81no—
365C3#2901⋊C311yescharacteristic, complemented
366C2—21no—
367C2—61no—
368C2—61no—
369C2—121no—
370C2—361no—
371C2A4×(C2×D3)⋊C411yescharacteristic
subgroupG/H
centerC4A4×D12
commutatorC3×#41C2×C3×C4
FrattiniC2×C4C2×D3×A4
FittingC3×C4×#41C2×C3
socleC2×C3A4×C22⋊C4
prime radical#5964⋊C3C2
cosocleC2×C3×#41C22×C3
hypercenterC42D3×A4
hypocenterC3×Q8C3×C22⋊C4
LCSG ▷ C3×#41 ▷ C3×Q8
UCSC1 ◁ C4 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C3×C4×#41⋊C3C2semidirect
3. #5819⋊C3C2semidirect
4. #82536C3semidirect
7. #1010⋊C3C4semidirect
9. C3⋊#994C4semidirect
16. C3×C4×#41C2×C3semidirect
17. #5819C2×C3semidirect
20. C4×#41⋊C3D3semidirect
36. #1010C3×C4semidirect
38. #1009C3×C4semidirect
78. C4×#41C3×D3semidirect
365. C3#2901⋊C3semidirect

Maximal subgroups

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

subgroupindex
#5964⋊C32
C3×C4×#41⋊C32
#5819⋊C32
#825363
#2901⋊C33
C3×#8054