Finite Group Databasebeta

← #82457⋊C3 #11921265

Subgroups — #82457⋊C3 #11921265

Order 1152.

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

normal subgroups only

This group has 509 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
1C22×C3⋊(#41⋊C3)C211yesmaximal, characteristic
2C2×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C2×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#82457C311yesmaximal, characteristic, complemented
5#2341⋊C3—31nomaximal
6C22×C3×SL(2,𝔽3)C2211yescharacteristic
7C2×C3⋊(#41⋊C3)—21no—
8C2×C3⋊(#41⋊C3)—21no—
9C2×C3⋊(#41⋊C3)C412yescomplemented
10C3×C4×SL(2,𝔽3)—21no—
11Dic3×SL(2,𝔽3)—21no—
12C3×C3⋊#95—41nomaximal
13C22×#239C2×C311yescharacteristic
14C22×#41⋊C3—31no—
15C2×C3×C4×Q8C2×C311yescharacteristic
16C2×Q8×Dic3C2×C311yescharacteristic
17C2×C4×SL(2,𝔽3)—21no—
18C2×C4×SL(2,𝔽3)—31no—
19C2×C4×SL(2,𝔽3)D311yescharacteristic, complemented
20#4988—31no—
21#4987—31no—
22#4939—31no—
23C4×(C2×D3)⋊C4—31no—
24C22×C32×C4—41no—
25C22×C3×Dic3—41no—
26C22×C3×Dic3—41no—
27C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
28C2×C3×SL(2,𝔽3)D411yescharacteristic
29C2×C3×SL(2,𝔽3)D411yescharacteristic
30C3⋊(#41⋊C3)—22no—
31#2341—31noSylow
32C22×C3×Q8C22×C311yescharacteristic
33C2×#239—21no—
34C2×#239—21no—
35C2×#239—32no—
36C2×#239—61no—
37C2×#239—61no—
38C2×#239C3×C412yescomplemented
39C22×D12—31no—
40C22×C4×D3—31no—
41C2×#41⋊C3—32no—
42C2×#41⋊C3—61no—
43C2×#41⋊C3—61no—
44C22×SL(2,𝔽3)—21no—
45C22×SL(2,𝔽3)C2×D311yescharacteristic
46C3×C4×Q8—21no—
47C3×C4×Q8—61no—
48C2×C3×C4⋊C4—31no—
49C2×C3×C42—31no—
50Q8×Dic3—21no—
51Q8×Dic3—61no—
52C2×(C2×D3)⋊C4—31no—
53C2×(C2×D3)⋊C4—31no—
54C2×(C2×D3)⋊C4—32no—
55C2×C4⋊Dic3—31no—
56C2×C4×Dic3—31no—
57C4×SL(2,𝔽3)—22no—
58C4×SL(2,𝔽3)—21no—
59C4×SL(2,𝔽3)—61no—
60C3×#95—121no—
61C3⋊#95—31no—
62C3⋊#95A411yescharacteristic
63C23×C32—41no—
64C2×C32×C4—42no—
65C2×C32×C4—81no—
66C2×C32×C4—81no—
67C2×C3×Dic3—42no—
68C2×C3×Dic3—42no—
69C2×C3×Dic3—81no—
70C2×C3×Dic3—81no—
71C2×C3×Dic3—81no—
72C2×C3×Dic3—81no—
73C3×SL(2,𝔽3)C22⋊C411yescharacteristic, complemented
74C22×#41—31no—
75C2×C4×Q8—31no—
76C2×C4×Q8C3×D311yescharacteristic
77#390—91no—
78#390—91no—
79#380—91no—
80C4×C22⋊C4—91no—
81C23×D3—31no—
82C2×C3×Q8—31no—
83C2×C3×Q8—31no—
84C2×C3×Q8—31no—
85C2×C3×Q8C2×C3×C411yescharacteristic
86C2×C3×Q8C3×D411yescharacteristic
87C2×C3×Q8C3×D411yescharacteristic
88C22×C3×C4—31no—
89C22×C3×C4—31no—
90C22×C3×C4—41no—
91C22×C3×C4—81no—
92C22×C3×C4—121no—
93C22×C3×C4—121no—
94C22×C3×C4C2×A411yescharacteristic
95C22×Dic3—31no—
96C22×Dic3C2×A411yescharacteristic
97C22×Dic3C2×A411yescharacteristic
98#239—22no—
99#239—64no—
100#239—62no—
101#239—62no—
102#239—62no—
103C2×D12—32no—
104C2×D12—32no—
105C2×D12—61no—
106C2×D12—61no—
107C2×D12—61no—
108C2×D12—61no—
109C2×C4×D3—32no—
110C2×C4×D3—32no—
111C2×C4×D3—61no—
112C2×C4×D3—61no—
113C2×C4×D3—61no—
114C2×C4×D3—61no—
115#41⋊C3—62no—
116C2×SL(2,𝔽3)—21no—
117C2×SL(2,𝔽3)—21no—
118C2×SL(2,𝔽3)—21no—
119C2×SL(2,𝔽3)C4×D311yescharacteristic
120C2×SL(2,𝔽3)D1211yescharacteristic
121C2×SL(2,𝔽3)#6711yescharacteristic
122C3×C4⋊C4—61no—
123C3×C4⋊C4—61no—
124C3×C42—61no—
125C3×C42—61no—
126(C2×D3)⋊C4—62no—
127(C2×D3)⋊C4—62no—
128(C2×D3)⋊C4—122no—
129C4⋊Dic3—61no—
130C4⋊Dic3—61no—
131C4×Dic3—61no—
132C4×Dic3—61no—
133C22×C32—41no—
134C22×C32—41no—
135C22×C32—41no—
136C22×C32—41no—
137C22×C32—41no—
138C22×C32—41no—
139C22×C32—41no—
140C32×C4—82no—
141C3×Dic3—82no—
142C3×Dic3—82no—
143C2×#41—32no—
144C2×#41—61no—
145C2×#41—61no—
146C2×#41—92no—
147C2×#41—181no—
148C2×#41—181no—
149C22×Q8C2×C3×D311yescharacteristic
150C22×D4—91no—
151C23×C4—91no—
152C4×Q8—21no—
153C4×Q8—61no—
154C4×Q8—61no—
155C4×Q8—181no—
156C2×C4⋊C4—31no—
157C2×C4⋊C4—91no—
158C2×C22⋊C4—91no—
159C2×C22⋊C4—91no—
160C2×C22⋊C4—92no—
161C2×C42—31no—
162C2×C42—91no—
163#95—31no—
164#95—91no—
165C23×C3—41no—
166C23×C3—81no—
167C23×C3C22×A411yescharacteristic
168C22×D3—32no—
169C22×D3—61no—
170C22×D3—61no—
171C22×D3—61no—
172C22×D3—61no—
173C22×D3—61no—
174C22×D3—61no—
175C3×Q8—31no—
176C3×Q8—31no—
177C3×Q8—31no—
178C3×Q8—32no—
179C3×Q8C3×C22⋊C411yescharacteristic
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—31no—
183C2×C3×C4—31no—
184C2×C3×C4—31no—
185C2×C3×C4—31no—
186C2×C3×C4—31no—
187C2×C3×C4—31no—
188C2×C3×C4—42no—
189C2×C3×C4—61no—
190C2×C3×C4—61no—
191C2×C3×C4—61no—
192C2×C3×C4—81no—
193C2×C3×C4—81no—
194C2×C3×C4—82no—
195C2×C3×C4—82no—
196C2×C3×C4—82no—
197C2×C3×C4—122no—
198C2×C3×C4—122no—
199C2×C3×C4—241no—
200C2×C3×C4—241no—
201C2×C3×C4—241no—
202C2×C3×C4—241no—
203C2×C3×C4#41⋊C312yescomplemented
204C2×Dic3—21no—
205C2×Dic3—21no—
206C2×Dic3—21no—
207C2×Dic3—21no—
208C2×Dic3—61no—
209C2×Dic3—61no—
210C2×Dic3—61no—
211C2×Dic3C4×A412yes—
212C2×Dic3#41⋊C312yescomplemented
213D12—62no—
214D12—62no—
215D12—62no—
216D12—62no—
217C4×D3—62no—
218C4×D3—62no—
219C4×D3—62no—
220C4×D3—62no—
221SL(2,𝔽3)—21no—
222SL(2,𝔽3)(C2×D3)⋊C411yescharacteristic, complemented
223C2×C32—41no—
224C2×C32—41no—
225C2×C32—41no—
226C2×C32—41no—
227C2×C32—41no—
228C2×C32—41no—
229C2×C32—41no—
230C24—91no—
231#41—62no—
232#41—184no—
233#41—182no—
234#41—182no—
235#41—182no—
236C2×Q8—31no—
237C2×Q8—31no—
238C2×Q8—31no—
239C2×Q8C3×C4×D311yescharacteristic
240C2×Q8C3×D1211yescharacteristic
241C2×Q8C3×#6711yescharacteristic
242C2×D4—92no—
243C2×D4—92no—
244C2×D4—181no—
245C2×D4—181no—
246C2×D4—181no—
247C2×D4—181no—
248C22×C4—31no—
249C22×C4—31no—
250C22×C4—31no—
251C22×C4—31no—
252C22×C4—92no—
253C22×C4—92no—
254C22×C4—91no—
255C22×C4—181no—
256C22×C4—181no—
257C22×C4—181no—
258C22×C4—181no—
259C22×C4D3×A411yescharacteristic
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—181no—
263C4⋊C4—181no—
264C22⋊C4—182no—
265C22⋊C4—182no—
266C22⋊C4—362no—
267C42—61no—
268C42—61no—
269C42—181no—
270C42—181no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—81no—
279C22×C3—81no—
280C22×C3—81no—
281C22×C3—81no—
282C22×C3—81no—
283C22×C3—81no—
284C22×C3—81no—
285C22×C3C2×C4×A411yescharacteristic
286C22×C3D4×A411yescharacteristic
287C22×C3D4×A411yescharacteristic
288C22×C3C2×#41⋊C311yescharacteristic
289C22×C3C2×#41⋊C311yescharacteristic
290C22×C3#99411yescharacteristic
291C22×C32−1+4⋊C311yescharacteristic
292C2×D3—62no—
293C2×D3—62no—
294C2×D3—62no—
295C2×D3—121no—
296C2×D3—121no—
297C2×D3—121no—
298C2×D3—121no—
299C3×C4—22no—
300C3×C4—31no—
301C3×C4—31no—
302C3×C4—31no—
303C3×C4—31no—
304C3×C4—82no—
305C3×C4—84no—
306C3×C4—121no—
307C3×C4—242no—
308C3×C4—242no—
309Dic3—22no—
310Dic3—22no—
311Dic3—121no—
312C32—41noSylow
313C23—92no—
314C23—181no—
315C23—181no—
316C23—181no—
317C23—181no—
318C23—181no—
319C23—181no—
320C23C2×D3×A411yescharacteristic
321Q8—31no—
322Q8—31no—
323Q8—31no—
324Q8—32no—
325Q8C3×(C2×D3)⋊C411yescharacteristic
326D4—182no—
327D4—182no—
328D4—182no—
329D4—182no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—31no—
333C2×C4—31no—
334C2×C4—31no—
335C2×C4—31no—
336C2×C4—31no—
337C2×C4—31no—
338C2×C4—32no—
339C2×C4—32no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—182no—
348C2×C4—182no—
349C2×C4—182no—
350C2×C4—182no—
351C2×C4—181no—
352C2×C4—181no—
353C2×C4—181no—
354C2×C4C3⋊(#41⋊C3)12yescomplemented
355C2×C3—41no—
356C2×C3—41no—
357C2×C3—41no—
358C2×C3—41no—
359C2×C3—41no—
360C2×C3—41no—
361C2×C3—41no—
362C2×C3—81no—
363C2×C3—81no—
364C2×C3—81no—
365C2×C3—81no—
366C2×C3—81no—
367C2×C3—81no—
368C2×C3—81no—
369C2×C3A4×C22⋊C411yescharacteristic
370C2×C3C4×#41⋊C311yescharacteristic
371C2×C3#520⋊C311yescharacteristic
372C2×C3#548⋊C311yescharacteristic
373C2×C3#548⋊C311yescharacteristic
374C2×C3C4⋊(#41⋊C3)11yescharacteristic
375C2×C3C4⋊(#41⋊C3)11yescharacteristic
376D3—122no—
377C22—182no—
378C22—182no—
379C22—182no—
380C22—361no—
381C22—361no—
382C22—361no—
383C22—361no—
384C22C4×D3×A411yescharacteristic
385C22A4×D1211yescharacteristic
386C22C2×C3⋊(#41⋊C3)11yescharacteristic
387C22#1007⋊C311yescharacteristic
388C22C3⋊#99411yescharacteristic
389C22D3×#41⋊C311yescharacteristic
390C22A4×#6711yescharacteristic
391C4—22no—
392C4—31no—
393C4—31no—
394C4—31no—
395C4—31no—
396C4—62no—
397C4—62no—
398C4—121no—
399C4—361no—
400C3—41no—
401C3—81no—
402C3#2341⋊C311yescharacteristic, complemented
403C2—362no—
404C2C4×C3⋊(#41⋊C3)11yescharacteristic
405C2#5574⋊C311yescharacteristic
406C2C4⋊(C3⋊(#41⋊C3))11yescharacteristic
407C2#5576⋊C311yescharacteristic
408C2A4×(C2×D3)⋊C411yescharacteristic
409C2#5817⋊C311yescharacteristic
410C2#5834⋊C311yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×A4
commutatorC2×C3×Q8C2×C3×C4
FittingC2×C3×C4×Q8C2×C3
socleC23×C3C22×A4
prime radicalC22×C3⋊(#41⋊C3)C2
cosocleC22×C3×Q8C22×C3
hypercenterC22×C4D3×A4
hypocenterC3×Q8C3×C22⋊C4
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4×SL(2,𝔽3)C2semidirect
3. C2×Dic3×SL(2,𝔽3)C2semidirect
4. #82457C3semidirect
9. C2×C3⋊(#41⋊C3)C4semidirect
19. C2×C4×SL(2,𝔽3)D3semidirect
38. C2×#239C3×C4semidirect
73. C3×SL(2,𝔽3)C22⋊C4semidirect
203. C2×C3×C4#41⋊C3semidirect
212. C2×Dic3#41⋊C3semidirect
222. SL(2,𝔽3)(C2×D3)⋊C4semidirect
354. C2×C4C3⋊(#41⋊C3)semidirect
402. C3#2341⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×C3⋊(#41⋊C3)2
C2×C3×C4×SL(2,𝔽3)2
C2×Dic3×SL(2,𝔽3)2
#824573
#2341⋊C33
C3×C3⋊#954