Finite Group Databasebeta

← #13165992

Subgroups — #13165992

Order 1728.

1114 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 1114 conjugacy classes of subgroups — too many to draw. Only its 103 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⋊(C3⋊S4)C211yesmaximal, characteristic
2C2×A4×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C2×C32×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
4#10593242—34nomaximal
5#10592657—31nomaximal
6C22×C32×A4C2211yescharacteristic
7C2×C3⋊(C3⋊S4)—21no—
8C2×C3⋊(C3⋊S4)—21no—
9C2×C3⋊(C3⋊S4)C412yescomplemented
10A4×C3⋊Dic3—21no—
11C32×SL(2,ℤ/4)—21no—
12#88804—41nomaximal
13C22×C3⋊S4—34no—
14C22×C3⋊S4—64no—
15C22×#638—31no—
16C23×C3⋊Dic3D311yescharacteristic, complemented
17C2×Dic3×A4—34no—
18C2×C3×SL(2,ℤ/4)D314yescomplemented
19C2×C32×C22⋊C4—31no—
20C2×#3752—31no—
21C2×#3752—31no—
22#63658—31no—
23#63658—31no—
24C22×C3⋊(C3⋊D3)—41no—
25C2×C32×A4C2×C411yescharacteristic
26C2×C32×A4D411yescharacteristic
27C2×C32×A4D411yescharacteristic
28C3⋊(C3⋊S4)—22no—
29C2×C3×C3⋊Dic3—41no—
30C2×C32×Dic3—41no—
31(C2×S4)⋊C4—91no—
32#4958—94no—
33C24×C32C2×D311yescharacteristic
34C23×C3⋊D3—31no—
35C22×C3×A4—24no—
36C22×C3×A4C2×D314yes—
37C2×C3⋊S4—38no—
38C2×C3⋊S4—68no—
39C2×C3⋊S4—68no—
40C2×C3⋊S4—68no—
41C2×C3⋊S4—64no—
42C2×C3⋊S4—64no—
43C22×C32×C4—31no—
44C22×C32×C4—31no—
45C2×#638—32no—
46C2×#638—32no—
47C2×#638—61no—
48C2×#638—61no—
49C2×#638—61no—
50C2×#638—61no—
51C22×C3⋊Dic3—21no—
52C22×C3⋊Dic3—31no—
53C22×C3⋊Dic3—31no—
54C22×C3⋊Dic3—31no—
55C22×C3⋊Dic3—61no—
56Dic3×A4—64no—
57C3×SL(2,ℤ/4)—24no—
58C32×C22⋊C4—61no—
59C32×C22⋊C4—61no—
60#3752—62no—
61#3752—61no—
62#3752—61no—
63#3722—124no—
64C22×C33—41no—
65C2×C3⋊(C3⋊D3)—42no—
66C2×C3⋊(C3⋊D3)—81no—
67C2×C3⋊(C3⋊D3)—81no—
68C32×A4C22⋊C411yescharacteristic, complemented
69C3×C3⋊Dic3—81no—
70C32×Dic3—81no—
71C22×S4—91no—
72C22×S4—184no—
73C22×#67—94no—
74C23×Dic3—34no—
75C2×C4×A4—91no—
76C2×SL(2,ℤ/4)C3⋊D311yescharacteristic, complemented
77C2×C3×C22⋊C4—34no—
78C2×(C2×D3)⋊C4—94no—
79C2×(C2×D3)⋊C4—94no—
80C3⋊#95—94no—
81C3⋊#95—94no—
82C23×C32—31no—
83C23×C32—31no—
84C23×C32—31no—
85C23×C32—31no—
86C23×C32C4×D311yescharacteristic
87C23×C32D1211yescharacteristic
88C23×C32#6711yescharacteristic
89C22×C3⋊D3—32no—
90C22×C3⋊D3—61no—
91C22×C3⋊D3—61no—
92C22×C3⋊D3—61no—
93C22×C3⋊D3—61no—
94C22×C3⋊D3—61no—
95C22×C3⋊D3—61no—
96C22×C3⋊D3—124no—
97C22×C3⋊D3—244no—
98C2×C3×A4—24no—
99C2×C3×A4—24no—
100C2×C3×A4—24no—
101C2×C3×A4C4×D314yes—
102C2×C3×A4D1214yes—
103C2×C3×A4#6714yes—
104C3⋊S4—616no—
105C3⋊S4—68no—
106C2×C32×C4—61no—
107C2×C32×C4—61no—
108C2×C32×C4—61no—
109C2×C32×C4—61no—
110C2×C32×C4—61no—
111C2×C32×C4—61no—
112#638—62no—
113#638—62no—
114#638—62no—
115#638—62no—
116C2×C3⋊Dic3—32no—
117C2×C3⋊Dic3—32no—
118C2×C3⋊Dic3—31no—
119C2×C3⋊Dic3—61no—
120C2×C3⋊Dic3—61no—
121C2×C3⋊Dic3—61no—
122C2×C3⋊Dic3—61no—
123C2×C3⋊Dic3—61no—
124C2×C3⋊Dic3S411yescharacteristic, complemented
125C2×C3×Dic3—44no—
126C2×C3×Dic3—124no—
127#386—271noSylow
128C2×C33—41no—
129C2×C33—41no—
130C2×C33—41no—
131C3⋊(C3⋊D3)—82no—
132C24×C3D3214yes—
133C23×D3—94no—
134C22×A4—24no—
135C22×A4C2×C3⋊D311yescharacteristic
136C2×S4—92no—
137C2×S4—188no—
138C2×S4—188no—
139C2×S4—188no—
140C2×S4—181no—
141C2×S4—181no—
142C22×C3×C4—34no—
143C22×C3×C4—34no—
144C2×#67—98no—
145C2×#67—98no—
146C2×#67—184no—
147C2×#67—184no—
148C2×#67—184no—
149C2×#67—184no—
150C22×Dic3—64no—
151C22×Dic3—94no—
152C22×Dic3—94no—
153C22×Dic3—94no—
154C22×Dic3—184no—
155C4×A4—181no—
156SL(2,ℤ/4)—21no—
157C3×C22⋊C4—64no—
158C3×C22⋊C4—64no—
159(C2×D3)⋊C4—184no—
160(C2×D3)⋊C4—188no—
161(C2×D3)⋊C4—184no—
162(C2×D3)⋊C4—361no—
163C22×C32—31no—
164C22×C32—31no—
165C22×C32—31no—
166C22×C32—31no—
167C22×C32—31no—
168C22×C32—31no—
169C22×C32—31no—
170C22×C32—31no—
171C22×C32—31no—
172C22×C32—44no—
173C22×C32—61no—
174C22×C32—84no—
175C22×C32(C2×D3)⋊C411yescharacteristic, complemented
176C22×C32C2×S411yescharacteristic
177C2×C3⋊D3—62no—
178C2×C3⋊D3—62no—
179C2×C3⋊D3—62no—
180C2×C3⋊D3—128no—
181C2×C3⋊D3—121no—
182C2×C3⋊D3—121no—
183C2×C3⋊D3—121no—
184C2×C3⋊D3—121no—
185C2×C3⋊D3—248no—
186C2×C3⋊D3—244no—
187C2×C3⋊D3—244no—
188C2×C3⋊D3—248no—
189C2×C3⋊D3—248no—
190C3×A4—24no—
191C3×A4(C2×D3)⋊C414yescomplemented
192C32×C4—121no—
193C32×C4—121no—
194C3⋊Dic3—21no—
195C3⋊Dic3—62no—
196C3⋊Dic3—61no—
197C3×Dic3—84no—
198C3×Dic3—244no—
199C22×D4—271no—
200C23×C4—91no—
201C2×C22⋊C4—31no—
202C2×C22⋊C4—271no—
203C2×C22⋊C4—271no—
204#95—271no—
205#95—271no—
206C33—41noSylow
207C23×C3—34no—
208C23×C3—34no—
209C23×C3—34no—
210C23×C3—34no—
211C23×C3#62414yes—
212C23×C3#62614yes—
213C23×C3#62614yes—
214C22×D3—98no—
215C22×D3—184no—
216C22×D3—184no—
217C22×D3—184no—
218C22×D3—184no—
219C22×D3—184no—
220C22×D3—184no—
221C22×D3—361no—
222C22×D3—724no—
223C2×A4—24no—
224C2×A4—24no—
225C2×A4—24no—
226C2×A4C4×C3⋊D311yescharacteristic
227C2×A4C4⋊(C3⋊D3)11yescharacteristic
228C2×A4#63811yescharacteristic
229S4—1816no—
230S4—182no—
231C2×C3×C4—64no—
232C2×C3×C4—64no—
233C2×C3×C4—64no—
234C2×C3×C4—64no—
235C2×C3×C4—64no—
236C2×C3×C4—64no—
237C2×C3×C4—361no—
238#67—188no—
239#67—188no—
240#67—188no—
241#67—188no—
242C2×Dic3—34no—
243C2×Dic3—41no—
244C2×Dic3—98no—
245C2×Dic3—98no—
246C2×Dic3—94no—
247C2×Dic3—184no—
248C2×Dic3—184no—
249C2×Dic3—184no—
250C2×Dic3—184no—
251C2×Dic3—184no—
252C2×C32—31no—
253C2×C32—31no—
254C2×C32—31no—
255C2×C32—31no—
256C2×C32—44no—
257C2×C32—44no—
258C2×C32—44no—
259C2×C32—84no—
260C2×C32—84no—
261C2×C32—84no—
262C2×C32C4×S411yescharacteristic
263C2×C32C4⋊S411yescharacteristic
264C2×C32GL(2,ℤ/4)11yescharacteristic
265C3⋊D3—122no—
266C3⋊D3—248no—
267C3⋊D3—2416no—
268C24—271no—
269C24D3×C3⋊D311yescharacteristic
270C2×D4—272no—
271C2×D4—272no—
272C2×D4—541no—
273C2×D4—541no—
274C2×D4—541no—
275C2×D4—541no—
276C22×C4—31no—
277C22×C4—31no—
278C22×C4—181no—
279C22×C4—271no—
280C22×C4—271no—
281C22×C4—271no—
282C22×C4—541no—
283C22⋊C4—61no—
284C22⋊C4—61no—
285C22⋊C4—541no—
286C22⋊C4—542no—
287C22⋊C4—541no—
288C22×C3—34no—
289C22×C3—34no—
290C22×C3—34no—
291C22×C3—34no—
292C22×C3—34no—
293C22×C3—34no—
294C22×C3—34no—
295C22×C3—34no—
296C22×C3—34no—
297C22×C3—41no—
298C22×C3—64no—
299C22×C3—84no—
300C22×C3#372214yescomplemented
301C22×C3D3×S414yes—
302C2×D3—188no—
303C2×D3—188no—
304C2×D3—188no—
305C2×D3—362no—
306C2×D3—364no—
307C2×D3—364no—
308C2×D3—364no—
309C2×D3—364no—
310C2×D3—728no—
311C2×D3—721no—
312C2×D3—721no—
313C2×D3—728no—
314C2×D3—728no—
315A4—24no—
316A4#375211yescharacteristic, complemented
317C3×C4—124no—
318C3×C4—124no—
319C3×C4—721no—
320Dic3—64no—
321Dic3—81no—
322Dic3—188no—
323Dic3—184no—
324C32—44no—
325C32—84no—
326C32(C2×S4)⋊C411yescharacteristic, complemented
327C23—31no—
328C23—31no—
329C23—31no—
330C23—31no—
331C23—272no—
332C23—541no—
333C23—541no—
334C23—541no—
335C23—541no—
336C23—541no—
337C23—541no—
338C23#628911yescharacteristic
339C23#629111yescharacteristic
340C23#629211yescharacteristic
341D4—542no—
342D4—542no—
343D4—542no—
344D4—542no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—61no—
350C2×C4—61no—
351C2×C4—91no—
352C2×C4—272no—
353C2×C4—272no—
354C2×C4—271no—
355C2×C4—541no—
356C2×C4—541no—
357C2×C4—541no—
358C2×C4—541no—
359C2×C4—541no—
360C2×C3—34no—
361C2×C3—34no—
362C2×C3—34no—
363C2×C3—34no—
364C2×C3—41no—
365C2×C3—41no—
366C2×C3—41no—
367C2×C3—84no—
368C2×C3—84no—
369C2×C3—84no—
370C2×C3(C3⋊S4)⋊C414yes—
371C2×C3#6420714yes—
372C2×C3#6421014yes—
373D3—368no—
374D3—722no—
375D3—7216no—
376C22—31no—
377C22—31no—
378C22—31no—
379C22—31no—
380C22—31no—
381C22—31no—
382C22—31no—
383C22—31no—
384C22—31no—
385C22—61no—
386C22—542no—
387C22—542no—
388C22—542no—
389C22—1081no—
390C22—1081no—
391C22—1081no—
392C22—1081no—
393C22#8880411yescharacteristic, complemented
394C22C3⋊D3×S411yescharacteristic
395C4—121no—
396C4—121no—
397C4—181no—
398C4—542no—
399C4—541no—
400C3—41no—
401C3—84no—
402C3#1059324214yescomplemented
403C2—31no—
404C2—31no—
405C2—31no—
406C2—31no—
407C2—1082no—
408C2#1172418611yescharacteristic
409C2#1172418711yescharacteristic
410C2#1172419011yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22C3⋊D3×S4
commutatorC2×C32×A4C2×C4
Fitting, socleC24×C32C2×D3
prime radicalC22×C3⋊(C3⋊S4)C2
cosocleC22×C32×A4C22
hypocenterC32×A4C22⋊C4
LCSG ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×A4×C3⋊Dic3C2semidirect
3. C2×C32×SL(2,ℤ/4)C2semidirect
9. C2×C3⋊(C3⋊S4)C4semidirect
16. C23×C3⋊Dic3D3semidirect
18. C2×C3×SL(2,ℤ/4)D3semidirect
68. C32×A4C22⋊C4semidirect
76. C2×SL(2,ℤ/4)C3⋊D3semidirect
124. C2×C3⋊Dic3S4semidirect
175. C22×C32(C2×D3)⋊C4semidirect
191. C3×A4(C2×D3)⋊C4semidirect
300. C22×C3#3722semidirect
316. A4#3752semidirect
326. C32(C2×S4)⋊C4semidirect
393. C22#88804semidirect
402. C3#10593242semidirect

Maximal subgroups

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

subgroupindex
C22×C3⋊(C3⋊S4)2
C2×A4×C3⋊Dic32
C2×C32×SL(2,ℤ/4)2
#105932423
#105926573
#888044