Finite Group Databasebeta

← #10593666⋊C3 #13146940

Subgroups — #10593666⋊C3 #13146940

Order 1728.

836 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 70 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
1C23×C22⋊H(3,𝔽3)C211yesmaximal, characteristic
2C2×(C3⋊SL(2,ℤ/4))⋊C3C212yesmaximal, complemented
3#10593666C311yesmaximal, characteristic, complemented
4C3×#5247—31nomaximal
5C22×C22⋊H(3,𝔽3)—22no—
6C22×C22⋊H(3,𝔽3)C412yescomplemented
7C22×C22⋊H(3,𝔽3)C2211yescharacteristic
8(C3⋊SL(2,ℤ/4))⋊C3—22no—
9#3757⋊C3—41nomaximal
10C25×C32D311yescharacteristic
11C23×C3×A4—21no—
12C23×C3×A4C2×C311yescharacteristic
13C2×C3⋊SL(2,ℤ/4)C2×C312yescomplemented
14C2×C3×#217—32no—
15C2×C3×#217—31no—
16C2×C3×#217—31no—
17C2×C3×#217—32no—
18C23×H(3,𝔽3)—41no—
19C2×C22⋊H(3,𝔽3)—22no—
20C2×C22⋊H(3,𝔽3)C2×C411yescharacteristic
21C2×C22⋊H(3,𝔽3)D412yes—
22C2×(C3⋊Dic3)⋊C3—42no—
23#5432—91no—
24C3×#379—91no—
25#5247—31no—
26C24×C32—22no—
27C24×C32—31no—
28C24×C32—31no—
29C24×C32—32no—
30C24×C32—62no—
31C24×C32Dic312yescomplemented
32C24×C32C2×D311yescharacteristic
33C22×C3×A4—24no—
34C22×C3×A4—22no—
35C22×C3×A4—21no—
36C22×C3×A4—22no—
37C22×C3×A4C3×C412yescomplemented
38C22×C3×A4C22×C311yescharacteristic
39C22×C3×Dic3—32no—
40C22×C3×Dic3—32no—
41C3⋊SL(2,ℤ/4)—22no—
42#3757—41no—
43C3×#217—62no—
44C3×#217—64no—
45C3×#217—62no—
46C3×#217—61no—
47C3×#217—62no—
48C3×#217—61no—
49C22×H(3,𝔽3)—41no—
50C22×H(3,𝔽3)—42no—
51C22×H(3,𝔽3)—82no—
52C22⋊H(3,𝔽3)C22⋊C411yescharacteristic, complemented
53(C3⋊Dic3)⋊C3—82no—
54C25×C3—31no—
55C25×C3C3×D311yescharacteristic
56C23×A4—31no—
57C23×A4—61no—
58C2×SL(2,ℤ/4)—92no—
59C2×C3×C22⋊C4—92no—
60C2×C3×C22⋊C4—91no—
61C2×C3×C22⋊C4—91no—
62C2×C3×C22⋊C4—92no—
63C2×#217—32no—
64C2×#217—31no—
65C2×#217—31no—
66C2×#217—32no—
67C23×C32—22no—
68C23×C32—31no—
69C23×C32—32no—
70C23×C32—31no—
71C23×C32—32no—
72C23×C32—34no—
73C23×C32—32no—
74C23×C32—32no—
75C23×C32—31no—
76C23×C32—41no—
77C23×C32—62no—
78C23×C32—62no—
79C23×C32—62no—
80C23×C32—61no—
81C23×C32—64no—
82C23×C32—62no—
83C23×C32—64no—
84C23×C32—81no—
85C23×C32C2×Dic311yescharacteristic
86C23×C32#6712yes—
87C23×C32S411yescharacteristic
88C2×C3×A4—24no—
89C2×C3×A4—22no—
90C2×C3×A4—21no—
91C2×C3×A4—22no—
92C2×C3×A4C2×C3×C411yescharacteristic
93C2×C3×A4C3×D412yes—
94C2×C3⋊Dic3—42no—
95C2×C3×Dic3—62no—
96C2×C3×Dic3—62no—
97C2×C3×Dic3—62no—
98C2×C3×Dic3—62no—
99C2×C3×Dic3—62no—
100C2×C3×Dic3—62no—
101#379—91noSylow
102C2×H(3,𝔽3)—41no—
103C2×H(3,𝔽3)—42no—
104C2×H(3,𝔽3)—82no—
105C24×C3—22no—
106C24×C3—31no—
107C24×C3—31no—
108C24×C3—31no—
109C24×C3—31no—
110C24×C3—32no—
111C24×C3—32no—
112C24×C3—32no—
113C24×C3—31no—
114C24×C3—62no—
115C24×C3—61no—
116C24×C3—61no—
117C24×C3—62no—
118C24×C3—62no—
119C24×C3—64no—
120C24×C3—62no—
121C24×C3C3×Dic312yes—
122C24×C3C2×C3×D311yescharacteristic
123C22×A4—32no—
124C22×A4—31no—
125C22×A4—62no—
126C22×A4—64no—
127C22×A4—62no—
128C22×A4—61no—
129C22×C3×C4—92no—
130C22×C3×C4—92no—
131C22×Dic3—32no—
132C22×Dic3—32no—
133SL(2,ℤ/4)—182no—
134C3×C22⋊C4—182no—
135C3×C22⋊C4—184no—
136C3×C22⋊C4—182no—
137C3×C22⋊C4—181no—
138C3×C22⋊C4—182no—
139C3×C22⋊C4—181no—
140#217—62no—
141#217—64no—
142#217—62no—
143#217—61no—
144#217—62no—
145#217—61no—
146#217—361no—
147C22×C32—22no—
148C22×C32—31no—
149C22×C32—32no—
150C22×C32—31no—
151C22×C32—31no—
152C22×C32—32no—
153C22×C32—32no—
154C22×C32—34no—
155C22×C32—32no—
156C22×C32—41no—
157C22×C32—42no—
158C22×C32—62no—
159C22×C32—64no—
160C22×C32—62no—
161C22×C32—61no—
162C22×C32—62no—
163C22×C32—64no—
164C22×C32—62no—
165C22×C32—82no—
166C22×C32—84no—
167C22×C32—81no—
168C22×C32—82no—
169C22×C32#21711yescharacteristic, complemented
170C22×C32SL(2,ℤ/4)12yescomplemented
171C22×C32C2×S411yescharacteristic
172C3×A4—21no—
173C3×A4C3×C22⋊C411yescharacteristic, complemented
174C3⋊Dic3—82no—
175C3×Dic3—122no—
176C3×Dic3—122no—
177C25(C3⋊D3)⋊C311yescharacteristic
178C2×C22⋊C4—92no—
179C2×C22⋊C4—91no—
180C2×C22⋊C4—91no—
181C2×C22⋊C4—92no—
182H(3,𝔽3)—41noSylow
183C23×C3—22no—
184C23×C3—31no—
185C23×C3—31no—
186C23×C3—32no—
187C23×C3—32no—
188C23×C3—31no—
189C23×C3—34no—
190C23×C3—32no—
191C23×C3—34no—
192C23×C3—32no—
193C23×C3—31no—
194C23×C3—32no—
195C23×C3—31no—
196C23×C3—32no—
197C23×C3—31no—
198C23×C3—32no—
199C23×C3—32no—
200C23×C3—31no—
201C23×C3—32no—
202C23×C3—31no—
203C23×C3—62no—
204C23×C3—61no—
205C23×C3—62no—
206C23×C3—62no—
207C23×C3—61no—
208C23×C3—62no—
209C23×C3—62no—
210C23×C3—61no—
211C23×C3—64no—
212C23×C3—62no—
213C23×C3—62no—
214C23×C3—62no—
215C23×C3—61no—
216C23×C3—64no—
217C23×C3—62no—
218C23×C3—62no—
219C23×C3—62no—
220C23×C3—64no—
221C23×C3—62no—
222C23×C3—61no—
223C23×C3—62no—
224C23×C3—62no—
225C23×C3—64no—
226C23×C3—64no—
227C23×C3—64no—
228C23×C3—64no—
229C23×C3—64no—
230C23×C3—68no—
231C23×C3—64no—
232C23×C3—68no—
233C23×C3—121no—
234C23×C3—241no—
235C23×C3C2×C3×Dic311yescharacteristic
236C23×C3C3×#6712yes—
237C23×C3C3×S411yescharacteristic
238C2×A4—31no—
239C2×A4—32no—
240C2×A4—62no—
241C2×A4—64no—
242C2×A4—61no—
243C2×A4—62no—
244C2×C3×C4—182no—
245C2×C3×C4—182no—
246C2×C3×C4—182no—
247C2×C3×C4—182no—
248C2×C3×C4—182no—
249C2×C3×C4—182no—
250C2×Dic3—62no—
251C2×Dic3—62no—
252C2×Dic3—62no—
253C2×Dic3—62no—
254C2×Dic3—62no—
255C2×Dic3—62no—
256C2×Dic3—362no—
257C2×C32—22no—
258C2×C32—32no—
259C2×C32—31no—
260C2×C32—31no—
261C2×C32—41no—
262C2×C32—42no—
263C2×C32—62no—
264C2×C32—82no—
265C2×C32—84no—
266C2×C32—81no—
267C2×C32—82no—
268C2×C32C2×SL(2,ℤ/4)11yescharacteristic
269C2×C32GL(2,ℤ/4)12yes—
270C24—22no—
271C24—31no—
272C24—31no—
273C24—32no—
274C24—62no—
275C24(C3⋊Dic3)⋊C312yescomplemented
276C24C2×(C3⋊D3)⋊C311yescharacteristic
277C22×C4—92no—
278C22×C4—92no—
279C22⋊C4—182no—
280C22⋊C4—184no—
281C22⋊C4—182no—
282C22⋊C4—181no—
283C22⋊C4—182no—
284C22⋊C4—181no—
285C22×C3—22no—
286C22×C3—31no—
287C22×C3—32no—
288C22×C3—31no—
289C22×C3—32no—
290C22×C3—32no—
291C22×C3—31no—
292C22×C3—31no—
293C22×C3—31no—
294C22×C3—32no—
295C22×C3—32no—
296C22×C3—31no—
297C22×C3—32no—
298C22×C3—34no—
299C22×C3—34no—
300C22×C3—32no—
301C22×C3—32no—
302C22×C3—32no—
303C22×C3—31no—
304C22×C3—31no—
305C22×C3—62no—
306C22×C3—61no—
307C22×C3—62no—
308C22×C3—61no—
309C22×C3—61no—
310C22×C3—62no—
311C22×C3—62no—
312C22×C3—62no—
313C22×C3—64no—
314C22×C3—62no—
315C22×C3—64no—
316C22×C3—62no—
317C22×C3—61no—
318C22×C3—62no—
319C22×C3—62no—
320C22×C3—64no—
321C22×C3—62no—
322C22×C3—62no—
323C22×C3—61no—
324C22×C3—62no—
325C22×C3—64no—
326C22×C3—64no—
327C22×C3—64no—
328C22×C3—62no—
329C22×C3—64no—
330C22×C3—68no—
331C22×C3—64no—
332C22×C3—64no—
333C22×C3—68no—
334C22×C3—62no—
335C22×C3—122no—
336C22×C3—121no—
337C22×C3—242no—
338C22×C3—244no—
339C22×C3—242no—
340C22×C3—241no—
341C22×C3C3×#21711yescharacteristic
342C22×C3C3×SL(2,ℤ/4)12yes—
343C22×C3C2×C3×S411yescharacteristic
344A4—31no—
345A4—61no—
346C3×C4—362no—
347C3×C4—362no—
348Dic3—122no—
349Dic3—122no—
350Dic3—722no—
351C32—41no—
352C32—81no—
353C32#543211yescharacteristic, complemented
354C23—22no—
355C23—31no—
356C23—32no—
357C23—32no—
358C23—31no—
359C23—34no—
360C23—32no—
361C23—32no—
362C23—31no—
363C23—62no—
364C23—62no—
365C23—62no—
366C23—61no—
367C23—64no—
368C23—62no—
369C23—64no—
370C23C2×(C3⋊Dic3)⋊C311yescharacteristic
371C23#638⋊C312yes—
372C23(C3⋊S4)⋊C311yescharacteristic
373C2×C4—182no—
374C2×C4—182no—
375C2×C4—182no—
376C2×C4—182no—
377C2×C4—182no—
378C2×C4—182no—
379C2×C3—22no—
380C2×C3—32no—
381C2×C3—31no—
382C2×C3—31no—
383C2×C3—32no—
384C2×C3—31no—
385C2×C3—31no—
386C2×C3—31no—
387C2×C3—32no—
388C2×C3—62no—
389C2×C3—62no—
390C2×C3—61no—
391C2×C3—61no—
392C2×C3—64no—
393C2×C3—62no—
394C2×C3—62no—
395C2×C3—121no—
396C2×C3—122no—
397C2×C3—242no—
398C2×C3—244no—
399C2×C3—241no—
400C2×C3—242no—
401C2×C3C2×C3×SL(2,ℤ/4)11yescharacteristic
402C2×C3C3×GL(2,ℤ/4)12yes—
403C22—22no—
404C22—31no—
405C22—32no—
406C22—31no—
407C22—31no—
408C22—32no—
409C22—32no—
410C22—34no—
411C22—32no—
412C22—62no—
413C22—64no—
414C22—62no—
415C22—61no—
416C22—62no—
417C22—64no—
418C22—62no—
419C22#3757⋊C311yescharacteristic, complemented
420C22(C3⋊SL(2,ℤ/4))⋊C312yescomplemented
421C22C2×(C3⋊S4)⋊C311yescharacteristic
422C4—362no—
423C4—362no—
424C3—31no—
425C3—121no—
426C3—241no—
427C3C3×#543211yescharacteristic
428C2—22no—
429C2—32no—
430C2—31no—
431C2—31no—
432C2—62no—
433C2C2×(C3⋊SL(2,ℤ/4))⋊C311yescharacteristic
434C2#64269⋊C312yes—
subgroupG/H
centerC22C2×(C3⋊S4)⋊C3
commutatorC2×C3×A4C2×C3×C4
FrattiniC22×C3C2×C3×S4
FittingC25×C32D3
socleC24×C3C2×C3×D3
prime radicalC23×C22⋊H(3,𝔽3)C2
cosocleC22×C3×A4C22×C3
hypercenterC23(C3⋊S4)⋊C3
hypocenterC3×A4C3×C22⋊C4
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×(C3⋊SL(2,ℤ/4))⋊C3C2semidirect
3. #10593666C3semidirect
6. C22×C22⋊H(3,𝔽3)C4semidirect
13. C2×C3⋊SL(2,ℤ/4)C2×C3semidirect
31. C24×C32Dic3semidirect
37. C22×C3×A4C3×C4semidirect
52. C22⋊H(3,𝔽3)C22⋊C4semidirect
169. C22×C32#217semidirect
170. C22×C32SL(2,ℤ/4)semidirect
173. C3×A4C3×C22⋊C4semidirect
275. C24(C3⋊Dic3)⋊C3semidirect
353. C32#5432semidirect
419. C22#3757⋊C3semidirect
420. C22(C3⋊SL(2,ℤ/4))⋊C3semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊H(3,𝔽3)2
C2×(C3⋊SL(2,ℤ/4))⋊C32
#105936663
C3×#52473
#3757⋊C34