Finite Group Databasebeta

← #13164481

Subgroups — #13164481

Order 1728.

846 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 846 conjugacy classes of subgroups — too many to draw. Only its 71 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
1D4×C22⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#11722398C212yesmaximal, complemented
3C2×#88613C212yesmaximal, complemented
4C4×#6258C211yesmaximal, characteristic, complemented
5#11722380C211yesmaximal, characteristic, complemented
6C3×#5914—33nomaximal
7C3×#5806—31nomaximal
8C22×C22⋊H(3,𝔽3)C2212yes—
9C2×#6258C2211yescharacteristic
10#88744—41nomaximal
11C4×C22⋊H(3,𝔽3)C2211yescharacteristic
12#88613C2212yes—
13#88613C2211yescharacteristic
14C22×C32×D4D311yescharacteristic, complemented
15C2×C3×#237—31no—
16C3×D4×A4D313yescomplemented
17C3×GL(2,ℤ/4)—36no—
18C2×C3×SL(2,ℤ/4)—36no—
19C3×C4×S4—33no—
20C3×A4⋊Q8—33no—
21C3×#953—32no—
22C2×C3×#217—32no—
23C3×#939—32no—
24C3×#938—31no—
25C3×#936—31no—
26C3×#935—32no—
27C3×D4×Dic3—31no—
28C3×C4×#67—31no—
29C3×#924—31no—
30C2×C22⋊H(3,𝔽3)—22no—
31C2×C22⋊H(3,𝔽3)C2311yescharacteristic
32#6258—21no—
33D4×H(3,𝔽3)—41no—
34#6235—42no—
35C2×#1084—42no—
36C4×#272—41no—
37#6229—41no—
38#5914—33no—
39C3×#546—91no—
40#5806—31no—
41C24×C32C2×D312yes—
42C22×C3×A4C2×D316yes—
43C2×C3×S4—33no—
44C2×C32×D4—31no—
45C2×C32×D4—32no—
46C2×C32×D4—31no—
47C22×C32×C4C2×D311yescharacteristic
48C2×C3×#67—32no—
49C2×C3×#67—31no—
50C2×C3×#67—32no—
51C22×C3×Dic3—32no—
52C22×C3×Dic3—32no—
53C3×#237—62no—
54C3×#237—61no—
55C3×#237—61no—
56C3×#237—123no—
57C2×C3×C4×D3—31no—
58C2×C3×Dic6—31no—
59C3×C4×A4C2×D313yes—
60C3×SL(2,ℤ/4)—33no—
61C3×SL(2,ℤ/4)—36no—
62C3×#217—32no—
63C3×#217—32no—
64C3×#217—32no—
65C3×#217—32no—
66C3×#217—32no—
67C3×#217—31no—
68C3×(C2×D3)⋊C4—31no—
69C3×C4⋊Dic3—31no—
70C3×Dic3⋊C4—32no—
71C3×Dic3⋊C4—31no—
72C3×C4×Dic3—31no—
73C22×H(3,𝔽3)—42no—
74C2×#272—41no—
75C22⋊H(3,𝔽3)#4111yescharacteristic, complemented
76C4×H(3,𝔽3)—41no—
77#1084—42no—
78#1084—41no—
79C2×C3×#41—91no—
80C22×C3×D4—31no—
81C22×C3×D4C3⋊D311yescharacteristic
82C2×#237—31no—
83D4×A4—33no—
84GL(2,ℤ/4)—36no—
85C2×SL(2,ℤ/4)—36no—
86C4×S4—33no—
87A4⋊Q8—33no—
88C3×#124—91no—
89C3×#123—92no—
90C3×C22⋊Q8—91no—
91C3×#121—91no—
92C3×#121—92no—
93C3×#120—92no—
94C3×C4×D4—91no—
95C3×C4×D4—91no—
96C2×C3×C22⋊C4—92no—
97#953—32no—
98C2×#217—32no—
99#939—32no—
100#938—31no—
101#936—31no—
102#935—32no—
103D4×Dic3—31no—
104C4×#67—31no—
105#924—31no—
106C23×C32—22no—
107C23×C32—32no—
108C23×C32—32no—
109C23×C32—62no—
110C23×C32C22×D311yescharacteristic
111C22×C3×D3—31no—
112C2×C3×A4—26no—
113C2×C3×A4C22×D313yes—
114C3×S4—63no—
115C32×D4—32no—
116C32×D4—31no—
117C32×D4—43no—
118C32×D4—61no—
119C32×D4S411yescharacteristic, complemented
120C2×C32×C4—31no—
121C2×C32×C4—31no—
122C3×#67—62no—
123C3×#67—61no—
124C3×#67—61no—
125C3×#67—62no—
126C3×#67—126no—
127C3×#67—122no—
128C2×C3×Dic3—31no—
129C2×C3×Dic3—31no—
130C2×C3×Dic3—32no—
131C2×C3×Dic3—32no—
132C2×C3×Dic3—31no—
133C2×C3×Dic3—62no—
134C2×C3×Dic3—62no—
135C2×C3×Dic3—126no—
136C2×C3×Dic3—122no—
137C3×C4×D3—61no—
138C3×C4×D3—61no—
139C3×C4×D3—123no—
140C3×Dic6—61no—
141C3×Dic6—61no—
142C3×Dic6—123no—
143#546—91noSylow
144C2×H(3,𝔽3)—41no—
145C2×H(3,𝔽3)—82no—
146#272—81no—
147C24×C3—32no—
148C24×C3C2×C3⋊D312yes—
149C22×A4—36no—
150C2×S4—33no—
151C3×#41—182no—
152C3×#41—181no—
153C3×#41—181no—
154C2×C3×Q8—91no—
155C2×C3×D4—31no—
156C2×C3×D4—31no—
157C2×C3×D4—32no—
158C2×C3×D4—31no—
159C2×C3×D4—32no—
160C2×C3×D4—31no—
161C2×C3×D4—61no—
162C2×C3×D4—62no—
163C2×C3×D4—61no—
164C2×C3×D4—92no—
165C2×C3×D4—91no—
166C2×C3×D4—92no—
167C22×C3×C4—31no—
168C22×C3×C4—91no—
169C22×C3×C4—92no—
170C22×C3×C4—92no—
171C22×C3×C4C2×C3⋊D311yescharacteristic
172C2×#67—31no—
173C2×#67—32no—
174C2×#67—32no—
175C22×Dic3—32no—
176C22×Dic3—32no—
177#237—62no—
178#237—61no—
179#237—61no—
180#237—123no—
181C2×C4×D3—31no—
182C2×Dic6—31no—
183C4×A4—33no—
184SL(2,ℤ/4)—36no—
185SL(2,ℤ/4)—33no—
186C3×C4⋊C4—91no—
187C3×C4⋊C4—92no—
188C3×C4⋊C4—91no—
189C3×C22⋊C4—91no—
190C3×C22⋊C4—92no—
191C3×C22⋊C4—92no—
192C3×C22⋊C4—92no—
193C3×C22⋊C4—92no—
194C3×C22⋊C4—91no—
195C3×C22⋊C4—92no—
196C3×C42—91no—
197#217—32no—
198#217—31no—
199#217—32no—
200#217—32no—
201#217—32no—
202#217—32no—
203(C2×D3)⋊C4—31no—
204C4⋊Dic3—31no—
205Dic3⋊C4—32no—
206Dic3⋊C4—31no—
207C4×Dic3—31no—
208C22×C32—31no—
209C22×C32—31no—
210C22×C32—32no—
211C22×C32—46no—
212C22×C32—62no—
213C22×C32—62no—
214C22×C32—62no—
215C22×C32—62no—
216C22×C32#23711yescharacteristic, complemented
217C22×C32C2×S412yes—
218C2×C3×D3—61no—
219C2×C3×D3—61no—
220C2×C3×D3—61no—
221C2×C3×D3—123no—
222C3×A4#23713yescomplemented
223C32×C4—31no—
224C32×C4—43no—
225C32×C4C2×S411yescharacteristic
226C3×Dic3—61no—
227C3×Dic3—62no—
228C3×Dic3—61no—
229C3×Dic3—62no—
230C3×Dic3—61no—
231C3×Dic3—126no—
232C3×Dic3—123no—
233C2×#41—91no—
234C22×D4#27211yescharacteristic, complemented
235#124—91no—
236#123—92no—
237C22⋊Q8—91no—
238#121—91no—
239#121—92no—
240#120—92no—
241C4×D4—91no—
242C4×D4—91no—
243C2×C22⋊C4—92no—
244H(3,𝔽3)—41noSylow
245C23×C3—22no—
246C23×C3—32no—
247C23×C3—32no—
248C23×C3—32no—
249C23×C3—32no—
250C23×C3—31no—
251C23×C3—62no—
252C23×C3—62no—
253C23×C3—62no—
254C23×C3—62no—
255C23×C3—62no—
256C23×C3—91no—
257C23×C3—122no—
258C23×C3C22×C3⋊D311yescharacteristic
259C22×D3—31no—
260C2×A4—33no—
261C2×A4—66no—
262S4—63no—
263C3×Q8—181no—
264C3×Q8—181no—
265C3×D4—32no—
266C3×D4—31no—
267C3×D4—32no—
268C3×D4—31no—
269C3×D4—31no—
270C3×D4—61no—
271C3×D4—62no—
272C3×D4—61no—
273C3×D4—61no—
274C3×D4—62no—
275C3×D4—123no—
276C3×D4—181no—
277C3×D4—182no—
278C3×D4—182no—
279C3×D4—181no—
280C3×D4—362no—
281C3×D4C3⋊S411yescharacteristic
282C2×C3×C4—31no—
283C2×C3×C4—31no—
284C2×C3×C4—31no—
285C2×C3×C4—31no—
286C2×C3×C4—61no—
287C2×C3×C4—61no—
288C2×C3×C4—91no—
289C2×C3×C4—91no—
290C2×C3×C4—92no—
291C2×C3×C4—91no—
292C2×C3×C4—92no—
293C2×C3×C4—181no—
294C2×C3×C4—182no—
295C2×C3×C4—182no—
296C2×C3×C4—181no—
297C2×C3×C4—362no—
298#67—62no—
299#67—62no—
300#67—61no—
301#67—61no—
302#67—126no—
303#67—122no—
304C2×Dic3—31no—
305C2×Dic3—31no—
306C2×Dic3—32no—
307C2×Dic3—32no—
308C2×Dic3—31no—
309C2×Dic3—62no—
310C2×Dic3—62no—
311C2×Dic3—126no—
312C2×Dic3—122no—
313C4×D3—61no—
314C4×D3—61no—
315C4×D3—123no—
316Dic6—61no—
317Dic6—61no—
318Dic6—123no—
319C2×C32—22no—
320C2×C32—31no—
321C2×C32—31no—
322C2×C32—43no—
323C2×C32—62no—
324C2×C32—86no—
325C2×C32C22×S411yescharacteristic
326C3×D3—121no—
327C3×D3—243no—
328C24C2×#27212yes—
329#41—182no—
330#41—181no—
331#41—181no—
332C2×Q8—91no—
333C2×D4—31no—
334C2×D4—32no—
335C2×D4—31no—
336C2×D4—92no—
337C2×D4—91no—
338C2×D4—92no—
339C22×C4—91no—
340C22×C4—92no—
341C22×C4—92no—
342C22×C4C2×#27211yescharacteristic
343C4⋊C4—91no—
344C4⋊C4—91no—
345C4⋊C4—92no—
346C22⋊C4—91no—
347C22⋊C4—92no—
348C22⋊C4—92no—
349C22⋊C4—92no—
350C22⋊C4—92no—
351C22⋊C4—91no—
352C22⋊C4—92no—
353C42—91no—
354C22×C3—31no—
355C22×C3—31no—
356C22×C3—31no—
357C22×C3—32no—
358C22×C3—31no—
359C22×C3—32no—
360C22×C3—32no—
361C22×C3—31no—
362C22×C3—61no—
363C22×C3—61no—
364C22×C3—62no—
365C22×C3—62no—
366C22×C3—62no—
367C22×C3—62no—
368C22×C3—62no—
369C22×C3—62no—
370C22×C3—62no—
371C22×C3—62no—
372C22×C3—62no—
373C22×C3—122no—
374C22×C3—122no—
375C22×C3—122no—
376C22×C3—122no—
377C22×C3—126no—
378C22×C3—181no—
379C22×C3—181no—
380C22×C3—181no—
381C22×C3#383011yescharacteristic
382C22×C3C2×C3⋊S412yes—
383C2×D3—61no—
384C2×D3—61no—
385C2×D3—61no—
386C2×D3—123no—
387A4—33no—
388C3×C4—31no—
389C3×C4—31no—
390C3×C4—31no—
391C3×C4—61no—
392C3×C4—123no—
393C3×C4—181no—
394C3×C4—182no—
395C3×C4—181no—
396C3×C4—181no—
397C3×C4—182no—
398C3×C4C2×C3⋊S411yescharacteristic
399Dic3—62no—
400Dic3—61no—
401Dic3—61no—
402Dic3—62no—
403Dic3—61no—
404Dic3—126no—
405Dic3—123no—
406C32—43no—
407C32#591411yescharacteristic, complemented
408C23—22no—
409C23—32no—
410C23—32no—
411C23—62no—
412C23—91no—
413C23C22×#27211yescharacteristic
414Q8—181no—
415Q8—181no—
416D4—32no—
417D4—31no—
418D4—61no—
419D4—182no—
420D4—181no—
421D4—182no—
422D4—181no—
423D4—362no—
424D4#625811yescharacteristic, complemented
425C2×C4—31no—
426C2×C4—31no—
427C2×C4—91no—
428C2×C4—91no—
429C2×C4—92no—
430C2×C4—91no—
431C2×C4—92no—
432C2×C4—181no—
433C2×C4—182no—
434C2×C4—182no—
435C2×C4—181no—
436C2×C4—362no—
437C2×C3—22no—
438C2×C3—31no—
439C2×C3—31no—
440C2×C3—31no—
441C2×C3—31no—
442C2×C3—31no—
443C2×C3—62no—
444C2×C3—61no—
445C2×C3—61no—
446C2×C3—62no—
447C2×C3—62no—
448C2×C3—122no—
449C2×C3—123no—
450C2×C3—246no—
451C2×C3—361no—
452C2×C3C22×C3⋊S411yescharacteristic
453D3—121no—
454D3—243no—
455C22—31no—
456C22—31no—
457C22—32no—
458C22—62no—
459C22—62no—
460C22—62no—
461C22—62no—
462C22—181no—
463C22—181no—
464C22—181no—
465C22#8874411yescharacteristic, complemented
466C22C2×#625812yes—
467C4—31no—
468C4—181no—
469C4—182no—
470C4—181no—
471C4—181no—
472C4—182no—
473C4C2×#625811yescharacteristic
474C3—31no—
475C3—123no—
476C3#1059657711yescharacteristic
477C2—22no—
478C2—31no—
479C2—31no—
480C2—62no—
481C2—361no—
482C2C22×#625811yescharacteristic
subgroupG/H
center, FrattiniC2×C3C22×C3⋊S4
commutator, cosocleC2×C22⋊H(3,𝔽3)C23
FittingC22×C32×D4D3
socleC23×C3C22×C3⋊D3
hypercenterC3×D4C3⋊S4
hypocenterC22⋊H(3,𝔽3)#41
LCSG ▷ C2×C22⋊H(3,𝔽3) ▷ C22⋊H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×D4

Complements

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

NH ≅ G/Ntype
1. D4×C22⋊H(3,𝔽3)C2semidirect
2. #11722398C2semidirect
3. C2×#88613C2semidirect
4. C4×#6258C2semidirect
5. #11722380C2semidirect
14. C22×C32×D4D3semidirect
16. C3×D4×A4D3semidirect
75. C22⋊H(3,𝔽3)#41semidirect
119. C32×D4S4semidirect
216. C22×C32#237semidirect
222. C3×A4#237semidirect
234. C22×D4#272semidirect
407. C32#5914semidirect
424. D4#6258semidirect
465. C22#88744semidirect

Maximal subgroups

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

subgroupindex
D4×C22⋊H(3,𝔽3)2
#117223982
C2×#886132
C4×#62582
#117223802
C3×#59143
C3×#58063
#887444