Finite Group Databasebeta

← #13164477

Subgroups — #13164477

Order 1728.

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

normal subgroups only

This group has 838 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
1C2×C4×C22⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#11722398C212yesmaximal, complemented
3C4⋊#6258C211yesmaximal, characteristic, complemented
4C4×#6258C212yesmaximal, complemented
5#11722380C211yesmaximal, characteristic, complemented
6C3×#5912—33nomaximal
7C3×#5791—31nomaximal
8C22×C22⋊H(3,𝔽3)C2211yescharacteristic
9C2×#6258C2212yes—
10#88740—41nomaximal
11C4×C22⋊H(3,𝔽3)C2211yescharacteristic
12C4×C22⋊H(3,𝔽3)C2211yescharacteristic
13#88613C2212yes—
14C23×C32×C4D311yescharacteristic, complemented
15C2×C3×#235—31no—
16C2×C3×C4×A4D313yescomplemented
17C3×GL(2,ℤ/4)—36no—
18C3×C4⋊S4—33no—
19C3×C4×S4—36no—
20C3×A4⋊Q8—33no—
21C3×#953—32no—
22C3×#930—31no—
23C3×#930—31no—
24C3×#929—32no—
25C3×C4×#67—32no—
26C3×C4×#67—32no—
27C3×#926—31no—
28C3×#924—31no—
29C3×#924—31no—
30C2×C22⋊H(3,𝔽3)—21no—
31C2×C22⋊H(3,𝔽3)C2311yescharacteristic
32#6258—22no—
33C2×C4×H(3,𝔽3)—41no—
34#6235—42no—
35C4⋊#272—41no—
36C4×#272—42no—
37#6229—41no—
38#5912—33no—
39C3×#525—91no—
40#5791—31no—
41C24×C32C2×D311yescharacteristic
42C22×C3×A4C2×D313yes—
43C2×C3×S4—36no—
44C22×C32×C4—31no—
45C22×C32×C4—31no—
46C22×C32×C4—31no—
47C22×C32×C4—31no—
48C22×C32×C4C2×D311yescharacteristic
49C22×C32×C4C2×D311yescharacteristic
50C2×C3×#67—32no—
51C2×C3×#67—32no—
52C2×C3×#67—32no—
53C3×#235—61no—
54C3×#235—61no—
55C3×#235—61no—
56C3×#235—61no—
57C3×#235—123no—
58C2×C3×D12—31no—
59C2×C3×C4×D3—32no—
60C2×C3×Dic6—31no—
61C3×C4×A4C2×D313yes—
62C3×C4×A4C2×D313yes—
63C3×SL(2,ℤ/4)—36no—
64C3×#217—32no—
65C3×#217—32no—
66C3×#217—32no—
67C3×(C2×D3)⋊C4—32no—
68C3×(C2×D3)⋊C4—32no—
69C3×C4⋊Dic3—31no—
70C3×C4⋊Dic3—31no—
71C3×Dic3⋊C4—32no—
72C3×Dic3⋊C4—32no—
73C3×C4×Dic3—32no—
74C22×H(3,𝔽3)—41no—
75C2×#272—42no—
76C22⋊H(3,𝔽3)#4111yescharacteristic, complemented
77C4×H(3,𝔽3)—41no—
78C4×H(3,𝔽3)—41no—
79#1084—42no—
80C2×C3×#41—91no—
81C23×C3×C4—31no—
82C23×C3×C4C3⋊D311yescharacteristic
83C2×#235—31no—
84C2×C4×A4—33no—
85GL(2,ℤ/4)—36no—
86C4⋊S4—33no—
87C4×S4—36no—
88A4⋊Q8—33no—
89C3×#123—92no—
90C3×C22⋊Q8—91no—
91C3×C22⋊Q8—91no—
92C3×#121—91no—
93C3×#121—91no—
94C3×#120—92no—
95C3×C4×D4—92no—
96C3×C4×D4—92no—
97C3×#117—91no—
98#953—32no—
99#930—31no—
100#930—31no—
101#929—32no—
102C4×#67—32no—
103C4×#67—32no—
104#926—31no—
105#924—31no—
106#924—31no—
107C23×C32—21no—
108C23×C32—31no—
109C23×C32—31no—
110C23×C32—61no—
111C23×C32C22×D311yescharacteristic
112C22×C3×D3—32no—
113C2×C3×A4—23no—
114C2×C3×A4C22×D313yes—
115C3×S4—66no—
116C2×C32×C4—31no—
117C2×C32×C4—31no—
118C2×C32×C4—31no—
119C2×C32×C4—31no—
120C2×C32×C4—31no—
121C2×C32×C4—31no—
122C2×C32×C4—31no—
123C2×C32×C4—43no—
124C2×C32×C4—61no—
125C2×C32×C4S411yescharacteristic, complemented
126C3×#67—62no—
127C3×#67—62no—
128C3×#67—64no—
129C3×#67—62no—
130C3×#67—62no—
131C3×#67—126no—
132C2×C3×Dic3—32no—
133C2×C3×Dic3—32no—
134C2×C3×Dic3—32no—
135C3×D12—61no—
136C3×D12—61no—
137C3×D12—123no—
138C3×C4×D3—62no—
139C3×C4×D3—62no—
140C3×C4×D3—126no—
141C3×Dic6—61no—
142C3×Dic6—61no—
143C3×Dic6—123no—
144#525—91noSylow
145C2×H(3,𝔽3)—41no—
146C2×H(3,𝔽3)—81no—
147#272—82no—
148C24×C3—31no—
149C24×C3C2×C3⋊D311yescharacteristic
150C22×A4—33no—
151C2×S4—36no—
152C3×#41—181no—
153C3×#41—181no—
154C3×#41—181no—
155C3×#41—181no—
156C2×C3×Q8—91no—
157C2×C3×D4—91no—
158C2×C3×D4—92no—
159C2×C3×D4—92no—
160C2×C3×D4—92no—
161C22×C3×C4—31no—
162C22×C3×C4—31no—
163C22×C3×C4—31no—
164C22×C3×C4—31no—
165C22×C3×C4—31no—
166C22×C3×C4—31no—
167C22×C3×C4—31no—
168C22×C3×C4—31no—
169C22×C3×C4—31no—
170C22×C3×C4—31no—
171C22×C3×C4—61no—
172C22×C3×C4—61no—
173C22×C3×C4—61no—
174C22×C3×C4—61no—
175C22×C3×C4—92no—
176C22×C3×C4C2×C3⋊D311yescharacteristic
177C22×C3×C4C2×C3⋊D311yescharacteristic
178C2×#67—32no—
179C2×#67—32no—
180C2×#67—32no—
181#235—61no—
182#235—61no—
183#235—61no—
184#235—61no—
185#235—123no—
186C2×D12—31no—
187C2×C4×D3—32no—
188C2×Dic6—31no—
189C4×A4—33no—
190C4×A4—33no—
191SL(2,ℤ/4)—36no—
192C3×C4⋊C4—92no—
193C3×C4⋊C4—92no—
194C3×C4⋊C4—91no—
195C3×C4⋊C4—91no—
196C3×C22⋊C4—92no—
197C3×C22⋊C4—92no—
198C3×C22⋊C4—92no—
199C3×C22⋊C4—92no—
200C3×C22⋊C4—92no—
201C3×C42—92no—
202#217—32no—
203#217—32no—
204#217—32no—
205(C2×D3)⋊C4—32no—
206(C2×D3)⋊C4—32no—
207C4⋊Dic3—31no—
208C4⋊Dic3—31no—
209Dic3⋊C4—32no—
210Dic3⋊C4—32no—
211C4×Dic3—32no—
212C22×C32—31no—
213C22×C32—31no—
214C22×C32—32no—
215C22×C32—31no—
216C22×C32—43no—
217C22×C32—61no—
218C22×C32—61no—
219C22×C32—61no—
220C22×C32#23511yescharacteristic, complemented
221C22×C32C2×S411yescharacteristic
222C2×C3×D3—62no—
223C2×C3×D3—62no—
224C2×C3×D3—62no—
225C2×C3×D3—126no—
226C3×A4#23513yescomplemented
227C32×C4—31no—
228C32×C4—31no—
229C32×C4—43no—
230C32×C4—43no—
231C32×C4C2×S411yescharacteristic
232C32×C4C2×S411yescharacteristic
233C3×Dic3—62no—
234C3×Dic3—62no—
235C3×Dic3—62no—
236C3×Dic3—126no—
237C2×#41—91no—
238C23×C4#27211yescharacteristic, complemented
239#123—92no—
240C22⋊Q8—91no—
241C22⋊Q8—91no—
242#121—91no—
243#121—91no—
244#120—92no—
245C4×D4—92no—
246C4×D4—92no—
247#117—91no—
248H(3,𝔽3)—41noSylow
249C23×C3—21no—
250C23×C3—31no—
251C23×C3—31no—
252C23×C3—31no—
253C23×C3—31no—
254C23×C3—31no—
255C23×C3—61no—
256C23×C3—61no—
257C23×C3—61no—
258C23×C3—61no—
259C23×C3—62no—
260C23×C3—61no—
261C23×C3—92no—
262C23×C3C22×C3⋊D311yescharacteristic
263C22×D3—32no—
264C2×A4—33no—
265C2×A4—63no—
266S4—66no—
267C3×Q8—181no—
268C3×Q8—181no—
269C3×D4—181no—
270C3×D4—182no—
271C3×D4—184no—
272C3×D4—182no—
273C3×D4—182no—
274C3×D4—182no—
275C3×D4—181no—
276C2×C3×C4—31no—
277C2×C3×C4—31no—
278C2×C3×C4—31no—
279C2×C3×C4—31no—
280C2×C3×C4—31no—
281C2×C3×C4—31no—
282C2×C3×C4—31no—
283C2×C3×C4—31no—
284C2×C3×C4—31no—
285C2×C3×C4—31no—
286C2×C3×C4—31no—
287C2×C3×C4—31no—
288C2×C3×C4—31no—
289C2×C3×C4—31no—
290C2×C3×C4—31no—
291C2×C3×C4—61no—
292C2×C3×C4—61no—
293C2×C3×C4—61no—
294C2×C3×C4—61no—
295C2×C3×C4—61no—
296C2×C3×C4—61no—
297C2×C3×C4—61no—
298C2×C3×C4—61no—
299C2×C3×C4—61no—
300C2×C3×C4—61no—
301C2×C3×C4—61no—
302C2×C3×C4—92no—
303C2×C3×C4—92no—
304C2×C3×C4—92no—
305C2×C3×C4—123no—
306C2×C3×C4—182no—
307C2×C3×C4—182no—
308C2×C3×C4C3⋊S411yescharacteristic
309#67—62no—
310#67—62no—
311#67—62no—
312#67—64no—
313#67—62no—
314#67—126no—
315C2×Dic3—32no—
316C2×Dic3—32no—
317C2×Dic3—32no—
318D12—61no—
319D12—61no—
320D12—123no—
321C4×D3—62no—
322C4×D3—62no—
323C4×D3—126no—
324Dic6—61no—
325Dic6—61no—
326Dic6—123no—
327C2×C32—21no—
328C2×C32—31no—
329C2×C32—31no—
330C2×C32—43no—
331C2×C32—61no—
332C2×C32—83no—
333C2×C32C22×S411yescharacteristic
334C3×D3—122no—
335C3×D3—246no—
336C24C2×#27211yescharacteristic
337#41—181no—
338#41—181no—
339#41—181no—
340#41—181no—
341C2×Q8—91no—
342C2×D4—91no—
343C2×D4—92no—
344C2×D4—92no—
345C2×D4—92no—
346C22×C4—31no—
347C22×C4—31no—
348C22×C4—31no—
349C22×C4—31no—
350C22×C4—92no—
351C22×C4C2×#27211yescharacteristic
352C22×C4C2×#27211yescharacteristic
353C4⋊C4—91no—
354C4⋊C4—91no—
355C4⋊C4—92no—
356C4⋊C4—92no—
357C22⋊C4—92no—
358C22⋊C4—92no—
359C22⋊C4—92no—
360C22⋊C4—92no—
361C22⋊C4—92no—
362C42—92no—
363C22×C3—31no—
364C22×C3—31no—
365C22×C3—31no—
366C22×C3—31no—
367C22×C3—31no—
368C22×C3—32no—
369C22×C3—32no—
370C22×C3—31no—
371C22×C3—31no—
372C22×C3—31no—
373C22×C3—61no—
374C22×C3—61no—
375C22×C3—61no—
376C22×C3—62no—
377C22×C3—61no—
378C22×C3—61no—
379C22×C3—61no—
380C22×C3—61no—
381C22×C3—61no—
382C22×C3—61no—
383C22×C3—62no—
384C22×C3—62no—
385C22×C3—62no—
386C22×C3—123no—
387C22×C3—182no—
388C22×C3—182no—
389C22×C3—182no—
390C22×C3#382811yescharacteristic
391C22×C3C2×C3⋊S411yescharacteristic
392C2×D3—62no—
393C2×D3—62no—
394C2×D3—62no—
395C2×D3—126no—
396A4—33no—
397C3×C4—31no—
398C3×C4—31no—
399C3×C4—31no—
400C3×C4—31no—
401C3×C4—31no—
402C3×C4—31no—
403C3×C4—61no—
404C3×C4—61no—
405C3×C4—123no—
406C3×C4—123no—
407C3×C4—182no—
408C3×C4—182no—
409C3×C4—182no—
410C3×C4C2×C3⋊S411yescharacteristic
411C3×C4C2×C3⋊S411yescharacteristic
412Dic3—62no—
413Dic3—62no—
414Dic3—62no—
415Dic3—126no—
416C32—43no—
417C32#591211yescharacteristic, complemented
418C23—21no—
419C23—31no—
420C23—31no—
421C23—61no—
422C23—92no—
423C23C22×#27211yescharacteristic
424Q8—181no—
425Q8—181no—
426D4—181no—
427D4—182no—
428D4—182no—
429D4—184no—
430D4—182no—
431D4—182no—
432D4—181no—
433C2×C4—31no—
434C2×C4—31no—
435C2×C4—31no—
436C2×C4—31no—
437C2×C4—31no—
438C2×C4—31no—
439C2×C4—31no—
440C2×C4—61no—
441C2×C4—92no—
442C2×C4—92no—
443C2×C4—92no—
444C2×C4—182no—
445C2×C4—182no—
446C2×C4#625811yescharacteristic, complemented
447C2×C3—21no—
448C2×C3—31no—
449C2×C3—31no—
450C2×C3—31no—
451C2×C3—31no—
452C2×C3—31no—
453C2×C3—61no—
454C2×C3—61no—
455C2×C3—61no—
456C2×C3—61no—
457C2×C3—62no—
458C2×C3—61no—
459C2×C3—123no—
460C2×C3—243no—
461C2×C3—362no—
462C2×C3C22×C3⋊S411yescharacteristic
463D3—122no—
464D3—246no—
465C22—31no—
466C22—31no—
467C22—32no—
468C22—31no—
469C22—61no—
470C22—61no—
471C22—61no—
472C22—182no—
473C22—182no—
474C22—182no—
475C22#8874011yescharacteristic, complemented
476C22C2×#625811yescharacteristic
477C4—31no—
478C4—31no—
479C4—182no—
480C4—182no—
481C4—182no—
482C4C2×#625811yescharacteristic
483C4C2×#625811yescharacteristic
484C3—31no—
485C3—123no—
486C3#1059657511yescharacteristic
487C2—21no—
488C2—31no—
489C2—31no—
490C2—61no—
491C2—362no—
492C2C22×#625811yescharacteristic
subgroupG/H
centerC3×C4C2×C3⋊S4
commutator, cosocleC2×C22⋊H(3,𝔽3)C23
FrattiniC2×C3C22×C3⋊S4
FittingC23×C32×C4D3
socleC23×C3C22×C3⋊D3
hypercenterC2×C3×C4C3⋊S4
hypocenterC22⋊H(3,𝔽3)#41
LCSG ▷ C2×C22⋊H(3,𝔽3) ▷ C22⋊H(3,𝔽3)
UCSC1 ◁ C3×C4 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C4×C22⋊H(3,𝔽3)C2semidirect
2. #11722398C2semidirect
3. C4⋊#6258C2semidirect
4. C4×#6258C2semidirect
5. #11722380C2semidirect
14. C23×C32×C4D3semidirect
16. C2×C3×C4×A4D3semidirect
76. C22⋊H(3,𝔽3)#41semidirect
125. C2×C32×C4S4semidirect
220. C22×C32#235semidirect
226. C3×A4#235semidirect
238. C23×C4#272semidirect
417. C32#5912semidirect
446. C2×C4#6258semidirect
475. C22#88740semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊H(3,𝔽3)2
#117223982
C4⋊#62582
C4×#62582
#117223802
C3×#59123
C3×#57913
#887404