Finite Group Databasebeta

← #13147084

Subgroups — #13147084

Order 1728.

908 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 908 conjugacy classes of subgroups — too many to draw. Only its 83 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×#88613C212yesmaximal, complemented
3C3×#5432—33nomaximal
4C3×#5247—31nomaximal
5C22×C22⋊H(3,𝔽3)—22no—
6C22×C22⋊H(3,𝔽3)C412yescomplemented
7C22×C22⋊H(3,𝔽3)C2211yescharacteristic
8#88613—22no—
9#88552—41nomaximal
10C25×C32D311yescharacteristic
11C23×C3×A4D313yes—
12C2×C3×SL(2,ℤ/4)—36no—
13C2×C3×#217—32no—
14C2×C3×#217—31no—
15C2×C3×#217—31no—
16C2×C3×#217—32no—
17C23×H(3,𝔽3)—41no—
18C2×C22⋊H(3,𝔽3)—22no—
19C2×C22⋊H(3,𝔽3)C2×C411yescharacteristic
20C2×C22⋊H(3,𝔽3)D412yes—
21C2×#1084—42no—
22#5432—33no—
23C3×#379—91no—
24#5247—31no—
25C24×C32—22no—
26C24×C32—31no—
27C24×C32—31no—
28C24×C32—32no—
29C24×C32—62no—
30C24×C32Dic312yescomplemented
31C24×C32C2×D311yescharacteristic
32C22×C3×A4—26no—
33C22×C3×A4Dic316yescomplemented
34C22×C3×A4C2×D313yes—
35C22×C3×Dic3—32no—
36C22×C3×Dic3—32no—
37C3×SL(2,ℤ/4)—66no—
38C3×#217—62no—
39C3×#217—64no—
40C3×#217—62no—
41C3×#217—61no—
42C3×#217—62no—
43C3×#217—61no—
44C3×#217—123no—
45C22×H(3,𝔽3)—41no—
46C22×H(3,𝔽3)—42no—
47C22×H(3,𝔽3)—82no—
48C22⋊H(3,𝔽3)C22⋊C411yescharacteristic, complemented
49#1084—82no—
50C25×C3—31no—
51C25×C3C3⋊D311yescharacteristic
52C23×A4—33no—
53C2×SL(2,ℤ/4)—36no—
54C2×C3×C22⋊C4—92no—
55C2×C3×C22⋊C4—91no—
56C2×C3×C22⋊C4—91no—
57C2×C3×C22⋊C4—92no—
58C2×#217—32no—
59C2×#217—31no—
60C2×#217—31no—
61C2×#217—32no—
62C23×C32—22no—
63C23×C32—31no—
64C23×C32—32no—
65C23×C32—31no—
66C23×C32—32no—
67C23×C32—34no—
68C23×C32—32no—
69C23×C32—32no—
70C23×C32—31no—
71C23×C32—43no—
72C23×C32—62no—
73C23×C32—62no—
74C23×C32—62no—
75C23×C32—61no—
76C23×C32—64no—
77C23×C32—62no—
78C23×C32—64no—
79C23×C32C2×Dic311yescharacteristic
80C23×C32#6712yes—
81C23×C32S411yescharacteristic
82C2×C3×A4—26no—
83C2×C3×A4C2×Dic313yes—
84C2×C3×A4#6716yes—
85C2×C3×Dic3—62no—
86C2×C3×Dic3—62no—
87C2×C3×Dic3—62no—
88C2×C3×Dic3—62no—
89C2×C3×Dic3—62no—
90C2×C3×Dic3—62no—
91C2×C3×Dic3—126no—
92#379—91noSylow
93C2×H(3,𝔽3)—41no—
94C2×H(3,𝔽3)—42no—
95C2×H(3,𝔽3)—82no—
96C24×C3—22no—
97C24×C3—31no—
98C24×C3—31no—
99C24×C3—31no—
100C24×C3—31no—
101C24×C3—32no—
102C24×C3—32no—
103C24×C3—32no—
104C24×C3—31no—
105C24×C3—62no—
106C24×C3—61no—
107C24×C3—61no—
108C24×C3—62no—
109C24×C3—62no—
110C24×C3—64no—
111C24×C3—62no—
112C24×C3C3⋊Dic312yes—
113C24×C3C2×C3⋊D311yescharacteristic
114C22×A4—36no—
115C22×A4—33no—
116C22×A4—66no—
117C22×C3×C4—92no—
118C22×C3×C4—92no—
119C22×Dic3—32no—
120C22×Dic3—32no—
121SL(2,ℤ/4)—66no—
122C3×C22⋊C4—182no—
123C3×C22⋊C4—184no—
124C3×C22⋊C4—182no—
125C3×C22⋊C4—181no—
126C3×C22⋊C4—182no—
127C3×C22⋊C4—181no—
128#217—62no—
129#217—64no—
130#217—62no—
131#217—61no—
132#217—62no—
133#217—61no—
134#217—123no—
135C22×C32—22no—
136C22×C32—31no—
137C22×C32—32no—
138C22×C32—31no—
139C22×C32—31no—
140C22×C32—32no—
141C22×C32—32no—
142C22×C32—34no—
143C22×C32—32no—
144C22×C32—43no—
145C22×C32—46no—
146C22×C32—62no—
147C22×C32—64no—
148C22×C32—62no—
149C22×C32—61no—
150C22×C32—62no—
151C22×C32—64no—
152C22×C32—62no—
153C22×C32—86no—
154C22×C32#21711yescharacteristic, complemented
155C22×C32SL(2,ℤ/4)12yescomplemented
156C22×C32C2×S411yescharacteristic
157C3×A4#21713yescomplemented
158C3×Dic3—122no—
159C3×Dic3—122no—
160C3×Dic3—246no—
161C25#27211yescharacteristic
162C2×C22⋊C4—92no—
163C2×C22⋊C4—91no—
164C2×C22⋊C4—91no—
165C2×C22⋊C4—92no—
166H(3,𝔽3)—41noSylow
167C23×C3—22no—
168C23×C3—31no—
169C23×C3—31no—
170C23×C3—32no—
171C23×C3—32no—
172C23×C3—31no—
173C23×C3—32no—
174C23×C3—34no—
175C23×C3—32no—
176C23×C3—31no—
177C23×C3—34no—
178C23×C3—32no—
179C23×C3—31no—
180C23×C3—32no—
181C23×C3—31no—
182C23×C3—32no—
183C23×C3—32no—
184C23×C3—31no—
185C23×C3—32no—
186C23×C3—31no—
187C23×C3—62no—
188C23×C3—61no—
189C23×C3—62no—
190C23×C3—62no—
191C23×C3—61no—
192C23×C3—62no—
193C23×C3—62no—
194C23×C3—61no—
195C23×C3—64no—
196C23×C3—62no—
197C23×C3—62no—
198C23×C3—62no—
199C23×C3—61no—
200C23×C3—64no—
201C23×C3—62no—
202C23×C3—62no—
203C23×C3—62no—
204C23×C3—64no—
205C23×C3—62no—
206C23×C3—61no—
207C23×C3—62no—
208C23×C3—62no—
209C23×C3—64no—
210C23×C3—64no—
211C23×C3—64no—
212C23×C3—64no—
213C23×C3—64no—
214C23×C3—68no—
215C23×C3—64no—
216C23×C3—68no—
217C23×C3—123no—
218C23×C3C2×C3⋊Dic311yescharacteristic
219C23×C3#63812yes—
220C23×C3C3⋊S411yescharacteristic
221C2×A4—33no—
222C2×A4—36no—
223C2×A4—66no—
224C2×C3×C4—182no—
225C2×C3×C4—182no—
226C2×C3×C4—182no—
227C2×C3×C4—182no—
228C2×C3×C4—182no—
229C2×C3×C4—182no—
230C2×Dic3—62no—
231C2×Dic3—62no—
232C2×Dic3—62no—
233C2×Dic3—62no—
234C2×Dic3—62no—
235C2×Dic3—62no—
236C2×Dic3—126no—
237C2×C32—22no—
238C2×C32—32no—
239C2×C32—31no—
240C2×C32—31no—
241C2×C32—43no—
242C2×C32—46no—
243C2×C32—62no—
244C2×C32—86no—
245C2×C32C2×SL(2,ℤ/4)11yescharacteristic
246C2×C32GL(2,ℤ/4)12yes—
247C24—22no—
248C24—31no—
249C24—31no—
250C24—32no—
251C24—62no—
252C24#108412yescomplemented
253C24C2×#27211yescharacteristic
254C22×C4—92no—
255C22×C4—92no—
256C22⋊C4—182no—
257C22⋊C4—184no—
258C22⋊C4—182no—
259C22⋊C4—181no—
260C22⋊C4—182no—
261C22⋊C4—181no—
262C22×C3—22no—
263C22×C3—31no—
264C22×C3—32no—
265C22×C3—31no—
266C22×C3—32no—
267C22×C3—32no—
268C22×C3—31no—
269C22×C3—31no—
270C22×C3—31no—
271C22×C3—32no—
272C22×C3—32no—
273C22×C3—31no—
274C22×C3—32no—
275C22×C3—34no—
276C22×C3—34no—
277C22×C3—32no—
278C22×C3—32no—
279C22×C3—32no—
280C22×C3—31no—
281C22×C3—31no—
282C22×C3—62no—
283C22×C3—61no—
284C22×C3—62no—
285C22×C3—61no—
286C22×C3—61no—
287C22×C3—62no—
288C22×C3—62no—
289C22×C3—62no—
290C22×C3—64no—
291C22×C3—62no—
292C22×C3—64no—
293C22×C3—62no—
294C22×C3—61no—
295C22×C3—62no—
296C22×C3—62no—
297C22×C3—64no—
298C22×C3—62no—
299C22×C3—62no—
300C22×C3—61no—
301C22×C3—62no—
302C22×C3—64no—
303C22×C3—64no—
304C22×C3—64no—
305C22×C3—62no—
306C22×C3—64no—
307C22×C3—68no—
308C22×C3—64no—
309C22×C3—64no—
310C22×C3—68no—
311C22×C3—62no—
312C22×C3—126no—
313C22×C3—123no—
314C22×C3—246no—
315C22×C3#375711yescharacteristic
316C22×C3C3⋊SL(2,ℤ/4)12yes—
317C22×C3C2×C3⋊S411yescharacteristic
318A4—33no—
319C3×C4—362no—
320C3×C4—362no—
321Dic3—122no—
322Dic3—122no—
323Dic3—246no—
324C32—43no—
325C32#543211yescharacteristic, complemented
326C23—22no—
327C23—31no—
328C23—32no—
329C23—32no—
330C23—31no—
331C23—34no—
332C23—32no—
333C23—32no—
334C23—31no—
335C23—62no—
336C23—62no—
337C23—62no—
338C23—61no—
339C23—64no—
340C23—62no—
341C23—64no—
342C23C2×#108411yescharacteristic
343C23#623512yes—
344C23#625811yescharacteristic
345C2×C4—182no—
346C2×C4—182no—
347C2×C4—182no—
348C2×C4—182no—
349C2×C4—182no—
350C2×C4—182no—
351C2×C3—22no—
352C2×C3—32no—
353C2×C3—31no—
354C2×C3—31no—
355C2×C3—32no—
356C2×C3—31no—
357C2×C3—31no—
358C2×C3—31no—
359C2×C3—32no—
360C2×C3—62no—
361C2×C3—62no—
362C2×C3—61no—
363C2×C3—61no—
364C2×C3—64no—
365C2×C3—62no—
366C2×C3—62no—
367C2×C3—123no—
368C2×C3—126no—
369C2×C3—246no—
370C2×C3C2×C3⋊SL(2,ℤ/4)11yescharacteristic
371C2×C3#6426912yes—
372C22—22no—
373C22—31no—
374C22—32no—
375C22—31no—
376C22—31no—
377C22—32no—
378C22—32no—
379C22—34no—
380C22—32no—
381C22—62no—
382C22—64no—
383C22—62no—
384C22—61no—
385C22—62no—
386C22—64no—
387C22—62no—
388C22#8855211yescharacteristic, complemented
389C22#8861312yescomplemented
390C22C2×#625811yescharacteristic
391C4—362no—
392C4—362no—
393C3—31no—
394C3—123no—
395C3#1059366611yescharacteristic
396C2—22no—
397C2—32no—
398C2—31no—
399C2—31no—
400C2—62no—
401C2C2×#8861311yescharacteristic
402C2#1172239812yes—
subgroupG/H
center, FrattiniC22×C3C2×C3⋊S4
commutatorC2×C22⋊H(3,𝔽3)C2×C4
FittingC25×C32D3
socleC24×C3C2×C3⋊D3
prime radicalC23×C22⋊H(3,𝔽3)C2
cosocleC22×C22⋊H(3,𝔽3)C22
hypercenterC23×C3C3⋊S4
hypocenterC22⋊H(3,𝔽3)C22⋊C4
LCSG ▷ C2×C22⋊H(3,𝔽3) ▷ C22⋊H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C2×#88613C2semidirect
6. C22×C22⋊H(3,𝔽3)C4semidirect
30. C24×C32Dic3semidirect
33. C22×C3×A4Dic3semidirect
48. C22⋊H(3,𝔽3)C22⋊C4semidirect
154. C22×C32#217semidirect
155. C22×C32SL(2,ℤ/4)semidirect
157. C3×A4#217semidirect
252. C24#1084semidirect
325. C32#5432semidirect
388. C22#88552semidirect
389. C22#88613semidirect

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×#886132
C3×#54323
C3×#52473
#885524