Finite Group Databasebeta

← #13167562

Subgroups — #13167562

Order 1728.

892 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 892 conjugacy classes of subgroups — too many to draw. Only its 95 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×#6294C211yesmaximal, characteristic
2C2×C3×C22⋊#157C211yesmaximal, characteristic
3#11723723C211yesmaximal, characteristic
4#11722791C214yesmaximal, complemented
5#10593488—31nomaximal
6C23×C3×C3⋊D3C2211yescharacteristic
7C22×C3×#157C2212yes—
8C22×C3×C3⋊Dic3—21no—
9C2×#6294—21no—
10C2×#6294—22no—
11C2×#6294C414yescomplemented
12C2×#6294C2212yes—
13C2×#6294C2212yes—
14C3×C22⋊#157—22no—
15C3×#3757—21no—
16#88814—21no—
17C22×#624—31no—
18C22×D3×Dic3—61no—
19C2×C22⋊#157D311yescharacteristic
20#63974—31no—
21D3×#217—61no—
22#63737—34no—
23C23×C33D411yescharacteristic
24C22×C3×C3⋊D3—24no—
25C22×C3×C3⋊D3C2×C412yes—
26C22×C3×C3⋊D3D411yescharacteristic
27C22×C3×C3⋊D3D412yes—
28C22×C3×C3⋊D3C2311yescharacteristic
29C2×C3×#157—22no—
30C2×C3×#157—24no—
31C2×C3×C3⋊Dic3—24no—
32C2×C3×C3⋊Dic3—22no—
33C2×C3×C3⋊Dic3—22no—
34#6294—24no—
35#6294—22no—
36#6294—24no—
37#6294C2×C414yes—
38#4943—91nomaximal
39C23×C3⋊D3C2×D311yescharacteristic
40C23×C3×D3—61no—
41C23×C3×D3—61no—
42C22×#157C2×D312yes—
43C22×C3⋊Dic3—21no—
44C22×C3×Dic3—61no—
45C22×C3×Dic3—61no—
46C22×C3×Dic3—121no—
47C2×#624—34no—
48C2×#624—32no—
49C2×#624—32no—
50C2×#624—61no—
51C2×#624—62no—
52C2×D3×Dic3—64no—
53C2×D3×Dic3—62no—
54C2×D3×Dic3—62no—
55C2×D3×Dic3—64no—
56C2×D3×Dic3—62no—
57C22⋊#157—22no—
58#3757—21no—
59C3×#217—61no—
60C3×#217—61no—
61C3×#217—121no—
62#3722—61no—
63(C2×D3)⋊Dic3—62no—
64C22×C33—22no—
65C22×C33C22⋊C412yes—
66C22×C33C2×D411yescharacteristic
67C2×C3×C3⋊D3—22no—
68C2×C3×C3⋊D3—24no—
69C2×C3×C3⋊D3—24no—
70C2×C3×C3⋊D3C22⋊C412yes—
71C2×C3×C3⋊D3C22×C411yescharacteristic
72C2×C3×C3⋊D3C2×D411yescharacteristic
73C2×C3×C3⋊D3#4114yes—
74C3×#157—42no—
75C3×C3⋊Dic3—24no—
76C3×C3⋊Dic3—42no—
77C23×Dic3—91no—
78C22×C4×D3—181no—
79C2×C3×C22⋊C4—91no—
80C2×#217—91no—
81D3×C22⋊C4—181no—
82C3⋊#95—94no—
83C23×C32—21no—
84C23×C32—21no—
85C23×C32—41no—
86C23×C32—41no—
87C23×C32D1211yescharacteristic
88C22×C3⋊D3—24no—
89C22×C3⋊D3C4×D312yes—
90C22×C3⋊D3D1211yescharacteristic
91C22×C3⋊D3#6712yes—
92C22×C3⋊D3C22×D311yescharacteristic
93C22×C3×D3—62no—
94C22×C3×D3—61no—
95C22×C3×D3—62no—
96C22×C3×D3—61no—
97C22×C3×D3—61no—
98C22×C3×D3—61no—
99C22×C3×D3—62no—
100C22×C3×D3—62no—
101C22×C3×D3—68no—
102C22×C3×D3—124no—
103C2×#157—24no—
104C2×#157—22no—
105C2×C3⋊Dic3—22no—
106C2×C3⋊Dic3—24no—
107C2×C3⋊Dic3—22no—
108C2×C3×Dic3—62no—
109C2×C3×Dic3—64no—
110C2×C3×Dic3—64no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—62no—
114C2×C3×Dic3—124no—
115C2×C3×Dic3—122no—
116C2×C3×Dic3—122no—
117#624—34no—
118#624—64no—
119#624—62no—
120#624—64no—
121D3×Dic3—64no—
122D3×Dic3—68no—
123D3×Dic3—64no—
124D3×Dic3—124no—
125#381—271noSylow
126C2×C33—22no—
127C2×C33C2×C22⋊C411yescharacteristic
128C2×C33#12312yes—
129C3×C3⋊D3—22no—
130C3×C3⋊D3#11712yes—
131C3×C3⋊D3#12312yes—
132C24×C3—91no—
133C23×D3—61no—
134C23×D3—61no—
135C22×C3×C4—92no—
136C22×C3×C4—181no—
137C22×Dic3—61no—
138C22×Dic3—61no—
139C22×Dic3—92no—
140C22×Dic3—94no—
141C22×Dic3—92no—
142C22×Dic3—121no—
143C22×Dic3—181no—
144C22×Dic3—182no—
145C2×C4×D3—182no—
146C2×C4×D3—184no—
147C2×C4×D3—182no—
148C2×C4×D3—182no—
149C2×C4×D3—184no—
150C3×C22⋊C4—181no—
151C3×C22⋊C4—182no—
152#217—61no—
153#217—61no—
154#217—121no—
155#217—181no—
156(C2×D3)⋊C4—182no—
157C22×C32—24no—
158C22×C32—22no—
159C22×C32—21no—
160C22×C32—21no—
161C22×C32—22no—
162C22×C32—22no—
163C22×C32—42no—
164C22×C32—41no—
165C22×C32—41no—
166C22×C32—42no—
167C22×C32—42no—
168C22×C32—44no—
169C22×C32—82no—
170C22×C32(C2×D3)⋊C412yes—
171C22×C32C2×D1211yescharacteristic
172C2×C3⋊D3—22no—
173C2×C3⋊D3—24no—
174C2×C3⋊D3—24no—
175C2×C3⋊D3(C2×D3)⋊C412yes—
176C2×C3⋊D3C2×C4×D311yescharacteristic
177C2×C3⋊D3#23714yes—
178C2×C3⋊D3C2×#6711yescharacteristic
179C2×C3×D3—62no—
180C2×C3×D3—61no—
181C2×C3×D3—61no—
182C2×C3×D3—61no—
183C2×C3×D3—61no—
184C2×C3×D3—64no—
185C2×C3×D3—64no—
186C2×C3×D3—62no—
187C2×C3×D3—68no—
188C2×C3×D3—68no—
189C2×C3×D3—64no—
190C2×C3×D3—124no—
191C2×C3×D3—124no—
192C2×C3×D3—122no—
193#157—42no—
194C3⋊Dic3—24no—
195C3⋊Dic3—42no—
196C3×Dic3—64no—
197C3×Dic3—64no—
198C3×Dic3—124no—
199C3×Dic3—122no—
200C3×Dic3—122no—
201C3×Dic3—242no—
202C23×C4—271no—
203C2×C22⋊C4—91no—
204C2×C22⋊C4—271no—
205#95—274no—
206C33#38111yescharacteristic, complemented, Sylow
207C23×C3—21no—
208C23×C3—21no—
209C23×C3—41no—
210C23×C3—41no—
211C23×C3—91no—
212C23×C3—91no—
213C23×C3—92no—
214C23×C3—92no—
215C23×C3—184no—
216C23×C3GO+(4,𝔽2)11yescharacteristic
217C22×D3—62no—
218C22×D3—61no—
219C22×D3—62no—
220C22×D3—61no—
221C22×D3—61no—
222C22×D3—62no—
223C22×D3—61no—
224C22×D3—62no—
225C22×D3—68no—
226C22×D3—124no—
227C2×C3×C4—182no—
228C2×C3×C4—184no—
229C2×C3×C4—182no—
230C2×C3×C4—182no—
231C2×C3×C4—184no—
232C2×Dic3—62no—
233C2×Dic3—64no—
234C2×Dic3—62no—
235C2×Dic3—64no—
236C2×Dic3—62no—
237C2×Dic3—62no—
238C2×Dic3—94no—
239C2×Dic3—122no—
240C2×Dic3—124no—
241C2×Dic3—122no—
242C2×Dic3—182no—
243C2×Dic3—184no—
244C2×Dic3—184no—
245C4×D3—184no—
246C4×D3—188no—
247C4×D3—184no—
248C4×D3—364no—
249C2×C32—24no—
250C2×C32—22no—
251C2×C32—21no—
252C2×C32—21no—
253C2×C32—22no—
254C2×C32—22no—
255C2×C32—41no—
256C2×C32—41no—
257C2×C32—42no—
258C2×C32—42no—
259C2×C32—42no—
260C2×C32—44no—
261C2×C32—82no—
262C2×C32#88612yes—
263C2×C32C2×(C2×D3)⋊C411yescharacteristic
264C3⋊D3—22no—
265C3⋊D3#87712yes—
266C3⋊D3#93512yes—
267C3×D3—62no—
268C3×D3—62no—
269C3×D3—62no—
270C3×D3—62no—
271C3×D3—64no—
272C3×D3—122no—
273C24—91no—
274C22×C4—92no—
275C22×C4—181no—
276C22×C4—272no—
277C22×C4—274no—
278C22×C4—272no—
279C22×C4—541no—
280C22×C4—542no—
281C22⋊C4—182no—
282C22⋊C4—181no—
283C22⋊C4—541no—
284C22×C3—24no—
285C22×C3—22no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—22no—
289C22×C3—22no—
290C22×C3—42no—
291C22×C3—44no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3—42no—
295C22×C3—42no—
296C22×C3—82no—
297C22×C3—91no—
298C22×C3—91no—
299C22×C3—94no—
300C22×C3—92no—
301C22×C3—184no—
302C22×C3—184no—
303C22×C3—182no—
304C22×C3D32⋊C412yes—
305C22×C3C2×GO+(4,𝔽2)11yescharacteristic
306C2×D3—62no—
307C2×D3—61no—
308C2×D3—61no—
309C2×D3—64no—
310C2×D3—61no—
311C2×D3—64no—
312C2×D3—61no—
313C2×D3—62no—
314C2×D3—68no—
315C2×D3—68no—
316C2×D3—64no—
317C2×D3—124no—
318C2×D3—124no—
319C2×D3—122no—
320C3×C4—184no—
321C3×C4—362no—
322C3×C4—362no—
323Dic3—64no—
324Dic3—64no—
325Dic3—124no—
326Dic3—122no—
327Dic3—122no—
328Dic3—242no—
329C32—21no—
330C32—21no—
331C32—41no—
332C32—41no—
333C32#494311yescharacteristic, complemented
334C23—91no—
335C23—91no—
336C23—92no—
337C23—92no—
338C23—184no—
339C23#157⋊D311yescharacteristic
340C2×C4—182no—
341C2×C4—184no—
342C2×C4—182no—
343C2×C4—184no—
344C2×C4—182no—
345C2×C4—274no—
346C2×C4—542no—
347C2×C4—544no—
348C2×C4—544no—
349C2×C3—24no—
350C2×C3—22no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—22no—
354C2×C3—22no—
355C2×C3—42no—
356C2×C3—44no—
357C2×C3—41no—
358C2×C3—41no—
359C2×C3—42no—
360C2×C3—42no—
361C2×C3—82no—
362C2×C3—92no—
363C2×C3—92no—
364C2×C3—182no—
365C2×C3C2×D32⋊C411yescharacteristic
366C2×C3#6423312yes—
367D3—62no—
368D3—62no—
369D3—62no—
370D3—62no—
371D3—64no—
372D3—122no—
373C22—22no—
374C22—91no—
375C22—91no—
376C22—94no—
377C22—92no—
378C22—184no—
379C22—184no—
380C22—182no—
381C22#8893912yes—
382C22C2×#157⋊D311yescharacteristic
383C4—184no—
384C4—362no—
385C4—362no—
386C3—21no—
387C3—21no—
388C3—41no—
389C3—41no—
390C3#1059348811yescharacteristic, complemented
391C2—22no—
392C2—92no—
393C2—92no—
394C2—182no—
395C2C2×#8893911yescharacteristic
396C2#1172434612yes—
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutatorC2×C3×C3⋊D3C22×C4
FittingC23×C33D4
socleC22×C33C2×D4
prime radicalC23×C3×C3⋊D3C22
cosocleC22×C3×C3⋊D3C23
hypercenterC23#157⋊D3
hypocenterC33#381
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
4. #11722791C2semidirect
11. C2×#6294C4semidirect
206. C33#381semidirect
333. C32#4943semidirect
390. C3#10593488semidirect

Maximal subgroups

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

subgroupindex
C22×#62942
C2×C3×C22⋊#1572
#117237232
#117227912
#105934883
#49439