Finite Group Databasebeta

← #13167571

Subgroups — #13167571

Order 1728.

644 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 79 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×#89046C211yesmaximal, characteristic
2C22×C3×#622C211yesmaximal, characteristic, complemented
3#11723715C211yesmaximal, characteristic, complemented
4#11722794C212yesmaximal, complemented
5#11722793C212yesmaximal, complemented
6#10593498—31nomaximal
7C2×C3×#638—21no—
8C22×C3×C3⋊Dic3C2211yescharacteristic
9C2×#6296C2211yescharacteristic
10C2×#6295C2211yescharacteristic
11#89046—22no—
12#89046C414yescomplemented
13C2×#6294—21no—
14C2×C3×#622—22no—
15C2×C3×#622C2212yes—
16C3×#3757—21no—
17#88815C2212yes—
18#88813—21no—
19C2×#3805—31no—
20C2×#3804—61no—
21C22×#622D311yescharacteristic, complemented
22#63957—31no—
23#63955—61no—
24#63740—32no—
25#63739—32no—
26C23×C33D411yescharacteristic
27C22×C3×C3⋊D3—21no—
28C3×#638—24no—
29C2×C3×C3⋊Dic3—21no—
30C2×C3×C3⋊Dic3—22no—
31C2×C3×C3⋊Dic3C2×C412yes—
32C2×C3×C3⋊Dic3D411yescharacteristic
33C2×C3×C3⋊Dic3D412yes—
34C2×C3×C3⋊Dic3C2311yescharacteristic
35#6296—21no—
36#6296C2×C412yes—
37#6295—21no—
38#6295C2×C412yes—
39#6294—24no—
40C3×#622—22no—
41#5113—91nomaximal
42C2×#638—21no—
43C22×C3⋊Dic3C2×D311yescharacteristic
44C2×C3×#67—61no—
45C2×C3×#67—61no—
46C2×C3×#67—121no—
47C22×C3×Dic3—61no—
48C22×C3×Dic3—61no—
49C2×#627—31no—
50C2×#627—61no—
51C2×#626—61no—
52C2×#625—31no—
53C2×#624—61no—
54#3805—34no—
55#3805—62no—
56#3804—64no—
57#3804—64no—
58C2×D3×Dic3—61no—
59C2×D3×Dic3—61no—
60C2×#622—22no—
61C2×#622C2×D312yes—
62#3757—21no—
63C3×#217—61no—
64C3×#217—61no—
65C3×#217—121no—
66#3724—32no—
67Dic3⋊Dic3—62no—
68Dic32—62no—
69Dic32—61no—
70C22×C33—22no—
71C22×C33C22⋊C412yes—
72C22×C33C2×D411yescharacteristic
73C2×C3×C3⋊D3—22no—
74C2×C3×C3⋊D3—42no—
75C3×C3⋊Dic3—22no—
76C3×C3⋊Dic3—42no—
77C3×C3⋊Dic3C22⋊C412yes—
78C3×C3⋊Dic3C22×C411yescharacteristic
79C3×C3⋊Dic3C2×D411yescharacteristic
80C2×#237—181no—
81C2×#235—91no—
82C22×C3×C8—91no—
83#926—91no—
84#877—181no—
85#821—92no—
86#816—92no—
87C23×C32—21no—
88C23×C32—21no—
89C23×C32—41no—
90C23×C32—41no—
91C23×C32D1211yescharacteristic
92C22×C3⋊D3—21no—
93C22×C3×D3—61no—
94C22×C3×D3—61no—
95C22×C3×D3—121no—
96#638—24no—
97C2×C3⋊Dic3—21no—
98C2×C3⋊Dic3—22no—
99C2×C3⋊Dic3C4×D312yes—
100C2×C3⋊Dic3D1211yescharacteristic
101C2×C3⋊Dic3#6712yes—
102C2×C3⋊Dic3C22×D311yescharacteristic
103C3×#67—64no—
104C3×#67—64no—
105C3×#67—124no—
106C2×C3×Dic3—61no—
107C2×C3×Dic3—61no—
108C2×C3×Dic3—62no—
109C2×C3×Dic3—62no—
110C2×C3×Dic3—61no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—61no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—62no—
117C2×C3×Dic3—62no—
118C2×C3×Dic3—121no—
119C2×C3×Dic3—122no—
120#627—32no—
121#627—62no—
122#627—62no—
123#627—61no—
124#626—62no—
125#626—62no—
126#625—32no—
127#625—61no—
128#624—64no—
129D3×Dic3—64no—
130D3×Dic3—64no—
131#622—22no—
132#416—271noSylow
133C2×C33—22no—
134C2×C33C2×C22⋊C411yescharacteristic
135C2×C33#13512yes—
136C3×C3⋊D3—42no—
137C2×C3×D4—181no—
138C22×C3×C4—91no—
139C2×#67—61no—
140C2×#67—61no—
141C2×#67—121no—
142C2×#67—181no—
143C22×Dic3—61no—
144C22×Dic3—61no—
145C22×Dic3—181no—
146#237—184no—
147#237—184no—
148#235—94no—
149#235—182no—
150C2×D12—91no—
151C2×C4×D3—181no—
152C2×C4×D3—181no—
153C2×Dic6—91no—
154C2×Dic6—181no—
155C2×C3×C8—92no—
156C2×C3×C8—182no—
157C3×C22⋊C4—181no—
158#217—61no—
159#217—61no—
160#217—121no—
161C4⋊Dic3—92no—
162Dic3⋊C4—182no—
163C4×Dic3—181no—
164C4×Dic3—182no—
165C22×C32—21no—
166C22×C32—21no—
167C22×C32—22no—
168C22×C32—22no—
169C22×C32—22no—
170C22×C32—41no—
171C22×C32—41no—
172C22×C32—42no—
173C22×C32—42no—
174C22×C32—42no—
175C22×C32—42no—
176C22×C32—82no—
177C22×C32—82no—
178C22×C32(C2×D3)⋊C412yes—
179C22×C32C2×D1211yescharacteristic
180C2×C3⋊D3—22no—
181C2×C3⋊D3—42no—
182C2×C3×D3—62no—
183C2×C3×D3—62no—
184C2×C3×D3—122no—
185C2×C3×D3—122no—
186C2×C3×D3—122no—
187C2×C3×D3—242no—
188C3⋊Dic3—22no—
189C3⋊Dic3—42no—
190C3⋊Dic3(C2×D3)⋊C412yes—
191C3⋊Dic3C2×C4×D311yescharacteristic
192C3⋊Dic3C2×#6711yescharacteristic
193C3×Dic3—62no—
194C3×Dic3—62no—
195C3×Dic3—61no—
196C3×Dic3—61no—
197C3×Dic3—61no—
198C3×Dic3—61no—
199C3×Dic3—62no—
200C3×Dic3—62no—
201C3×Dic3—122no—
202C3×Dic3—122no—
203C3×Dic3—122no—
204C3×Dic3—242no—
205C2×#41—271no—
206C22×C8—91no—
207#117—271no—
208#103—272no—
209#102—272no—
210C33#41611yescharacteristic, complemented, Sylow
211C23×C3—21no—
212C23×C3—21no—
213C23×C3—41no—
214C23×C3—41no—
215C23×C3—181no—
216C23×C3GO+(4,𝔽2)11yescharacteristic
217C22×D3—61no—
218C22×D3—61no—
219C22×D3—121no—
220C3×D4—184no—
221C2×C3×C4—91no—
222C2×C3×C4—91no—
223C2×C3×C4—92no—
224C2×C3×C4—92no—
225C2×C3×C4—181no—
226C2×C3×C4—182no—
227#67—64no—
228#67—64no—
229#67—124no—
230#67—182no—
231#67—182no—
232C2×Dic3—62no—
233C2×Dic3—62no—
234C2×Dic3—61no—
235C2×Dic3—61no—
236C2×Dic3—62no—
237C2×Dic3—62no—
238C2×Dic3—61no—
239C2×Dic3—61no—
240C2×Dic3—61no—
241C2×Dic3—61no—
242C2×Dic3—62no—
243C2×Dic3—62no—
244C2×Dic3—121no—
245C2×Dic3—122no—
246C2×Dic3—184no—
247D12—92no—
248D12—181no—
249C4×D3—184no—
250C4×D3—184no—
251Dic6—92no—
252Dic6—182no—
253Dic6—181no—
254Dic6—182no—
255C3×C8—182no—
256C2×C32—21no—
257C2×C32—21no—
258C2×C32—22no—
259C2×C32—22no—
260C2×C32—22no—
261C2×C32—41no—
262C2×C32—41no—
263C2×C32—42no—
264C2×C32—42no—
265C2×C32—42no—
266C2×C32—42no—
267C2×C32—82no—
268C2×C32—82no—
269C2×C32#90412yes—
270C2×C32C2×(C2×D3)⋊C411yescharacteristic
271C3⋊D3—42no—
272C3×D3—122no—
273C3×D3—122no—
274C3×D3—242no—
275#41—274no—
276#41—542no—
277C2×Q8—271no—
278C2×D4—181no—
279C2×D4—271no—
280C22×C4—91no—
281C22×C4—541no—
282C2×C8—92no—
283C2×C8—182no—
284C4⋊C4—272no—
285C22⋊C4—181no—
286C42—541no—
287C22×C3—22no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—22no—
291C22×C3—22no—
292C22×C3—42no—
293C22×C3—42no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—42no—
297C22×C3—42no—
298C22×C3—82no—
299C22×C3—82no—
300C22×C3—182no—
301C22×C3—362no—
302C22×C3D32⋊C412yes—
303C22×C3C2×GO+(4,𝔽2)11yescharacteristic
304C2×D3—62no—
305C2×D3—62no—
306C2×D3—122no—
307C2×D3—122no—
308C2×D3—122no—
309C2×D3—242no—
310C3×C4—91no—
311C3×C4—91no—
312C3×C4—92no—
313C3×C4—182no—
314C3×C4—362no—
315Dic3—61no—
316Dic3—61no—
317Dic3—62no—
318Dic3—62no—
319Dic3—61no—
320Dic3—61no—
321Dic3—62no—
322Dic3—62no—
323Dic3—122no—
324Dic3—122no—
325Dic3—122no—
326Dic3—242no—
327C32—21no—
328C32—21no—
329C32—41no—
330C32—41no—
331C32#511311yescharacteristic, complemented
332C23—181no—
333C23#157⋊D311yescharacteristic
334Q8—272no—
335Q8—541no—
336D4—184no—
337D4—272no—
338D4—541no—
339C2×C4—91no—
340C2×C4—91no—
341C2×C4—92no—
342C2×C4—92no—
343C2×C4—181no—
344C2×C4—182no—
345C2×C4—544no—
346C8—182no—
347C2×C3—22no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—22no—
351C2×C3—22no—
352C2×C3—42no—
353C2×C3—42no—
354C2×C3—41no—
355C2×C3—41no—
356C2×C3—42no—
357C2×C3—42no—
358C2×C3—82no—
359C2×C3—82no—
360C2×C3—362no—
361C2×C3C2×D32⋊C411yescharacteristic
362C2×C3#6423712yes—
363D3—122no—
364D3—122no—
365D3—242no—
366C22—22no—
367C22—182no—
368C22—362no—
369C22#8893912yes—
370C22C2×#157⋊D311yescharacteristic
371C4—91no—
372C4—92no—
373C4—91no—
374C4—182no—
375C4—362no—
376C3—21no—
377C3—21no—
378C3—41no—
379C3—41no—
380C3#1059349811yescharacteristic, complemented
381C2—22no—
382C2—362no—
383C2C2×#8893911yescharacteristic
384C2#1172435112yes—
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutatorC3×C3⋊Dic3C22×C4
FittingC23×C33D4
socleC22×C33C2×D4
prime radicalC2×#89046C2
cosocleC2×C3×C3⋊Dic3C23
hypercenterC23#157⋊D3
hypocenterC33#416
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C22×C3×#622C2semidirect
3. #11723715C2semidirect
4. #11722794C2semidirect
5. #11722793C2semidirect
12. #89046C4semidirect
21. C22×#622D3semidirect
210. C33#416semidirect
331. C32#5113semidirect
380. C3#10593498semidirect

Maximal subgroups

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

subgroupindex
C2×#890462
C22×C3×#6222
#117237152
#117227942
#117227932
#105934983
#51139