Finite Group Databasebeta

← #13165512

Subgroups — #13165512

Order 1728.

484 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 68 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×C3×#3775C211yesmaximal, characteristic
2Dic3×#627C211yesmaximal, characteristic, complemented
3#11722925C211yesmaximal, characteristic
4#11722797C211yesmaximal, characteristic, complemented
5#11722775C211yesmaximal, characteristic, complemented
6#11722773C211yesmaximal, characteristic
7#11722724C211yesmaximal, characteristic, complemented
8#10593432—31nomaximal
9C2×D3×C3⋊Dic3—21no—
10C2×C3×#627C2211yescharacteristic
11C2×C3×#625C2211yescharacteristic
12C2×C3⋊#622C2211yescharacteristic
13C2×C3×#622C2211yescharacteristic
14C3×#3775C414yescomplemented
15#88815C2211yescharacteristic
16#88815C2211yescharacteristic
17Dic3⋊(C3⋊Dic3)—21no—
18Dic3×C3⋊Dic3C2211yescharacteristic
19C3×(C2×D3)⋊Dic3—21no—
20C3×Dic32—21no—
21C2×#3775D311yescharacteristic
22#67⋊Dic3—61no—
23C4×#627—31no—
24#63902—31no—
25Dic3×Dic6—61no—
26#63777—31no—
27#63743—31no—
28#63740—31no—
29#63739—31no—
30C22×C32×D3—21no—
31C2×C3⋊(C3⋊Dic3)D411yescharacteristic
32C2×C3×C3⋊Dic3—21no—
33C2×C3×C3⋊Dic3—21no—
34C2×C3×C3⋊Dic3C2311yescharacteristic
35C2×C32×Dic3—21no—
36C2×C32×Dic3D411yescharacteristic
37D3×C3⋊Dic3—42no—
38C3×#627—21no—
39C3×#627C2×C412yescomplemented
40C3×#625—21no—
41C3×#625C2×C412yes—
42C3⋊#622—21no—
43C3×#622C2×C412yescomplemented
44#5164—91nomaximal
45C22×C3⋊Dic3—61no—
46C2×C3×#67—61no—
47C2×C3×Dic6—61no—
48C2×#627C2×D311yescharacteristic
49C2×#625C2×D311yescharacteristic
50C2×D3×Dic3—61no—
51C2×D3×Dic3—61no—
52C2×D3×Dic3—121no—
53C2×#622—31no—
54C2×#622C2×D311yescharacteristic
55#3775Dic314yescomplemented
56C4⋊(C3⋊Dic3)—61no—
57C4×C3⋊Dic3—31no—
58C3×#217—61no—
59C3×(C2×D3)⋊C4—61no—
60C3×C4×Dic3—61no—
61C3×C4×Dic3—61no—
62#3724—31no—
63#3724—31no—
64Dic3⋊Dic3—61no—
65Dic3⋊Dic3—61no—
66Dic3⋊Dic3—61no—
67Dic3⋊Dic3—61no—
68Dic3⋊Dic3—121no—
69(C2×D3)⋊Dic3—21no—
70Dic32—21no—
71Dic32—61no—
72Dic32—61no—
73C22×C33C2×D411yescharacteristic
74C2×C32×D3—22no—
75C2×C32×D3—41no—
76C2×C32×D3—41no—
77C3⋊(C3⋊Dic3)—21no—
78C3×C3⋊Dic3—41no—
79C3×C3⋊Dic3—41no—
80C3×C3⋊Dic3C22×C411yescharacteristic
81C3×C3⋊Dic3#4111yescharacteristic
82C32×Dic3—22no—
83C32×Dic3—21no—
84C2×C3×SD16—91no—
85Q8×Dic3—91no—
86D4×Dic3—91no—
87#67⋊C4—181no—
88C4×Dic6—181no—
89#835—91no—
90#832—91no—
91#818—91no—
92#815—91no—
93C23×C32—61no—
94C22×C3×D3—21no—
95C22×C3×D3—21no—
96C22×C3×D3—41no—
97C2×C32×C4—31no—
98C2×C32×C4—61no—
99C2×C3⋊Dic3—21no—
100C2×C3⋊Dic3—21no—
101C2×C3⋊Dic3—31no—
102C2×C3⋊Dic3—61no—
103C2×C3⋊Dic3—61no—
104C2×C3⋊Dic3—121no—
105C2×C3⋊Dic3—121no—
106C2×C3⋊Dic3—122no—
107C2×C3⋊Dic3C22×D311yescharacteristic
108C3×#67—62no—
109C3×#67—62no—
110C2×C3×Dic3—21no—
111C2×C3×Dic3—21no—
112C2×C3×Dic3—21no—
113C2×C3×Dic3—21no—
114C2×C3×Dic3—41no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3—61no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—121no—
122C2×C3×Dic3—121no—
123C3×Dic6—62no—
124C3×Dic6—62no—
125#627—21no—
126#627C2×Dic312yescomplemented
127#625—21no—
128#625C2×Dic312yes—
129D3×Dic3—122no—
130D3×Dic3—122no—
131D3×Dic3—124no—
132#622—61no—
133#622C2×Dic312yescomplemented
134#440—271noSylow
135C2×C33C4×D411yescharacteristic
136C2×C33AGL(1,ℤ/8)11yescharacteristic
137C2×C33#13711yescharacteristic
138C32×D3—42no—
139C2×C3×Q8—91no—
140C2×C3×D4—91no—
141C2×#67—61no—
142C22×Dic3—181no—
143C22×Dic3—181no—
144C22×Dic3—361no—
145C2×C4×D3—181no—
146C2×Dic6—61no—
147C3×SD16—94no—
148C2×C3×C8—91no—
149C3×C22⋊C4—181no—
150C3×C42—181no—
151#217—61no—
152(C2×D3)⋊C4—61no—
153C4⋊Dic3—91no—
154C4⋊Dic3—91no—
155C4⋊Dic3—181no—
156C4⋊Dic3—181no—
157C4⋊Dic3—361no—
158Dic3⋊C4—181no—
159Dic3⋊C4—181no—
160Dic3⋊C4—181no—
161C4×Dic3—61no—
162C4×Dic3—61no—
163C4×Dic3—91no—
164C4×Dic3—181no—
165C4×Dic3—181no—
166C2×C3⋊C8—91no—
167C22×C32—21no—
168C22×C32—21no—
169C22×C32—41no—
170C22×C32—41no—
171C22×C32—62no—
172C22×C32—121no—
173C22×C32—121no—
174C22×C32D3×D411yescharacteristic
175C2×C3×D3—22no—
176C2×C3×D3—22no—
177C2×C3×D3—42no—
178C2×C3×D3—41no—
179C2×C3×D3—41no—
180C2×C3×D3—42no—
181C2×C3×D3—42no—
182C2×C3×D3—41no—
183C2×C3×D3—41no—
184C32×C4—62no—
185C32×C4—61no—
186C3⋊Dic3—41no—
187C3⋊Dic3—41no—
188C3⋊Dic3—62no—
189C3⋊Dic3—61no—
190C3⋊Dic3—121no—
191C3⋊Dic3—122no—
192C3⋊Dic3—241no—
193C3⋊Dic3#23711yescharacteristic
194C3⋊Dic3C22×Dic311yescharacteristic
195C3×Dic3—22no—
196C3×Dic3—22no—
197C3×Dic3—22no—
198C3×Dic3—41no—
199C3×Dic3—42no—
200C3×Dic3—61no—
201C3×Dic3—61no—
202C3×Dic3—61no—
203C3×Dic3—61no—
204C3×Dic3—121no—
205C3×Dic3—121no—
206C3×Dic3—122no—
207C3×Dic3—121no—
208C3×Dic3—121no—
209C3×Dic3—241no—
210C2×SD16—91no—
211C4×Q8—271no—
212C4×D4—271no—
213#107—271no—
214#103—271no—
215#102—271no—
216#97—271no—
217C33#44011yescharacteristic, complemented, Sylow
218C23×C3—61no—
219C23×C3—61no—
220C23×C3—121no—
221C22×D3—21no—
222C3×Q8—92no—
223C3×Q8—181no—
224C3×D4—92no—
225C3×D4—181no—
226C2×C3×C4—61no—
227C2×C3×C4—61no—
228C2×C3×C4—61no—
229C2×C3×C4—61no—
230C2×C3×C4—91no—
231C2×C3×C4—121no—
232C2×C3×C4—181no—
233C2×C3×C4—181no—
234#67—62no—
235#67—62no—
236C2×Dic3—21no—
237C2×Dic3—61no—
238C2×Dic3—61no—
239C2×Dic3—61no—
240C2×Dic3—61no—
241C2×Dic3—61no—
242C2×Dic3—61no—
243C2×Dic3—91no—
244C2×Dic3—121no—
245C2×Dic3—121no—
246C2×Dic3—181no—
247C2×Dic3—181no—
248C2×Dic3—361no—
249C2×Dic3—361no—
250C2×Dic3—362no—
251C2×Dic3—362no—
252C2×Dic3—364no—
253C2×Dic3GO+(4,𝔽2)11yescharacteristic
254C4×D3—362no—
255Dic6—62no—
256Dic6—62no—
257C3×C8—92no—
258C3⋊C8—181no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32—21no—
262C2×C32—21no—
263C2×C32—21no—
264C2×C32—21no—
265C2×C32—41no—
266C2×C32—41no—
267C2×C32—41no—
268C2×C32—41no—
269C2×C32—41no—
270C2×C32—41no—
271C2×C32—122no—
272C2×C32#91411yescharacteristic
273C2×C32#91511yescharacteristic
274C2×C32D4×Dic311yescharacteristic
275C3×D3—42no—
276C3×D3—44no—
277C3×D3—42no—
278C2×Q8—91no—
279C2×D4—91no—
280C22×C4—541no—
281SD16—94no—
282C2×C8—91no—
283C2×C8—271no—
284C4⋊C4—271no—
285C4⋊C4—271no—
286C4⋊C4—541no—
287C22⋊C4—181no—
288C42—181no—
289C42—271no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3—62no—
295C22×C3—62no—
296C22×C3—122no—
297C22×C3—121no—
298C22×C3—121no—
299C22×C3—122no—
300C22×C3—122no—
301C22×C3—121no—
302C22×C3—121no—
303C22×C3C2×GO+(4,𝔽2)11yescharacteristic
304C2×D3—22no—
305C2×D3—41no—
306C2×D3—41no—
307C3×C4—62no—
308C3×C4—62no—
309C3×C4—62no—
310C3×C4—91no—
311C3×C4—91no—
312C3×C4—121no—
313C3×C4—122no—
314C3×C4—361no—
315C3×C4—361no—
316Dic3—22no—
317Dic3—21no—
318Dic3—61no—
319Dic3—61no—
320Dic3—61no—
321Dic3—61no—
322Dic3—121no—
323Dic3—121no—
324Dic3—122no—
325Dic3—121no—
326Dic3—121no—
327Dic3—182no—
328Dic3—181no—
329Dic3—241no—
330Dic3—361no—
331Dic3—362no—
332Dic3—721no—
333C32—21no—
334C32—21no—
335C32—41no—
336C32—41no—
337C32#516411yescharacteristic, complemented
338C23—61no—
339Q8—92no—
340Q8—181no—
341D4—92no—
342D4—181no—
343C2×C4—31no—
344C2×C4—61no—
345C2×C4—91no—
346C2×C4—181no—
347C2×C4—181no—
348C2×C4—271no—
349C2×C4—1082no—
350C8—92no—
351C8—541no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3—21no—
357C2×C3—21no—
358C2×C3—41no—
359C2×C3—41no—
360C2×C3—41no—
361C2×C3—41no—
362C2×C3—41no—
363C2×C3—41no—
364C2×C3—122no—
365C2×C3—124no—
366C2×C3—122no—
367C2×C3#6422411yescharacteristic
368C2×C3C4×GO+(4,𝔽2)11yescharacteristic
369C2×C3#6423911yescharacteristic
370D3—42no—
371C22—62no—
372C22—121no—
373C22—121no—
374C22D3×GO+(4,𝔽2)11yescharacteristic
375C4—62no—
376C4—61no—
377C4—91no—
378C4—91no—
379C4—361no—
380C4—361no—
381C4—541no—
382C3—21no—
383C3—21no—
384C3—41no—
385C3—41no—
386C3#1059343211yescharacteristic, complemented
387C2—122no—
388C2Dic3×GO+(4,𝔽2)11yescharacteristic
389C2#1172414111yescharacteristic
390C2#1172414211yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutatorC3×C3⋊Dic3C22×C4
Fitting, socleC22×C33C2×D4
prime radicalC2×C3×#625C22
cosocleC2×C3×C3⋊Dic3C23
hypocenterC33#440
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. Dic3×#627C2semidirect
4. #11722797C2semidirect
5. #11722775C2semidirect
7. #11722724C2semidirect
14. C3×#3775C4semidirect
39. C3×#627C2×C4semidirect
43. C3×#622C2×C4semidirect
55. #3775Dic3semidirect
126. #627C2×Dic3semidirect
133. #622C2×Dic3semidirect
217. C33#440semidirect
337. C32#5164semidirect
386. C3#10593432semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37752
Dic3×#6272
#117229252
#117227972
#117227752
#117227732
#117227242
#105934323
#51649