Finite Group Databasebeta

← #13167502

Subgroups — #13167502

Order 1728.

484 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 63 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×#88942C211yesmaximal, characteristic
2C4×#6295C211yesmaximal, characteristic
3#11723665C211yesmaximal, characteristic, complemented
4C3×#63777C211yesmaximal, characteristic, complemented
5#11722797C211yesmaximal, characteristic, complemented
6#11722794C211yesmaximal, characteristic, complemented
7#11722793C211yesmaximal, characteristic
8#10593431—31nomaximal
9C2×C3×C4×C3⋊D3—21no—
10C2×#6296C2211yescharacteristic
11C2×#6295C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13C2×C3×#622C2211yescharacteristic
14#88942C414yescomplemented
15C3×#3750—21no—
16C3×C4×C3⋊Dic3C2211yescharacteristic
17#88815C2211yescharacteristic
18#88815C2211yescharacteristic
19#88814—21no—
20#88813—21no—
21C2×#3775—31no—
22C4×#626—61no—
23C4×#625—31no—
24#63838—31no—
25#63837—61no—
26#63777D311yescharacteristic, complemented
27#63743—31no—
28#63740—31no—
29#63739—31no—
30C22×C3×C3⋊D3—21no—
31C2×C33×C4D411yescharacteristic
32C2×C3×C3⋊Dic3—21no—
33C2×C3×C3⋊Dic3—21no—
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3D411yescharacteristic
36C2×C3×C3⋊Dic3C2311yescharacteristic
37C3×C4×C3⋊D3—42no—
38#6296—21no—
39#6296C2×C412yescomplemented
40#6295—21no—
41#6295C2×C412yes—
42C3×#622—21no—
43C3×#622C2×C412yescomplemented
44#4710—91nomaximal
45C2×C4×C3⋊D3—21no—
46C2×C3×C4×D3—61no—
47C2×C3×C4×D3—61no—
48C2×C3×C4×D3—121no—
49C2×#627—31no—
50C2×#627—61no—
51C2×#626—61no—
52C2×#625—31no—
53C2×#622C2×D311yescharacteristic
54C2×#622C2×D311yescharacteristic
55#3775—34no—
56#3750—21no—
57C4×C3⋊Dic3C2×D311yescharacteristic
58C3×Dic3⋊C4—61no—
59C3×Dic3⋊C4—61no—
60C3×Dic3⋊C4—121no—
61C3×C4×Dic3—61no—
62C3×C4×Dic3—61no—
63#3724—31no—
64#3724—31no—
65Dic3⋊Dic3—61no—
66Dic3⋊Dic3—61no—
67#3722—61no—
68(C2×D3)⋊Dic3—61no—
69(C2×D3)⋊Dic3—61no—
70Dic32—61no—
71Dic32—61no—
72Dic32—61no—
73C22×C33C2×D411yescharacteristic
74C2×C3×C3⋊D3—22no—
75C2×C3×C3⋊D3—41no—
76C2×C3×C3⋊D3—41no—
77C33×C4—21no—
78C3×C3⋊Dic3—22no—
79C3×C3⋊Dic3—21no—
80C3×C3⋊Dic3—41no—
81C3×C3⋊Dic3—41no—
82C3×C3⋊Dic3C22×C411yescharacteristic
83C3×C3⋊Dic3#4111yescharacteristic
84C4×#67—181no—
85C2×#204—91no—
86#887—181no—
87C4×D12—91no—
88C4×Dic6—91no—
89C3×#97—91no—
90#821—91no—
91#818—91no—
92#816—91no—
93C22×C3⋊D3—21no—
94C22×C3×D3—61no—
95C22×C3×D3—61no—
96C22×C3×D3—121no—
97C2×C32×C4—21no—
98C2×C32×C4—21no—
99C2×C32×C4—41no—
100C2×C32×C4—41no—
101C2×C32×C4D1211yescharacteristic
102C2×C3⋊Dic3—21no—
103C2×C3⋊Dic3—21no—
104C2×C3⋊Dic3—21no—
105C2×C3⋊Dic3D1211yescharacteristic
106C2×C3⋊Dic3C22×D311yescharacteristic
107C4×C3⋊D3—42no—
108C2×C3×Dic3—61no—
109C2×C3×Dic3—61no—
110C2×C3×Dic3—61no—
111C2×C3×Dic3—61no—
112C2×C3×Dic3—61no—
113C2×C3×Dic3—61no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3—61no—
118C2×C3×Dic3—121no—
119C2×C3×Dic3—121no—
120C2×C3×Dic3—121no—
121C3×C4×D3—122no—
122C3×C4×D3—122no—
123C3×C4×D3—124no—
124#627—32no—
125#627—62no—
126#627—62no—
127#627—61no—
128#626—62no—
129#626—62no—
130#625—32no—
131#625—61no—
132#622—21no—
133#622C4×D312yescomplemented
134#440—271noSylow
135C2×C33C4×D411yescharacteristic
136C2×C33AGL(1,ℤ/8)11yescharacteristic
137C2×C33#13711yescharacteristic
138C3×C3⋊D3—42no—
139C22×C3×C4—181no—
140C2×#67—181no—
141C2×D12—91no—
142C2×C4×D3—61no—
143C2×C4×D3—61no—
144C2×C4×D3—121no—
145C2×Dic6—91no—
146C2×Dic6—181no—
147C2×C3×C8—91no—
148C2×C3×C8—91no—
149C3×C4⋊C4—181no—
150C3×C42—91no—
151#217—181no—
152(C2×D3)⋊C4—181no—
153(C2×D3)⋊C4—181no—
154C4⋊Dic3—91no—
155C4⋊Dic3—91no—
156Dic3⋊C4—61no—
157Dic3⋊C4—61no—
158Dic3⋊C4—121no—
159Dic3⋊C4—181no—
160Dic3⋊C4—181no—
161C4×Dic3—61no—
162C4×Dic3—61no—
163C4×Dic3—181no—
164C4×Dic3—181no—
165C4×Dic3—181no—
166#204—94no—
167C22×C32—21no—
168C22×C32—21no—
169C22×C32—41no—
170C22×C32—41no—
171C22×C32C2×D1211yescharacteristic
172C2×C3⋊D3—22no—
173C2×C3⋊D3—41no—
174C2×C3⋊D3—41no—
175C2×C3×D3—62no—
176C2×C3×D3—62no—
177C2×C3×D3—122no—
178C2×C3×D3—121no—
179C2×C3×D3—121no—
180C2×C3×D3—122no—
181C2×C3×D3—122no—
182C2×C3×D3—121no—
183C2×C3×D3—121no—
184C32×C4—22no—
185C32×C4—21no—
186C32×C4—41no—
187C32×C4—42no—
188C32×C4—81no—
189C3⋊Dic3—22no—
190C3⋊Dic3—21no—
191C3⋊Dic3—41no—
192C3⋊Dic3—41no—
193C3⋊Dic3C2×C4×D311yescharacteristic
194C3⋊Dic3#23511yescharacteristic
195C3×Dic3—61no—
196C3×Dic3—61no—
197C3×Dic3—61no—
198C3×Dic3—61no—
199C3×Dic3—62no—
200C3×Dic3—62no—
201C3×Dic3—62no—
202C3×Dic3—121no—
203C3×Dic3—122no—
204C3×Dic3—121no—
205C3×Dic3—121no—
206C3×Dic3—122no—
207C3×Dic3—121no—
208C3×Dic3—121no—
209C3×Dic3—241no—
210C2×SD16—271no—
211C4×Q8—271no—
212C4×D4—271no—
213#107—271no—
214#103—271no—
215#102—271no—
216#97—91no—
217C33#44011yescharacteristic, complemented, Sylow
218C23×C3—181no—
219C22×D3—61no—
220C22×D3—61no—
221C22×D3—121no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—91no—
227C2×C3×C4—91no—
228C2×C3×C4—181no—
229C2×C3×C4—181no—
230C2×C3×C4—181no—
231C2×C3×C4—362no—
232C2×C3×C4GO+(4,𝔽2)11yescharacteristic
233#67—182no—
234#67—182no—
235C2×Dic3—61no—
236C2×Dic3—61no—
237C2×Dic3—61no—
238C2×Dic3—61no—
239C2×Dic3—61no—
240C2×Dic3—61no—
241C2×Dic3—61no—
242C2×Dic3—61no—
243C2×Dic3—61no—
244C2×Dic3—61no—
245C2×Dic3—121no—
246C2×Dic3—121no—
247C2×Dic3—121no—
248D12—92no—
249D12—181no—
250C4×D3—122no—
251C4×D3—122no—
252C4×D3—124no—
253Dic6—92no—
254Dic6—182no—
255Dic6—181no—
256Dic6—182no—
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×C32C4×D1211yescharacteristic
272C2×C32#90811yescharacteristic
273C2×C32#90911yescharacteristic
274C3⋊D3—42no—
275C3×D3—122no—
276C3×D3—124no—
277C3×D3—122no—
278C2×Q8—271no—
279C2×D4—271no—
280C22×C4—181no—
281SD16—274no—
282C2×C8—91no—
283C2×C8—91no—
284C4⋊C4—181no—
285C4⋊C4—271no—
286C4⋊C4—271no—
287C22⋊C4—541no—
288C42—91no—
289C42—541no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3—182no—
295C22×C3—361no—
296C22×C3—361no—
297C22×C3C2×GO+(4,𝔽2)11yescharacteristic
298C2×D3—62no—
299C2×D3—62no—
300C2×D3—122no—
301C2×D3—121no—
302C2×D3—121no—
303C2×D3—122no—
304C2×D3—122no—
305C2×D3—121no—
306C2×D3—121no—
307C3×C4—22no—
308C3×C4—21no—
309C3×C4—41no—
310C3×C4—42no—
311C3×C4—81no—
312C3×C4—91no—
313C3×C4—91no—
314C3×C4—182no—
315C3×C4—181no—
316C3×C4—361no—
317C3×C4—361no—
318Dic3—61no—
319Dic3—61no—
320Dic3—61no—
321Dic3—61no—
322Dic3—62no—
323Dic3—62no—
324Dic3—62no—
325Dic3—121no—
326Dic3—122no—
327Dic3—121no—
328Dic3—121no—
329Dic3—122no—
330Dic3—121no—
331Dic3—121no—
332Dic3—241no—
333C32—21no—
334C32—21no—
335C32—41no—
336C32—41no—
337C32#471011yescharacteristic, complemented
338C23—181no—
339Q8—272no—
340Q8—541no—
341D4—272no—
342D4—541no—
343C2×C4—91no—
344C2×C4—91no—
345C2×C4—181no—
346C2×C4—181no—
347C2×C4—181no—
348C2×C4—362no—
349C2×C4#157⋊D311yescharacteristic
350C8—92no—
351C8—181no—
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—362no—
365C2×C3#6422611yescharacteristic
366C2×C3C4×GO+(4,𝔽2)11yescharacteristic
367C2×C3#6423611yescharacteristic
368D3—122no—
369D3—124no—
370D3—122no—
371C22—182no—
372C22—361no—
373C22—361no—
374C22C2×#157⋊D311yescharacteristic
375C4—21no—
376C4—91no—
377C4—91no—
378C4—181no—
379C4—182no—
380C4—361no—
381C4—361no—
382C3—21no—
383C3—21no—
384C3—41no—
385C3—41no—
386C3#1059343111yescharacteristic, complemented
387C2—362no—
388C2#1172433811yescharacteristic
389C2C4×#157⋊D311yescharacteristic
390C2#1172435011yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutatorC3×C3⋊Dic3C22×C4
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×#6295C22
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4#157⋊D3
hypocenterC33#440
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. #11723665C2semidirect
4. C3×#63777C2semidirect
5. #11722797C2semidirect
6. #11722794C2semidirect
14. #88942C4semidirect
26. #63777D3semidirect
39. #6296C2×C4semidirect
43. C3×#622C2×C4semidirect
133. #622C4×D3semidirect
217. C33#440semidirect
337. C32#4710semidirect
386. C3#10593431semidirect

Maximal subgroups

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

subgroupindex
C2×#889422
C4×#62952
#117236652
C3×#637772
#117227972
#117227942
#117227932
#105934313
#47109