Finite Group Databasebeta

← #13165490

Subgroups — #13165490

Order 1728.

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

normal subgroups only

This group has 492 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×#88926C211yesmaximal, characteristic
2#11722918C211yesmaximal, characteristic
3#11722912C211yesmaximal, characteristic
4#11722797C211yesmaximal, characteristic, complemented
5#11722779C211yesmaximal, characteristic, complemented
6#11722778C211yesmaximal, characteristic, complemented
7C3×#63742C211yesmaximal, characteristic, complemented
8#10593394—31nomaximal
9C2×#6289—21no—
10C2×#6289C2211yescharacteristic
11C2×C3⋊#622C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13#88926C414yescomplemented
14#88815C2211yescharacteristic
15#88815C2211yescharacteristic
16#88804—21no—
17C3×#3724C2211yescharacteristic
18C3×#3724C2211yescharacteristic
19C3×Dic3⋊Dic3—21no—
20C3×Dic32—21no—
21C2×#3788—31no—
22#63900—31no—
23#63883—61no—
24#63874—31no—
25#63836—61no—
26#63743—31no—
27#63743—31no—
28#63742—31no—
29#63742D311yescharacteristic, complemented
30C22×C3⋊(C3⋊D3)D411yescharacteristic
31C2×C3×C3⋊Dic3—21no—
32C2×C3×C3⋊Dic3—21no—
33C2×C3×C3⋊Dic3C2311yescharacteristic
34C2×C32×Dic3—21no—
35C2×C32×Dic3—21no—
36C2×C32×Dic3D411yescharacteristic
37#6289—42no—
38#6289C2×C412yes—
39#6289Q812yes—
40C3⋊#622C2×C412yescomplemented
41C3×#622C2×C412yescomplemented
42#4861—91nomaximal
43C2×C4×C3⋊D3—31no—
44C2×C4×C3⋊D3—61no—
45C2×#624—61no—
46C2×#624—61no—
47C2×#624—61no—
48C2×#624—61no—
49C2×#624—121no—
50C2×#622—31no—
51C2×#622C2×D311yescharacteristic
52#3788—34no—
53#3752—61no—
54C3×C4⋊Dic3—61no—
55C3×Dic3⋊C4—61no—
56C3×Dic3⋊C4—61no—
57C3×Dic3⋊C4—61no—
58C3×C4×Dic3—61no—
59C3×C4×Dic3—61no—
60#3724—31no—
61#3724—31no—
62#3724C2×D311yescharacteristic
63#3724C2×D311yescharacteristic
64Dic3⋊Dic3—21no—
65Dic3⋊Dic3—61no—
66Dic3⋊Dic3—61no—
67#3722—61no—
68#3722—61no—
69#3722—121no—
70Dic32—21no—
71C22×C33C2×D411yescharacteristic
72C2×C3⋊(C3⋊D3)—21no—
73C2×C3⋊(C3⋊D3)—21no—
74C2×C3⋊(C3⋊D3)C4⋊C412yes—
75C3×C3⋊Dic3—41no—
76C3×C3⋊Dic3—41no—
77C3×C3⋊Dic3C22×C411yescharacteristic
78C3×C3⋊Dic3C2×Q811yescharacteristic
79C32×Dic3—41no—
80C32×Dic3—41no—
81C32×Dic3C4⋊C412yes—
82C2×ΓL(1,𝔽25)—91no—
83#892—91no—
84D3×C4⋊C4—91no—
85D3×C4⋊C4—181no—
86#872—181no—
87C3×#106—91no—
88#818—91no—
89#808—91no—
90#807—91no—
91C22×C3⋊D3—31no—
92C22×C3⋊D3—61no—
93C22×C3⋊D3—61no—
94C22×C3⋊D3—121no—
95C22×C3⋊D3—121no—
96C2×C32×C4—31no—
97C2×C32×C4—61no—
98C2×C32×C4—61no—
99C2×C3⋊Dic3—21no—
100C2×C3⋊Dic3—21no—
101C2×C3⋊Dic3C22×D311yescharacteristic
102C4×C3⋊D3—32no—
103C4×C3⋊D3—32no—
104C4×C3⋊D3—122no—
105C2×C3×Dic3—21no—
106C2×C3×Dic3—21no—
107C2×C3×Dic3—21no—
108C2×C3×Dic3—21no—
109C2×C3×Dic3—21no—
110C2×C3×Dic3—21no—
111C2×C3×Dic3—41no—
112C2×C3×Dic3—41no—
113C2×C3×Dic3—61no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3—61no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3—121no—
120C2×C3×Dic3—121no—
121#624—62no—
122#624—62no—
123#624—62no—
124#624—62no—
125#624—122no—
126#624—122no—
127#624—124no—
128#622—32no—
129#622C4×D312yescomplemented
130#428—271noSylow
131C2×C33C2×C4⋊C411yescharacteristic
132C2×C33AGL(1,ℤ/8)11yescharacteristic
133C2×C33#13711yescharacteristic
134C3⋊(C3⋊D3)—22no—
135C2×C4×D3—91no—
136C2×C4×D3—181no—
137C2×C4×D3—181no—
138C2×C4×D3—181no—
139C2×C4×D3—181no—
140C2×C4×D3—181no—
141C2×C4×D3—361no—
142C2×C3×C8—91no—
143C3×C4⋊C4—91no—
144C3×C4⋊C4—91no—
145C3×C4⋊C4—181no—
146C3×C42—181no—
147(C2×D3)⋊C4—181no—
148(C2×D3)⋊C4—181no—
149(C2×D3)⋊C4—181no—
150(C2×D3)⋊C4—361no—
151C4⋊Dic3—61no—
152C4⋊Dic3—91no—
153C4⋊Dic3—91no—
154Dic3⋊C4—61no—
155Dic3⋊C4—61no—
156Dic3⋊C4—61no—
157Dic3⋊C4—181no—
158Dic3⋊C4—181no—
159C4×Dic3—61no—
160C4×Dic3—61no—
161C2×C3⋊C8—91no—
162ΓL(1,𝔽25)—94no—
163C22×C32—21no—
164C22×C32—21no—
165C22×C32—41no—
166C22×C32—41no—
167C22×C32D3×D411yescharacteristic
168C2×C3⋊D3—32no—
169C2×C3⋊D3—62no—
170C2×C3⋊D3—62no—
171C2×C3⋊D3—62no—
172C2×C3⋊D3—62no—
173C2×C3⋊D3—61no—
174C2×C3⋊D3—61no—
175C2×C3⋊D3—122no—
176C2×C3⋊D3—122no—
177C2×C3⋊D3—121no—
178C2×C3⋊D3—121no—
179C2×C3⋊D3—122no—
180C2×C3⋊D3—122no—
181C2×C3⋊D3—241no—
182C2×C3⋊D3—241no—
183C32×C4—32no—
184C32×C4—121no—
185C32×C4—121no—
186C3⋊Dic3—41no—
187C3⋊Dic3—41no—
188C3⋊Dic3C2×C4×D311yescharacteristic
189C3⋊Dic3D3×Q811yescharacteristic
190C3×Dic3—22no—
191C3×Dic3—22no—
192C3×Dic3—41no—
193C3×Dic3—41no—
194C3×Dic3—42no—
195C3×Dic3—41no—
196C3×Dic3—41no—
197C3×Dic3—61no—
198C3×Dic3—61no—
199C3×Dic3—61no—
200C3×Dic3—61no—
201C3×Dic3—81no—
202C3×Dic3—121no—
203C3×Dic3—121no—
204C3×Dic3—122no—
205C3×Dic3—121no—
206C3×Dic3—121no—
207C3×Dic3—241no—
208C2×M16—271no—
209#117—271no—
210C2×C4⋊C4—271no—
211#107—271no—
212#107—271no—
213#106—91no—
214#106—271no—
215C33#42811yescharacteristic, complemented, Sylow
216C22×D3—91no—
217C22×D3—181no—
218C22×D3—181no—
219C22×D3—361no—
220C22×D3—361no—
221C2×C3×C4—61no—
222C2×C3×C4—61no—
223C2×C3×C4—61no—
224C2×C3×C4—61no—
225C2×C3×C4—61no—
226C2×C3×C4—61no—
227C2×C3×C4—91no—
228C2×C3×C4—121no—
229C2×C3×C4—121no—
230C2×C3×C4—181no—
231C2×C3×C4—181no—
232C2×Dic3—21no—
233C2×Dic3—21no—
234C2×Dic3—61no—
235C2×Dic3—61no—
236C2×Dic3—61no—
237C2×Dic3—61no—
238C2×Dic3—61no—
239C2×Dic3—61no—
240C2×Dic3—121no—
241C2×Dic3—121no—
242C2×Dic3GO+(4,𝔽2)11yescharacteristic
243C4×D3—92no—
244C4×D3—92no—
245C4×D3—182no—
246C4×D3—182no—
247C4×D3—182no—
248C4×D3—182no—
249C4×D3—362no—
250C4×D3—362no—
251C4×D3—362no—
252C4×D3—364no—
253C3×C8—92no—
254C3⋊C8—92no—
255C2×C32—21no—
256C2×C32—21no—
257C2×C32—21no—
258C2×C32—21no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32—41no—
262C2×C32—41no—
263C2×C32—41no—
264C2×C32—41no—
265C2×C32—41no—
266C2×C32—41no—
267C2×C32D3×C4⋊C411yescharacteristic
268C2×C32#91411yescharacteristic
269C2×C32#91511yescharacteristic
270C3⋊D3—64no—
271C3⋊D3—62no—
272C3⋊D3—122no—
273C3⋊D3—124no—
274C3⋊D3—242no—
275C22×C4—271no—
276C22×C4—541no—
277M16—274no—
278C2×C8—91no—
279C2×C8—271no—
280C4⋊C4—91no—
281C4⋊C4—91no—
282C4⋊C4—181no—
283C4⋊C4—271no—
284C4⋊C4—271no—
285C22⋊C4—541no—
286C42—181no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—41no—
290C22×C3—41no—
291C22×C3C2×GO+(4,𝔽2)11yescharacteristic
292C2×D3—92no—
293C2×D3—182no—
294C2×D3—182no—
295C2×D3—182no—
296C2×D3—182no—
297C2×D3—181no—
298C2×D3—181no—
299C2×D3—362no—
300C2×D3—362no—
301C2×D3—361no—
302C2×D3—361no—
303C2×D3—362no—
304C2×D3—362no—
305C2×D3—721no—
306C2×D3—721no—
307C3×C4—62no—
308C3×C4—62no—
309C3×C4—91no—
310C3×C4—91no—
311C3×C4—121no—
312C3×C4—121no—
313C3×C4—122no—
314C3×C4—121no—
315C3×C4—121no—
316C3×C4—241no—
317C3×C4—361no—
318C3×C4—361no—
319Dic3—41no—
320Dic3—41no—
321Dic3—61no—
322Dic3—61no—
323Dic3—61no—
324Dic3—61no—
325Dic3—121no—
326Dic3—121no—
327Dic3—122no—
328Dic3—121no—
329Dic3—121no—
330Dic3—241no—
331Dic3#377312yes—
332C32—21no—
333C32—21no—
334C32—41no—
335C32—41no—
336C32#486111yescharacteristic, complemented
337C23—271no—
338C2×C4—31no—
339C2×C4—61no—
340C2×C4—61no—
341C2×C4—91no—
342C2×C4—181no—
343C2×C4—181no—
344C2×C4—272no—
345C2×C4—272no—
346C2×C4—1082no—
347C8—92no—
348C8—272no—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3—41no—
356C2×C3—41no—
357C2×C3—41no—
358C2×C3—41no—
359C2×C3—41no—
360C2×C3—41no—
361C2×C3#6422411yescharacteristic
362C2×C3#6422611yescharacteristic
363C2×C3C2×#377311yescharacteristic
364D3—184no—
365D3—182no—
366D3—362no—
367D3—364no—
368D3—722no—
369C22—272no—
370C22—541no—
371C22—541no—
372C22D3×GO+(4,𝔽2)11yescharacteristic
373C4—32no—
374C4—91no—
375C4—91no—
376C4—121no—
377C4—121no—
378C4—361no—
379C4—361no—
380C3—21no—
381C3—21no—
382C3—41no—
383C3—41no—
384C3#1059339411yescharacteristic, complemented
385C2—542no—
386C2D3×#377311yescharacteristic
387C2#1172414111yescharacteristic
388C2#1172414411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutatorC3×C3⋊Dic3C22×C4
Fitting, socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
cosocleC2×C3×C3⋊Dic3C23
hypocenterC33#428
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
4. #11722797C2semidirect
5. #11722779C2semidirect
6. #11722778C2semidirect
7. C3×#63742C2semidirect
13. #88926C4semidirect
29. #63742D3semidirect
40. C3⋊#622C2×C4semidirect
41. C3×#622C2×C4semidirect
129. #622C4×D3semidirect
215. C33#428semidirect
336. C32#4861semidirect
384. C3#10593394semidirect

Maximal subgroups

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

subgroupindex
C2×#889262
#117229182
#117229122
#117227972
#117227792
#117227782
C3×#637422
#105933943
#48619