Finite Group Databasebeta

← #13165513

Subgroups — #13165513

Order 1728.

468 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 60 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×#88937C211yesmaximal, characteristic
2#11722945C211yesmaximal, characteristic, complemented
3Dic3×#625C211yesmaximal, characteristic
4#11722794C211yesmaximal, characteristic, complemented
5#11722779C211yesmaximal, characteristic, complemented
6C3×#63739C211yesmaximal, characteristic
7#11722724C211yesmaximal, characteristic, complemented
8#10593431—31nomaximal
9C2×#6296C2211yescharacteristic
10C2×C3×#625C2211yescharacteristic
11C2×C3×D3×Dic3—21no—
12C2×C3⋊#622C2211yescharacteristic
13C2×C3×#622C2211yescharacteristic
14#88937C414yescomplemented
15#88815C2211yescharacteristic
16(C2×D3)⋊(C3⋊Dic3)—21no—
17Dic3×C3⋊Dic3—21no—
18Dic3×C3⋊Dic3C2211yescharacteristic
19C3×#3724—21no—
20C3×#3724C2211yescharacteristic
21C2×#3775—31no—
22Dic3×#67—61no—
23C4×#625—31no—
24#63838—31no—
25#63838—61no—
26#63777—31no—
27#63743—31no—
28#63740—31no—
29#63739D311yescharacteristic
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
37#6296—21no—
38#6296C2×C412yescomplemented
39C3×#625—21no—
40C3×#625C2×C412yes—
41C3×D3×Dic3—42no—
42C3⋊#622C2×C412yescomplemented
43C3×#622—21no—
44#4757—91nomaximal
45C2×C3×#67—61no—
46C22×C3×Dic3—61no—
47C2×C3×C4×D3—61no—
48C2×#627—31no—
49C2×#627—61no—
50C2×#625C2×D311yescharacteristic
51C2×D3×Dic3—21no—
52C2×#622—31no—
53C2×#622C2×D311yescharacteristic
54#3775—34no—
55#3757—61no—
56C4×C3⋊Dic3—31no—
57C4×C3⋊Dic3—61no—
58C3×Dic3⋊C4—61no—
59C3×Dic3⋊C4—61no—
60C3×Dic3⋊C4—61no—
61#3724—21no—
62#3724—31no—
63#3724C2×D311yescharacteristic
64Dic3⋊Dic3—61no—
65(C2×D3)⋊Dic3—61no—
66(C2×D3)⋊Dic3—61no—
67(C2×D3)⋊Dic3—121no—
68Dic32—61no—
69Dic32—61no—
70Dic32—61no—
71Dic32—61no—
72Dic32—121no—
73C22×C33C2×D411yescharacteristic
74C2×C32×D3—22no—
75C2×C32×D3—41no—
76C2×C32×D3—41no—
77C3⋊(C3⋊Dic3)—21no—
78C3×C3⋊Dic3—22no—
79C3×C3⋊Dic3—41no—
80C3×C3⋊Dic3C22×C411yescharacteristic
81C3×C3⋊Dic3#4111yescharacteristic
82C32×Dic3—21no—
83C32×Dic3—41no—
84D4×Dic3—91no—
85C2×#214—91no—
86C4×#67—181no—
87#887—91no—
88#887—181no—
89C3×#102—91no—
90#816—91no—
91#815—91no—
92#807—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⋊Dic3C22×D311yescharacteristic
107C3×#67—62no—
108C3×#67—62no—
109C2×C3×Dic3—21no—
110C2×C3×Dic3—21no—
111C2×C3×Dic3—21no—
112C2×C3×Dic3—21no—
113C2×C3×Dic3—41no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3—61no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—121no—
121C2×C3×Dic3—121no—
122C2×C3×Dic3—122no—
123C3×C4×D3—122no—
124#627—32no—
125#627—62no—
126#627—62no—
127#627—61no—
128#625—21no—
129#625C4×D312yes—
130D3×Dic3—42no—
131#622—21no—
132#622—32no—
133#440—271noSylow
134C2×C33C4×D411yescharacteristic
135C2×C33AGL(1,ℤ/8)11yescharacteristic
136C2×C33#13711yescharacteristic
137C32×D3—42no—
138C2×C3×D4—91no—
139C22×C3×C4—181no—
140C2×#67—61no—
141C22×Dic3—61no—
142C2×C4×D3—61no—
143C2×Dic6—91no—
144C2×Dic6—181no—
145C2×C3×C8—91no—
146C3×C4⋊C4—91no—
147C3×C4⋊C4—181no—
148#217—181no—
149#217—181no—
150#217—361no—
151#214—94no—
152(C2×D3)⋊C4—181no—
153C4⋊Dic3—91no—
154Dic3⋊C4—61no—
155Dic3⋊C4—61no—
156Dic3⋊C4—61no—
157Dic3⋊C4—181no—
158C4×Dic3—91no—
159C4×Dic3—181no—
160C4×Dic3—181no—
161C4×Dic3—181no—
162C4×Dic3—181no—
163C4×Dic3—181no—
164C4×Dic3—361no—
165C2×C3⋊C8—91no—
166C22×C32—21no—
167C22×C32—21no—
168C22×C32—41no—
169C22×C32—41no—
170C22×C32—62no—
171C22×C32—121no—
172C22×C32—121no—
173C22×C32D3×D411yescharacteristic
174C2×C3×D3—22no—
175C2×C3×D3—22no—
176C2×C3×D3—42no—
177C2×C3×D3—41no—
178C2×C3×D3—41no—
179C2×C3×D3—41no—
180C2×C3×D3—41no—
181C2×C3×D3—81no—
182C2×C3×D3—81no—
183C32×C4—61no—
184C32×C4—121no—
185C3⋊Dic3—22no—
186C3⋊Dic3—41no—
187C3⋊Dic3—62no—
188C3⋊Dic3—61no—
189C3⋊Dic3—121no—
190C3⋊Dic3—122no—
191C3⋊Dic3—241no—
192C3⋊Dic3C2×C4×D311yescharacteristic
193C3⋊Dic3#23711yescharacteristic
194C3×Dic3—22no—
195C3×Dic3—41no—
196C3×Dic3—41no—
197C3×Dic3—42no—
198C3×Dic3—41no—
199C3×Dic3—61no—
200C3×Dic3—61no—
201C3×Dic3—61no—
202C3×Dic3—61no—
203C3×Dic3—62no—
204C3×Dic3—62no—
205C3×Dic3—122no—
206C3×Dic3—121no—
207C3×Dic3—121no—
208C3×Dic3—241no—
209C2×SD16—271no—
210C4×Q8—271no—
211C4×D4—271no—
212#107—271no—
213#103—271no—
214#102—91no—
215#97—271no—
216C33#44011yescharacteristic, complemented, Sylow
217C23×C3—61no—
218C23×C3—61no—
219C23×C3—121no—
220C22×D3—21no—
221C3×D4—92no—
222C3×D4—181no—
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—181no—
230C2×C3×C4—181no—
231C2×C3×C4—362no—
232#67—62no—
233#67—62no—
234C2×Dic3—21no—
235C2×Dic3—61no—
236C2×Dic3—61no—
237C2×Dic3—61no—
238C2×Dic3—61no—
239C2×Dic3—61no—
240C2×Dic3—61no—
241C2×Dic3—91no—
242C2×Dic3—121no—
243C2×Dic3—121no—
244C2×Dic3—122no—
245C2×Dic3—181no—
246C2×Dic3—181no—
247C2×Dic3—361no—
248C2×Dic3—361no—
249C2×Dic3GO+(4,𝔽2)11yescharacteristic
250C4×D3—122no—
251Dic6—92no—
252Dic6—182no—
253Dic6—181no—
254Dic6—182no—
255C3×C8—181no—
256C3⋊C8—92no—
257C2×C32—21no—
258C2×C32—21no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32—21no—
262C2×C32—21no—
263C2×C32—41no—
264C2×C32—41no—
265C2×C32—41no—
266C2×C32—41no—
267C2×C32—41no—
268C2×C32—41no—
269C2×C32—122no—
270C2×C32#67⋊C411yescharacteristic
271C2×C32#91111yescharacteristic
272C2×C32#91511yescharacteristic
273C3×D3—42no—
274C3×D3—42no—
275C3×D3—82no—
276C2×Q8—271no—
277C2×D4—91no—
278C22×C4—181no—
279SD16—274no—
280C2×C8—91no—
281C2×C8—271no—
282C4⋊C4—91no—
283C4⋊C4—181no—
284C4⋊C4—271no—
285C22⋊C4—541no—
286C42—271no—
287C42—541no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—62no—
293C22×C3—62no—
294C22×C3—122no—
295C22×C3—121no—
296C22×C3—121no—
297C22×C3—121no—
298C22×C3—121no—
299C22×C3—241no—
300C22×C3—241no—
301C22×C3C2×GO+(4,𝔽2)11yescharacteristic
302C2×D3—22no—
303C2×D3—41no—
304C2×D3—41no—
305C3×C4—62no—
306C3×C4—91no—
307C3×C4—91no—
308C3×C4—121no—
309C3×C4—121no—
310C3×C4—122no—
311C3×C4—121no—
312C3×C4—182no—
313C3×C4—361no—
314Dic3—21no—
315Dic3—41no—
316Dic3—61no—
317Dic3—61no—
318Dic3—61no—
319Dic3—61no—
320Dic3—62no—
321Dic3—62no—
322Dic3—122no—
323Dic3—121no—
324Dic3—121no—
325Dic3—182no—
326Dic3—181no—
327Dic3—241no—
328Dic3—361no—
329Dic3—362no—
330Dic3—721no—
331C32—21no—
332C32—21no—
333C32—41no—
334C32—41no—
335C32#475711yescharacteristic, complemented
336C23—61no—
337Q8—272no—
338Q8—541no—
339D4—92no—
340D4—181no—
341C2×C4—31no—
342C2×C4—61no—
343C2×C4—91no—
344C2×C4—181no—
345C2×C4—181no—
346C2×C4—271no—
347C2×C4—362no—
348C8—181no—
349C8—272no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3—41no—
357C2×C3—41no—
358C2×C3—41no—
359C2×C3—41no—
360C2×C3—41no—
361C2×C3—41no—
362C2×C3—122no—
363C2×C3—122no—
364C2×C3—242no—
365C2×C3#6422611yescharacteristic
366C2×C3C4×GO+(4,𝔽2)11yescharacteristic
367C2×C3#6423611yescharacteristic
368D3—42no—
369C22—62no—
370C22—121no—
371C22—121no—
372C22D3×GO+(4,𝔽2)11yescharacteristic
373C4—61no—
374C4—91no—
375C4—91no—
376C4—121no—
377C4—182no—
378C4—361no—
379C4—541no—
380C3—21no—
381C3—21no—
382C3—41no—
383C3—41no—
384C3#1059343111yescharacteristic, complemented
385C2—122no—
386C2(D32⋊D3)⋊C411yescharacteristic
387C2#1172413811yescharacteristic
388C2#1172414411yescharacteristic
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. #11722945C2semidirect
4. #11722794C2semidirect
5. #11722779C2semidirect
7. #11722724C2semidirect
14. #88937C4semidirect
38. #6296C2×C4semidirect
42. C3⋊#622C2×C4semidirect
216. C33#440semidirect
335. C32#4757semidirect
384. C3#10593431semidirect

Maximal subgroups

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

subgroupindex
C2×#889372
#117229452
Dic3×#6252
#117227942
#117227792
C3×#637392
#117227242
#105934313
#47579