Finite Group Databasebeta

← #13165656

Subgroups — #13165656

Order 1728.

450 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 56 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
1#11723037C211yesmaximal, characteristic, complemented
2Dic3×#625C211yesmaximal, characteristic
3#11722796C211yesmaximal, characteristic, complemented
4#11722778C211yesmaximal, characteristic, complemented
5#11722773C211yesmaximal, characteristic
6C3×#63739C211yesmaximal, characteristic
7Dic3⋊#622C211yesmaximal, characteristic, complemented
8#10593359—31nomaximal
9C2×#6293—21no—
10C2×C3×#625C2211yescharacteristic
11C2×C3×D3×Dic3—21no—
12C2×C3⋊#622C2211yescharacteristic
13C2×C3×#622C2211yescharacteristic
14#88815C2211yescharacteristic
15#88815C2211yescharacteristic
16(C2×D3)⋊(C3⋊Dic3)—21no—
17Dic3×C3⋊Dic3C2211yescharacteristic
18C3×#3724C2211yescharacteristic
19C3×Dic3⋊Dic3—21no—
20Dic3×#67—61no—
21#63919—31no—
22C4×#625—31no—
23#63840—61no—
24C4⋊#622—31no—
25#63742—31no—
26#63742—31no—
27#63739—31no—
28#63739D311yescharacteristic
29C22×C32×D3—21no—
30C2×C3⋊(C3⋊Dic3)D411yescharacteristic
31C2×C3×C3⋊Dic3—21no—
32C2×C3×C3⋊Dic3—21no—
33C2×C3×C3⋊Dic3C2311yescharacteristic
34C2×C32×Dic3—21no—
35C2×C32×Dic3D411yescharacteristic
36#6293—42no—
37C3×#625—21no—
38C3×#625Q812yescomplemented
39C3×D3×Dic3—42no—
40C3⋊#622—21no—
41C3×#622—21no—
42#4761—91nomaximal
43C2×#634—61no—
44C2×C3×#67—61no—
45C22×C3×Dic3—61no—
46C2×C3×C4×D3—61no—
47C2×#627—61no—
48C2×#627—61no—
49C2×#627—121no—
50C2×#625C2×D311yescharacteristic
51C2×D3×Dic3—21no—
52C2×#622—31no—
53C2×#622C2×D311yescharacteristic
54#3757—61no—
55C4×C3⋊Dic3—31no—
56C3×C4⋊Dic3—61no—
57C3×Dic3⋊C4—61no—
58C3×Dic3⋊C4—61no—
59#3724—31no—
60#3724—31no—
61#3724C2×D311yescharacteristic
62Dic3⋊Dic3—21no—
63Dic3⋊Dic3—61no—
64Dic3⋊Dic3—61no—
65(C2×D3)⋊Dic3—61no—
66(C2×D3)⋊Dic3—61no—
67(C2×D3)⋊Dic3—121no—
68Dic32—61no—
69Dic32—61no—
70C22×C33C2×D411yescharacteristic
71C2×C32×D3—22no—
72C2×C32×D3—41no—
73C2×C32×D3—41no—
74C3⋊(C3⋊Dic3)—21no—
75C3×C3⋊Dic3—41no—
76C3×C3⋊Dic3—41no—
77C3×C3⋊Dic3C2×Q811yescharacteristic
78C3×C3⋊Dic3#4111yescharacteristic
79C32×Dic3—41no—
80C32×Dic3SD1612yes—
81D4×Dic3—91no—
82C4×#67—181no—
83#888—91no—
84#888—181no—
85C3×#102—91no—
86#832—91no—
87#817—91no—
88Dic3⋊C8—91no—
89#808—91no—
90C23×C32—61no—
91C22×C3×D3—21no—
92C22×C3×D3—21no—
93C22×C3×D3—41no—
94C2×C32×C4—31no—
95C2×C32×C4—61no—
96C2×C3⋊Dic3—21no—
97C2×C3⋊Dic3—21no—
98C2×C3⋊Dic3—31no—
99C2×C3⋊Dic3—61no—
100C2×C3⋊Dic3—61no—
101C2×C3⋊Dic3—121no—
102C2×C3⋊Dic3—121no—
103C2×C3⋊Dic3C22×D311yescharacteristic
104#634—122no—
105C3×#67—62no—
106C3×#67—62no—
107C2×C3×Dic3—21no—
108C2×C3×Dic3—21no—
109C2×C3×Dic3—21no—
110C2×C3×Dic3—21no—
111C2×C3×Dic3—41no—
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—122no—
121C3×C4×D3—122no—
122#627—122no—
123#627—122no—
124#627—124no—
125#625—21no—
126#625Dic612yescomplemented
127D3×Dic3—42no—
128#622—21no—
129#622—61no—
130#476—271noSylow
131C2×C33C22⋊Q811yescharacteristic
132C2×C33C2×SD1611yescharacteristic
133C2×C33AGL(1,ℤ/8)11yescharacteristic
134C32×D3—42no—
135C2×C3×D4—91no—
136C22×C3×C4—181no—
137C2×#67—61no—
138C22×Dic3—61no—
139C2×C4×D3—61no—
140C2×Dic6—181no—
141C2×Dic6—181no—
142C2×Dic6—181no—
143C2×Dic6—361no—
144C2×C3×C8—91no—
145C3×C4⋊C4—91no—
146C3×C4⋊C4—181no—
147#217—181no—
148#217—181no—
149#217—361no—
150(C2×D3)⋊C4—181no—
151C4⋊Dic3—61no—
152C4⋊Dic3—91no—
153C4⋊Dic3—91no—
154Dic3⋊C4—61no—
155Dic3⋊C4—61no—
156Dic3⋊C4—181no—
157Dic3⋊C4—181no—
158C4×Dic3—91no—
159C4×Dic3—181no—
160C4×Dic3—181no—
161C2×C3⋊C8—91no—
162C22×C32—21no—
163C22×C32—21no—
164C22×C32—41no—
165C22×C32—41no—
166C22×C32—62no—
167C22×C32—121no—
168C22×C32—121no—
169C22×C32D3×D411yescharacteristic
170C2×C3×D3—22no—
171C2×C3×D3—22no—
172C2×C3×D3—42no—
173C2×C3×D3—41no—
174C2×C3×D3—41no—
175C2×C3×D3—41no—
176C2×C3×D3—41no—
177C2×C3×D3—81no—
178C2×C3×D3—81no—
179C32×C4—32no—
180C32×C4—121no—
181C3⋊Dic3—41no—
182C3⋊Dic3—41no—
183C3⋊Dic3—62no—
184C3⋊Dic3—61no—
185C3⋊Dic3—121no—
186C3⋊Dic3—122no—
187C3⋊Dic3—241no—
188C3⋊Dic3C2×Dic611yescharacteristic
189C3⋊Dic3#23711yescharacteristic
190C3×Dic3—22no—
191C3×Dic3—22no—
192C3×Dic3—41no—
193C3×Dic3—42no—
194C3×Dic3—41no—
195C3×Dic3—61no—
196C3×Dic3—61no—
197C3×Dic3—61no—
198C3×Dic3—61no—
199C3×Dic3—121no—
200C3×Dic3—121no—
201C3×Dic3—122no—
202C3×Dic3—121no—
203C3×Dic3—121no—
204C3×Dic3—241no—
205C4⋊Q8—271no—
206C4×D4—271no—
207#106—271no—
208#106—271no—
209C4⋊C8—271no—
210#102—91no—
211#102—271no—
212C33#47611yescharacteristic, complemented, Sylow
213C23×C3—61no—
214C23×C3—61no—
215C23×C3—121no—
216C22×D3—21no—
217C3×D4—92no—
218C3×D4—181no—
219C2×C3×C4—61no—
220C2×C3×C4—61no—
221C2×C3×C4—61no—
222C2×C3×C4—61no—
223C2×C3×C4—91no—
224C2×C3×C4—121no—
225C2×C3×C4—181no—
226C2×C3×C4—181no—
227C2×C3×C4—362no—
228#67—62no—
229#67—62no—
230C2×Dic3—21no—
231C2×Dic3—61no—
232C2×Dic3—61no—
233C2×Dic3—61no—
234C2×Dic3—61no—
235C2×Dic3—61no—
236C2×Dic3—61no—
237C2×Dic3—91no—
238C2×Dic3—121no—
239C2×Dic3—121no—
240C2×Dic3—122no—
241C2×Dic3—181no—
242C2×Dic3—181no—
243C2×Dic3—361no—
244C2×Dic3—361no—
245C2×Dic3GO+(4,𝔽2)11yescharacteristic
246C4×D3—122no—
247Dic6—362no—
248Dic6—362no—
249Dic6—362no—
250Dic6—364no—
251C3×C8—181no—
252C3⋊C8—181no—
253C2×C32—21no—
254C2×C32—21no—
255C2×C32—21no—
256C2×C32—21no—
257C2×C32—21no—
258C2×C32—21no—
259C2×C32—41no—
260C2×C32—41no—
261C2×C32—41no—
262C2×C32—41no—
263C2×C32—41no—
264C2×C32—41no—
265C2×C32—122no—
266C2×C32#87811yescharacteristic
267C2×C32#91111yescharacteristic
268C2×C32D3×SD1611yescharacteristic
269C3×D3—42no—
270C3×D3—42no—
271C3×D3—82no—
272C2×Q8—541no—
273C2×D4—91no—
274C22×C4—181no—
275C2×C8—91no—
276C2×C8—271no—
277C4⋊C4—91no—
278C4⋊C4—181no—
279C4⋊C4—271no—
280C4⋊C4—271no—
281C22⋊C4—541no—
282C42—271no—
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3—62no—
288C22×C3—62no—
289C22×C3—122no—
290C22×C3—121no—
291C22×C3—121no—
292C22×C3—121no—
293C22×C3—121no—
294C22×C3—241no—
295C22×C3—241no—
296C22×C3C2×GO+(4,𝔽2)11yescharacteristic
297C2×D3—22no—
298C2×D3—41no—
299C2×D3—41no—
300C3×C4—62no—
301C3×C4—62no—
302C3×C4—91no—
303C3×C4—91no—
304C3×C4—121no—
305C3×C4—122no—
306C3×C4—121no—
307C3×C4—361no—
308C3×C4—361no—
309Dic3—41no—
310Dic3—61no—
311Dic3—61no—
312Dic3—61no—
313Dic3—61no—
314Dic3—121no—
315Dic3—121no—
316Dic3—122no—
317Dic3—121no—
318Dic3—121no—
319Dic3—182no—
320Dic3—181no—
321Dic3—241no—
322Dic3—361no—
323Dic3—362no—
324Dic3—721no—
325Dic3#377512yes—
326C32—21no—
327C32—21no—
328C32—41no—
329C32—41no—
330C32#476111yescharacteristic, complemented
331C23—61no—
332Q8—1082no—
333D4—92no—
334D4—181no—
335C2×C4—31no—
336C2×C4—61no—
337C2×C4—91no—
338C2×C4—181no—
339C2×C4—181no—
340C2×C4—271no—
341C2×C4—362no—
342C8—181no—
343C8—541no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—41no—
351C2×C3—41no—
352C2×C3—41no—
353C2×C3—41no—
354C2×C3—41no—
355C2×C3—41no—
356C2×C3—122no—
357C2×C3—122no—
358C2×C3—242no—
359C2×C3D32⋊Q811yescharacteristic
360C2×C3#6422411yescharacteristic
361C2×C3C2×#377511yescharacteristic
362D3—42no—
363C22—62no—
364C22—121no—
365C22—121no—
366C22D3×GO+(4,𝔽2)11yescharacteristic
367C4—32no—
368C4—91no—
369C4—91no—
370C4—121no—
371C4—361no—
372C4—361no—
373C4—541no—
374C3—21no—
375C3—21no—
376C3—41no—
377C3—41no—
378C3#1059335911yescharacteristic, complemented
379C2—122no—
380C2#1172411911yescharacteristic
381C2D3×#377511yescharacteristic
382C2#1172414011yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socleC22×C33C2×D4
prime radicalC2×C3×#625C22
hypocenterC33#476
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. #11723037C2semidirect
3. #11722796C2semidirect
4. #11722778C2semidirect
7. Dic3⋊#622C2semidirect
38. C3×#625Q8semidirect
126. #625Dic6semidirect
212. C33#476semidirect
330. C32#4761semidirect
378. C3#10593359semidirect

Maximal subgroups

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

subgroupindex
#117230372
Dic3×#6252
#117227962
#117227782
#117227732
C3×#637392
Dic3⋊#6222
#105933593
#47619