Finite Group Databasebeta

← #13167412

Subgroups — #13167412

Order 1728.

414 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 53 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#11723685C211yesmaximal, characteristic
2#11723679C211yesmaximal, characteristic, complemented
3C3×C4⋊#622C211yesmaximal, characteristic, complemented
4#11722797C211yesmaximal, characteristic, complemented
5#11722796C211yesmaximal, characteristic, complemented
6#11722793C211yesmaximal, characteristic
7#11722793C211yesmaximal, characteristic
8#10593364—31nomaximal
9C2×#6295C2211yescharacteristic
10C2×#6294—21no—
11C2×C3×#622C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13C3×#3752—21no—
14C3×#3750—21no—
15C3×C4×C3⋊Dic3C2211yescharacteristic
16#88815—21no—
17#88815C2211yescharacteristic
18#88815C2211yescharacteristic
19#88815C2211yescharacteristic
20#63902—31no—
21#63880—61no—
22#63870—31no—
23#63867—61no—
24C4⋊#622D311yescharacteristic, complemented
25#63743—31no—
26#63742—31no—
27#63739—31no—
28#63739—31no—
29C22×C3×C3⋊D3—21no—
30C2×C33×C4D411yescharacteristic
31C2×C3×C3⋊Dic3—21no—
32C2×C3×C3⋊Dic3—21no—
33C2×C3×C3⋊Dic3—21no—
34C2×C3×C3⋊Dic3D411yescharacteristic
35C2×C3×C3⋊Dic3C2311yescharacteristic
36#6295—21no—
37#6295Q812yes—
38#6294—42no—
39C3×#622—21no—
40C3×#622—21no—
41#4847—91nomaximal
42C2×#626—61no—
43C2×#625—31no—
44C2×#624—61no—
45C2×D3×Dic3—61no—
46C2×D3×Dic3—61no—
47C2×#622C2×D311yescharacteristic
48C2×#622C2×D311yescharacteristic
49#3752—21no—
50#3750—21no—
51C4×C3⋊Dic3C2×D311yescharacteristic
52C3×(C2×D3)⋊C4—61no—
53C3×(C2×D3)⋊C4—61no—
54C3×(C2×D3)⋊C4—121no—
55C3×Dic3⋊C4—61no—
56C3×Dic3⋊C4—61no—
57C3×Dic3⋊C4—121no—
58C3×C4×Dic3—61no—
59C3×C4×Dic3—61no—
60#3724—31no—
61#3724—31no—
62#3724—31no—
63#3724—61no—
64Dic3⋊Dic3—61no—
65Dic3⋊Dic3—61no—
66Dic3⋊Dic3—61no—
67Dic3⋊Dic3—61no—
68Dic3⋊Dic3—61no—
69C22×C33C2×D411yescharacteristic
70C2×C3×C3⋊D3—22no—
71C2×C3×C3⋊D3—41no—
72C2×C3×C3⋊D3—41no—
73C33×C4—21no—
74C3×C3⋊Dic3—21no—
75C3×C3⋊Dic3—41no—
76C3×C3⋊Dic3—41no—
77C3×C3⋊Dic3—41no—
78C3×C3⋊Dic3C2×Q811yescharacteristic
79C3×C3⋊Dic3#4111yescharacteristic
80#889—181no—
81#67⋊C4—181no—
82C4×D12—91no—
83#870—91no—
84C3×C4⋊C8—91no—
85#821—91no—
86#821—91no—
87#818—91no—
88#817—91no—
89C22×C3⋊D3—21no—
90C22×C3×D3—61no—
91C22×C3×D3—61no—
92C22×C3×D3—121no—
93C2×C32×C4—21no—
94C2×C32×C4—21no—
95C2×C32×C4—41no—
96C2×C32×C4—41no—
97C2×C32×C4D1211yescharacteristic
98C2×C3⋊Dic3—21no—
99C2×C3⋊Dic3—21no—
100C2×C3⋊Dic3—21no—
101C2×C3⋊Dic3D1211yescharacteristic
102C2×C3⋊Dic3C22×D311yescharacteristic
103C2×C3×Dic3—61no—
104C2×C3×Dic3—61no—
105C2×C3×Dic3—61no—
106C2×C3×Dic3—61no—
107C2×C3×Dic3—61no—
108C2×C3×Dic3—61no—
109C2×C3×Dic3—61no—
110C2×C3×Dic3—61no—
111C2×C3×Dic3—61no—
112C2×C3×Dic3—61no—
113C2×C3×Dic3—121no—
114C2×C3×Dic3—121no—
115C2×C3×Dic3—121no—
116#626—62no—
117#626—62no—
118#625—32no—
119#625—61no—
120#624—122no—
121D3×Dic3—122no—
122D3×Dic3—122no—
123#622—21no—
124#622—21no—
125#478—271noSylow
126C2×C33C22⋊Q811yescharacteristic
127C2×C33#13511yescharacteristic
128C2×C33AGL(1,ℤ/8)11yescharacteristic
129C3×C3⋊D3—42no—
130C2×#67—181no—
131C22×Dic3—181no—
132C2×D12—91no—
133C2×C4×D3—181no—
134C2×C4×D3—181no—
135C2×C3×C8—91no—
136C2×C3×C8—91no—
137C3×C4⋊C4—181no—
138C3×C22⋊C4—181no—
139C3×C42—91no—
140(C2×D3)⋊C4—61no—
141(C2×D3)⋊C4—61no—
142(C2×D3)⋊C4—121no—
143C4⋊Dic3—91no—
144C4⋊Dic3—91no—
145C4⋊Dic3—91no—
146C4⋊Dic3—181no—
147Dic3⋊C4—61no—
148Dic3⋊C4—61no—
149Dic3⋊C4—121no—
150Dic3⋊C4—181no—
151Dic3⋊C4—181no—
152Dic3⋊C4—181no—
153Dic3⋊C4—181no—
154Dic3⋊C4—181no—
155C4×Dic3—61no—
156C4×Dic3—61no—
157C22×C32—21no—
158C22×C32—21no—
159C22×C32—41no—
160C22×C32—41no—
161C22×C32C2×D1211yescharacteristic
162C2×C3⋊D3—22no—
163C2×C3⋊D3—41no—
164C2×C3⋊D3—41no—
165C2×C3×D3—62no—
166C2×C3×D3—62no—
167C2×C3×D3—122no—
168C2×C3×D3—121no—
169C2×C3×D3—121no—
170C2×C3×D3—121no—
171C2×C3×D3—121no—
172C2×C3×D3—241no—
173C2×C3×D3—241no—
174C32×C4—21no—
175C32×C4—41no—
176C32×C4—41no—
177C32×C4—81no—
178C32×C4—81no—
179C3⋊Dic3—21no—
180C3⋊Dic3—41no—
181C3⋊Dic3—41no—
182C3⋊Dic3—41no—
183C3⋊Dic3#23711yescharacteristic
184C3⋊Dic3D3×Q811yescharacteristic
185C3×Dic3—61no—
186C3×Dic3—61no—
187C3×Dic3—61no—
188C3×Dic3—61no—
189C3×Dic3—62no—
190C3×Dic3—121no—
191C3×Dic3—121no—
192C3×Dic3—121no—
193C3×Dic3—121no—
194C3×Dic3—122no—
195C3×Dic3—121no—
196C3×Dic3—121no—
197C3×Dic3—121no—
198C3×Dic3—241no—
199C3×Dic3—241no—
200#125—271no—
201C4×D4—271no—
202#107—271no—
203#106—271no—
204C4⋊C8—91no—
205#102—271no—
206#102—271no—
207C33#47811yescharacteristic, complemented, Sylow
208C23×C3—181no—
209C22×D3—61no—
210C22×D3—61no—
211C22×D3—121no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—91no—
217C2×C3×C4—91no—
218C2×C3×C4—181no—
219C2×C3×C4—181no—
220C2×C3×C4—181no—
221C2×C3×C4GO+(4,𝔽2)11yescharacteristic
222#67—182no—
223#67—182no—
224C2×Dic3—61no—
225C2×Dic3—61no—
226C2×Dic3—61no—
227C2×Dic3—61no—
228C2×Dic3—61no—
229C2×Dic3—61no—
230C2×Dic3—61no—
231C2×Dic3—61no—
232C2×Dic3—61no—
233C2×Dic3—61no—
234C2×Dic3—121no—
235C2×Dic3—121no—
236C2×Dic3—121no—
237C2×Dic3—362no—
238D12—92no—
239D12—181no—
240C4×D3—362no—
241C4×D3—362no—
242C3×C8—181no—
243C3×C8—181no—
244C2×C32—21no—
245C2×C32—21no—
246C2×C32—21no—
247C2×C32—21no—
248C2×C32—21no—
249C2×C32—21no—
250C2×C32—41no—
251C2×C32—41no—
252C2×C32—41no—
253C2×C32—41no—
254C2×C32—41no—
255C2×C32—41no—
256C2×C32#89711yescharacteristic
257C2×C32#90411yescharacteristic
258C2×C32#90811yescharacteristic
259C3⋊D3—42no—
260C3×D3—122no—
261C3×D3—122no—
262C3×D3—242no—
263C2×D4—271no—
264C22×C4—541no—
265C2×C8—91no—
266C2×C8—91no—
267C4⋊C4—181no—
268C4⋊C4—271no—
269C4⋊C4—271no—
270C4⋊C4—271no—
271C4⋊C4—541no—
272C22⋊C4—181no—
273C42—91no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—182no—
279C22×C3—361no—
280C22×C3—361no—
281C22×C3C2×GO+(4,𝔽2)11yescharacteristic
282C2×D3—62no—
283C2×D3—62no—
284C2×D3—122no—
285C2×D3—121no—
286C2×D3—121no—
287C2×D3—121no—
288C2×D3—121no—
289C2×D3—241no—
290C2×D3—241no—
291C3×C4—21no—
292C3×C4—41no—
293C3×C4—41no—
294C3×C4—81no—
295C3×C4—81no—
296C3×C4—91no—
297C3×C4—91no—
298C3×C4—181no—
299C3×C4—361no—
300C3×C4—361no—
301C3×C4—361no—
302Dic3—61no—
303Dic3—61no—
304Dic3—61no—
305Dic3—61no—
306Dic3—62no—
307Dic3—121no—
308Dic3—121no—
309Dic3—121no—
310Dic3—121no—
311Dic3—122no—
312Dic3—121no—
313Dic3—121no—
314Dic3—121no—
315Dic3—241no—
316Dic3—241no—
317C32—21no—
318C32—21no—
319C32—41no—
320C32—41no—
321C32#484711yescharacteristic, complemented
322C23—181no—
323D4—272no—
324D4—541no—
325C2×C4—91no—
326C2×C4—91no—
327C2×C4—181no—
328C2×C4—181no—
329C2×C4—181no—
330C2×C4—1082no—
331C2×C4#157⋊D311yescharacteristic
332C8—181no—
333C8—181no—
334C2×C3—21no—
335C2×C3—21no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—41no—
341C2×C3—41no—
342C2×C3—41no—
343C2×C3—41no—
344C2×C3—41no—
345C2×C3—41no—
346C2×C3—362no—
347C2×C3D32⋊Q811yescharacteristic
348C2×C3#6422411yescharacteristic
349C2×C3#6423711yescharacteristic
350D3—122no—
351D3—122no—
352D3—242no—
353C22—182no—
354C22—361no—
355C22—361no—
356C22C2×#157⋊D311yescharacteristic
357C4—21no—
358C4—91no—
359C4—91no—
360C4—181no—
361C4—361no—
362C4—361no—
363C4—361no—
364C3—21no—
365C3—21no—
366C3—41no—
367C3—41no—
368C3#1059336411yescharacteristic, complemented
369C2—362no—
370C2#1172433311yescharacteristic
371C2#1172434011yescharacteristic
372C2#1172435111yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×#6295C22
hypercenterC2×C4#157⋊D3
hypocenterC33#478
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723679C2semidirect
3. C3×C4⋊#622C2semidirect
4. #11722797C2semidirect
5. #11722796C2semidirect
24. C4⋊#622D3semidirect
207. C33#478semidirect
321. C32#4847semidirect
368. C3#10593364semidirect

Maximal subgroups

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

subgroupindex
#117236852
#117236792
C3×C4⋊#6222
#117227972
#117227962
#117227932
#105933643
#48479