Finite Group Databasebeta

← #13167482

Subgroups — #13167482

Order 1728.

536 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 77 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×#88940C212yesmaximal
2#11723689C211yesmaximal, characteristic
3#11723682C211yesmaximal, characteristic
4C3×#63781C211yesmaximal, characteristic
5#11722791C212yesmaximal
6#10593339—31nomaximal
7C22×C3×#157C2212yes—
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×#6294C2211yescharacteristic
10C2×#6294C2212yes—
11C2×#6294C2211yescharacteristic
12#88940—24no—
13C3×C4⋊(C3⋊Dic3)—21no—
14C3×#3750—21no—
15#88815—21no—
16#88815—21no—
17C2×#3773—32no—
18#63900—31no—
19D3×C4⋊Dic3—61no—
20#63883—31no—
21D3×Dic3⋊C4—61no—
22#63781D311yescharacteristic
23#63737—32no—
24C22×C3×C3⋊D3C2311yescharacteristic
25C2×C3×#157—22no—
26C2×C3×#157—22no—
27C2×C3×#157Q814yes—
28C2×C33×C4D411yescharacteristic
29C2×C3×C3⋊Dic3—21no—
30C2×C3×C3⋊Dic3—21no—
31C2×C3×C3⋊Dic3—22no—
32C2×C3×C3⋊Dic3D411yescharacteristic
33C3×C4×C3⋊D3—22no—
34#6294—21no—
35#6294—22no—
36#6294—22no—
37#6294—22no—
38#6294D412yes—
39#4661—91nomaximal
40C22×#157C2×D312yes—
41C2×C4×C3⋊D3C2×D311yescharacteristic
42C2×C3×C4×D3—61no—
43C2×C3×C4×D3—61no—
44C2×#624—31no—
45C2×#624—31no—
46C2×#624—32no—
47C2×D3×Dic3—61no—
48C2×D3×Dic3—61no—
49C2×D3×Dic3—62no—
50#3773—64no—
51C4⋊(C3⋊Dic3)—21no—
52#3750—21no—
53C3×C4⋊Dic3—61no—
54C3×C4⋊Dic3—61no—
55C3×C4⋊Dic3—121no—
56C3×Dic3⋊C4—61no—
57C3×Dic3⋊C4—61no—
58C3×Dic3⋊C4—121no—
59#3724—61no—
60#3724—61no—
61Dic3⋊Dic3—61no—
62Dic3⋊Dic3—62no—
63Dic3⋊Dic3—61no—
64C22×C33C2×D411yescharacteristic
65C2×C3×C3⋊D3C2×D411yescharacteristic
66C2×C3×C3⋊D3C2×Q812yes—
67C2×C3×C3⋊D3#4112yes—
68C2×C3×C3⋊D3#4111yescharacteristic
69C3×#157—24no—
70C33×C4—21no—
71C3×C3⋊Dic3—22no—
72C3×C3⋊Dic3—21no—
73C3×C3⋊Dic3—42no—
74C3×C3⋊Dic3—41no—
75C2×C4⋊Dic3—91no—
76C2×C4⋊Dic3—92no—
77C2×Dic3⋊C4—91no—
78D3×C4⋊C4—181no—
79D3×C4⋊C4—181no—
80C3×#95—91no—
81C3⋊#95—92no—
82C22×C3⋊D3C22×D311yescharacteristic
83C22×C3×D3—61no—
84C22×C3×D3—61no—
85C2×#157—22no—
86C2×#157—22no—
87C2×#157Dic614yes—
88C2×C32×C4—21no—
89C2×C32×C4—21no—
90C2×C32×C4—41no—
91C2×C32×C4—41no—
92C2×C32×C4D1211yescharacteristic
93C2×C3⋊Dic3—21no—
94C2×C3⋊Dic3—21no—
95C2×C3⋊Dic3—22no—
96C2×C3⋊Dic3D1211yescharacteristic
97C4×C3⋊D3—22no—
98C2×C3×Dic3—61no—
99C2×C3×Dic3—61no—
100C2×C3×Dic3—61no—
101C2×C3×Dic3—61no—
102C2×C3×Dic3—62no—
103C2×C3×Dic3—61no—
104C2×C3×Dic3—61no—
105C2×C3×Dic3—62no—
106C2×C3×Dic3—121no—
107C2×C3×Dic3—121no—
108C2×C3×Dic3—122no—
109C3×C4×D3—64no—
110C3×C4×D3—122no—
111#624—32no—
112#624—61no—
113#624—62no—
114#624—62no—
115#624—62no—
116D3×Dic3—62no—
117D3×Dic3—62no—
118D3×Dic3—122no—
119D3×Dic3—122no—
120D3×Dic3—122no—
121#398—271noSylow
122C2×C33#12111yescharacteristic
123C2×C33#12311yescharacteristic
124C2×C33C4⋊Q811yescharacteristic
125C3×C3⋊D3C22⋊Q812yes—
126C3×C3⋊D3#12512yes—
127C22×C3×C4—92no—
128C22×C3×C4—91no—
129C22×Dic3—91no—
130C22×Dic3—91no—
131C22×Dic3—92no—
132C2×C4×D3—61no—
133C2×C4×D3—61no—
134C2×C4×D3—181no—
135C2×C4×D3—181no—
136C2×C4×D3—182no—
137C3×C4⋊C4—181no—
138C3×C4⋊C4—181no—
139C4⋊Dic3—61no—
140C4⋊Dic3—61no—
141C4⋊Dic3—121no—
142C4⋊Dic3—181no—
143C4⋊Dic3—181no—
144C4⋊Dic3—184no—
145Dic3⋊C4—61no—
146Dic3⋊C4—61no—
147Dic3⋊C4—121no—
148Dic3⋊C4—181no—
149Dic3⋊C4—182no—
150Dic3⋊C4—181no—
151C22×C32—21no—
152C22×C32—21no—
153C22×C32—41no—
154C22×C32—41no—
155C22×C32C2×D1211yescharacteristic
156C2×C3⋊D3C2×Dic612yes—
157C2×C3⋊D3D3×D411yescharacteristic
158C2×C3⋊D3#23712yes—
159C2×C3⋊D3#23911yescharacteristic
160C2×C3×D3—61no—
161C2×C3×D3—61no—
162C2×C3×D3—62no—
163C2×C3×D3—62no—
164C2×C3×D3—61no—
165C2×C3×D3—61no—
166C2×C3×D3—62no—
167C2×C3×D3—62no—
168#157—24no—
169C32×C4—22no—
170C32×C4—21no—
171C32×C4—41no—
172C32×C4—42no—
173C32×C4—81no—
174C3⋊Dic3—22no—
175C3⋊Dic3—21no—
176C3⋊Dic3—42no—
177C3⋊Dic3—41no—
178C3×Dic3—62no—
179C3×Dic3—62no—
180C3×Dic3—62no—
181C3×Dic3—121no—
182C3×Dic3—122no—
183C3×Dic3—122no—
184C3×Dic3—121no—
185C3×Dic3—121no—
186C3×Dic3—122no—
187C3×Dic3—242no—
188C3×Dic3—241no—
189C2×C4⋊C4—271no—
190C2×C4⋊C4—271no—
191C2×C4⋊C4—272no—
192#95—91no—
193#95—272no—
194C33#39811yescharacteristic, complemented, Sylow
195C23×C3—91no—
196C22×D3—61no—
197C22×D3—61no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—94no—
203C2×C3×C4—91no—
204C2×C3×C4—182no—
205C2×C3×C4—181no—
206C2×C3×C4—182no—
207C2×C3×C4—182no—
208C2×C3×C4—182no—
209C2×C3×C4—181no—
210C2×C3×C4GO+(4,𝔽2)11yescharacteristic
211C2×Dic3—61no—
212C2×Dic3—61no—
213C2×Dic3—61no—
214C2×Dic3—61no—
215C2×Dic3—61no—
216C2×Dic3—62no—
217C2×Dic3—61no—
218C2×Dic3—62no—
219C2×Dic3—92no—
220C2×Dic3—121no—
221C2×Dic3—121no—
222C2×Dic3—122no—
223C2×Dic3—181no—
224C2×Dic3—182no—
225C2×Dic3—182no—
226C2×Dic3—182no—
227C4×D3—64no—
228C4×D3—122no—
229C4×D3—182no—
230C4×D3—182no—
231C4×D3—362no—
232C4×D3—362no—
233C4×D3—362no—
234C2×C32—21no—
235C2×C32—21no—
236C2×C32—21no—
237C2×C32—21no—
238C2×C32—21no—
239C2×C32—21no—
240C2×C32—41no—
241C2×C32—41no—
242C2×C32—41no—
243C2×C32—41no—
244C2×C32—41no—
245C2×C32—41no—
246C2×C32#86911yescharacteristic
247C2×C32#88611yescharacteristic
248C2×C32#89511yescharacteristic
249C3⋊D3#87812yes—
250C3⋊D3#89012yes—
251C3×D3—62no—
252C3×D3—62no—
253C3×D3—62no—
254C3×D3—62no—
255C22×C4—92no—
256C22×C4—91no—
257C22×C4—271no—
258C22×C4—271no—
259C22×C4—272no—
260C4⋊C4—181no—
261C4⋊C4—181no—
262C4⋊C4—541no—
263C4⋊C4—541no—
264C4⋊C4—544no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3—92no—
270C22×C3—91no—
271C22×C3—91no—
272C22×C3—92no—
273C22×C3C2×GO+(4,𝔽2)11yescharacteristic
274C2×D3—62no—
275C2×D3—62no—
276C2×D3—61no—
277C2×D3—61no—
278C2×D3—61no—
279C2×D3—61no—
280C2×D3—62no—
281C2×D3—62no—
282C3×C4—22no—
283C3×C4—21no—
284C3×C4—41no—
285C3×C4—42no—
286C3×C4—81no—
287C3×C4—182no—
288C3×C4—184no—
289C3×C4—181no—
290C3×C4—362no—
291C3×C4—361no—
292Dic3—62no—
293Dic3—62no—
294Dic3—62no—
295Dic3—121no—
296Dic3—122no—
297Dic3—122no—
298Dic3—121no—
299Dic3—122no—
300Dic3—121no—
301Dic3—242no—
302Dic3—241no—
303C32—21no—
304C32—21no—
305C32—41no—
306C32—41no—
307C32#466111yescharacteristic, complemented
308C23—91no—
309C2×C4—94no—
310C2×C4—91no—
311C2×C4—182no—
312C2×C4—182no—
313C2×C4—182no—
314C2×C4—181no—
315C2×C4—182no—
316C2×C4—181no—
317C2×C4—272no—
318C2×C4—541no—
319C2×C4—542no—
320C2×C4—542no—
321C2×C4—542no—
322C2×C4#157⋊D311yescharacteristic
323C2×C3—21no—
324C2×C3—21no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—41no—
330C2×C3—41no—
331C2×C3—41no—
332C2×C3—41no—
333C2×C3—41no—
334C2×C3—41no—
335C2×C3—92no—
336C2×C3—92no—
337C2×C3#6422211yescharacteristic
338C2×C3#6423011yescharacteristic
339C2×C3#6423311yescharacteristic
340D3—62no—
341D3—62no—
342D3—62no—
343D3—62no—
344C22—92no—
345C22—91no—
346C22—91no—
347C22—92no—
348C22C2×#157⋊D311yescharacteristic
349C4—21no—
350C4—184no—
351C4—182no—
352C4—181no—
353C4—362no—
354C4—361no—
355C3—21no—
356C3—21no—
357C3—41no—
358C3—41no—
359C3#1059333911yescharacteristic, complemented
360C2—92no—
361C2—92no—
362C2#1172433511yescharacteristic
363C2#1172434311yescharacteristic
364C2#1172434611yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, prime radical, cosocleC22×C3×C3⋊D3C23
FittingC2×C33×C4D4
socleC22×C33C2×D4
hypercenterC2×C4#157⋊D3
hypocenterC33#398
LCSG ▷ C22×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
194. C33#398semidirect
307. C32#4661semidirect
359. C3#10593339semidirect

Maximal subgroups

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

subgroupindex
C2×#889402
#117236892
#117236822
C3×#637812
#117227912
#105933393
#46619