Finite Group Databasebeta

← C3×#10593339 #13166775

Subgroups — C3×#10593339 #13166775

Order 1728.

536 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 90 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×C3×#3773C212yesmaximal
2C3×#63900C211yesmaximal, characteristic
3C3×#63883C211yesmaximal, characteristic
4C3×#63781C211yesmaximal, characteristic
5C3×#63737C212yesmaximal
6#10593339C311yesmaximal, characteristic, direct product factor
7C22×C3×#157C2212yes—
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C3×#624C2212yes—
10C2×C3×#624C2211yescharacteristic
11C2×C3×#624C2211yescharacteristic
12C3×#3773—24no—
13C32×C4⋊Dic3—21no—
14C32×Dic3⋊C4—21no—
15C3×#3724—21no—
16C3×Dic3⋊Dic3—21no—
17C2×#3773C2×C312yes—
18C3×D3×C4⋊C4—61no—
19C3×D3×C4⋊C4—61no—
20#63900C2×C311yescharacteristic
21#63883C2×C311yescharacteristic
22#63781C2×C311yescharacteristic
23#63737C2×C312yes—
24C22×C3×C3⋊D3C2311yescharacteristic
25C2×C3×#157—22no—
26C2×C3×#157—22no—
27C2×C3×#157Q814yes—
28C2×C33×C4D411yescharacteristic
29C2×C3×C3⋊Dic3D411yescharacteristic
30C3×C4×C3⋊D3—22no—
31C2×C32×Dic3—21no—
32C2×C32×Dic3—21no—
33C2×C32×Dic3—22no—
34C3×#624—21no—
35C3×#624—22no—
36C3×#624—22no—
37C3×#624—22no—
38C3×#624D412yes—
39C3×#398—91nomaximal
40C22×#157C22×C312yes—
41C2×C4×C3⋊D3C22×C311yescharacteristic
42C2×C3×C4×D3—61no—
43C2×C3×C4×D3—61no—
44C2×C3×C4×D3—61no—
45C2×C3×C4×D3—62no—
46C2×C3×C4×D3—61no—
47C2×#624C22×C311yescharacteristic
48C2×#624C22×C312yes—
49C2×#624C22×C311yescharacteristic
50#3773—24no—
51C32×C4⋊C4—61no—
52C32×C4⋊C4—61no—
53C3×C4⋊Dic3—21no—
54C3×C4⋊Dic3—21no—
55C3×C4⋊Dic3—41no—
56C3×C4⋊Dic3—61no—
57C3×Dic3⋊C4—21no—
58C3×Dic3⋊C4—21no—
59C3×Dic3⋊C4—41no—
60C3×Dic3⋊C4—61no—
61C3×Dic3⋊C4—62no—
62#3724—21no—
63Dic3⋊Dic3—21no—
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—21no—
72C32×Dic3—22no—
73C32×Dic3—42no—
74C32×Dic3—41no—
75C2×C3×C4⋊C4—91no—
76C2×C3×C4⋊C4—91no—
77C2×C3×C4⋊C4—92no—
78D3×C4⋊C4—61no—
79D3×C4⋊C4—61no—
80C3×#95—91no—
81C3×#95—92no—
82C22×C3⋊D3C23×C311yescharacteristic
83C22×C3×D3—61no—
84C22×C3×D3—61no—
85C2×#157—22no—
86C2×#157—22no—
87C2×#157C3×Q814yes—
88C2×C32×C4—21no—
89C2×C32×C4—21no—
90C2×C32×C4—41no—
91C2×C32×C4—41no—
92C2×C32×C4—61no—
93C2×C32×C4—61no—
94C2×C32×C4—62no—
95C2×C32×C4C3×D411yescharacteristic
96C2×C3⋊Dic3C3×D411yescharacteristic
97C4×C3⋊D3—22no—
98C2×C3×Dic3—21no—
99C2×C3×Dic3—21no—
100C2×C3×Dic3—21no—
101C2×C3×Dic3—22no—
102C2×C3×Dic3—21no—
103C2×C3×Dic3—22no—
104C2×C3×Dic3—41no—
105C2×C3×Dic3—41no—
106C2×C3×Dic3—42no—
107C2×C3×Dic3—61no—
108C2×C3×Dic3—61no—
109C3×C4×D3—62no—
110C3×C4×D3—62no—
111C3×C4×D3—64no—
112C3×C4×D3—122no—
113C3×C4×D3—122no—
114C3×C4×D3—122no—
115C3×C4×D3—122no—
116#624—21no—
117#624—22no—
118#624—22no—
119#624—22no—
120#624C3×D412yes—
121#398—91noSylow
122C2×C33#12111yescharacteristic
123C2×C33#12311yescharacteristic
124C2×C33C4⋊Q811yescharacteristic
125C3×C3⋊D3C22⋊Q812yes—
126C3×C3⋊D3#12512yes—
127C22×C3×C4—91no—
128C22×C3×C4—91no—
129C22×C3×C4—92no—
130C22×C3×C4—92no—
131C22×C3×C4—91no—
132C2×C4×D3—61no—
133C2×C4×D3—61no—
134C2×C4×D3—61no—
135C2×C4×D3—62no—
136C2×C4×D3—61no—
137C3×C4⋊C4—61no—
138C3×C4⋊C4—61no—
139C3×C4⋊C4—61no—
140C3×C4⋊C4—61no—
141C3×C4⋊C4—121no—
142C3×C4⋊C4—121no—
143C3×C4⋊C4—181no—
144C3×C4⋊C4—181no—
145C3×C4⋊C4—184no—
146C4⋊Dic3—21no—
147C4⋊Dic3—61no—
148Dic3⋊C4—21no—
149Dic3⋊C4—61no—
150Dic3⋊C4—62no—
151C22×C32—21no—
152C22×C32—21no—
153C22×C32—41no—
154C22×C32—41no—
155C22×C32C2×C3×D411yescharacteristic
156C2×C3⋊D3C2×C3×D411yescharacteristic
157C2×C3⋊D3C2×C3×Q812yes—
158C2×C3⋊D3C3×#4112yes—
159C2×C3⋊D3C3×#4111yescharacteristic
160C2×C3×D3—61no—
161C2×C3×D3—61no—
162C2×C3×D3—62no—
163C2×C3×D3—62no—
164C2×C3×D3—61no—
165C2×C3×D3—62no—
166C2×C3×D3—61no—
167C2×C3×D3—62no—
168#157—24no—
169C32×C4—22no—
170C32×C4—21no—
171C32×C4—41no—
172C32×C4—42no—
173C32×C4—62no—
174C32×C4—81no—
175C32×C4—122no—
176C32×C4—121no—
177C3⋊Dic3—21no—
178C3×Dic3—22no—
179C3×Dic3—22no—
180C3×Dic3—42no—
181C3×Dic3—42no—
182C3×Dic3—41no—
183C3×Dic3—42no—
184C3×Dic3—41no—
185C3×Dic3—62no—
186C3×Dic3—82no—
187C3×Dic3—81no—
188C3×Dic3—121no—
189C2×C4⋊C4—91no—
190C2×C4⋊C4—91no—
191C2×C4⋊C4—92no—
192#95—91no—
193#95—92no—
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—61no—
203C2×C3×C4—61no—
204C2×C3×C4—61no—
205C2×C3×C4—62no—
206C2×C3×C4—62no—
207C2×C3×C4—61no—
208C2×C3×C4—92no—
209C2×C3×C4—94no—
210C2×C3×C4—91no—
211C2×C3×C4—121no—
212C2×C3×C4—121no—
213C2×C3×C4—122no—
214C2×C3×C4—182no—
215C2×C3×C4—181no—
216C2×C3×C4—182no—
217C2×C3×C4—182no—
218C2×C3×C4—182no—
219C2×C3×C4—182no—
220C2×C3×C4—182no—
221C2×C3×C4GO+(4,𝔽2)11yescharacteristic
222C2×Dic3—21no—
223C2×Dic3—21no—
224C2×Dic3—22no—
225C2×Dic3—61no—
226C2×Dic3—61no—
227C4×D3—62no—
228C4×D3—62no—
229C4×D3—64no—
230C4×D3—122no—
231C4×D3—122no—
232C4×D3—122no—
233C4×D3—122no—
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×C32C3×#12111yescharacteristic
247C2×C32C3×#12311yescharacteristic
248C2×C32C3×C4⋊Q811yescharacteristic
249C3⋊D3C3×C22⋊Q812yes—
250C3⋊D3C3×#12512yes—
251C3×D3—62no—
252C3×D3—62no—
253C3×D3—62no—
254C3×D3—62no—
255C22×C4—91no—
256C22×C4—91no—
257C22×C4—92no—
258C22×C4—92no—
259C22×C4—91no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—181no—
263C4⋊C4—181no—
264C4⋊C4—184no—
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—61no—
275C2×D3—62no—
276C2×D3—61no—
277C2×D3—61no—
278C2×D3—62no—
279C2×D3—61no—
280C2×D3—62no—
281C2×D3—62no—
282C3×C4—22no—
283C3×C4—21no—
284C3×C4—41no—
285C3×C4—42no—
286C3×C4—62no—
287C3×C4—62no—
288C3×C4—81no—
289C3×C4—122no—
290C3×C4—121no—
291C3×C4—122no—
292C3×C4—122no—
293C3×C4—121no—
294C3×C4—184no—
295C3×C4—181no—
296C3×C4—241no—
297C3×C4—242no—
298Dic3—22no—
299Dic3—41no—
300Dic3—42no—
301Dic3—62no—
302Dic3—121no—
303C32—21no—
304C32—21no—
305C32—41no—
306C32—41no—
307C32C3×#39811yescharacteristic, complemented
308C23—91no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—62no—
312C2×C4—92no—
313C2×C4—94no—
314C2×C4—91no—
315C2×C4—182no—
316C2×C4—181no—
317C2×C4—182no—
318C2×C4—182no—
319C2×C4—182no—
320C2×C4—182no—
321C2×C4—182no—
322C2×C4C3×GO+(4,𝔽2)11yescharacteristic
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—92no—
346C22—91no—
347C22—91no—
348C22C2×C3×GO+(4,𝔽2)11yescharacteristic
349C4—21no—
350C4—62no—
351C4—121no—
352C4—122no—
353C4—184no—
354C4—181no—
355C3—21no—
356C3—21no—
357C3—41no—
358C3—41no—
359C3#1059333911yescharacteristic, direct product factor
360C2—92no—
361C2—92no—
362C2C3×#6422211yescharacteristic
363C2C3×#6423011yescharacteristic
364C2C3×#6423311yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC22×C3⋊D3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3C23
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#398
LCSG ▷ C22×C3⋊D3 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
6. #10593339C3direct (N × H)
194. C33#398semidirect
307. C32C3×#398semidirect
359. C3#10593339direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37732
C3×#639002
C3×#638832
C3×#637812
C3×#637372
#105933393
C3×#3989