Finite Group Databasebeta

← #13155116

Subgroups — #13155116

Order 1728.

398 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 45 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#11723721C211yesmaximal, characteristic
2C3×C22⋊#622C211yesmaximal, characteristic
3#11722082C211yesmaximal, characteristic
4#10590072—31nomaximal
5C22×C3×C3⋊Dic3—21no—
6C22×C3×C3⋊Dic3C2211yescharacteristic
7C2×#6296C411yescharacteristic, complemented
8C2×C3×#622—21no—
9C3×#3757—21no—
10#88815—21no—
11#88815C411yescharacteristic, complemented
12#63982—31no—
13#63964—61no—
14C22⋊#622D311yescharacteristic
15#63579—31no—
16#63579—61no—
17C23×C33D411yescharacteristic
18C2×C3×C3⋊Dic3—22no—
19C2×C3×C3⋊Dic3—21no—
20C2×C3×C3⋊Dic3—21no—
21C2×C3×C3⋊Dic3—41no—
22C2×C3×C3⋊Dic3—41no—
23C2×C3×C3⋊Dic3—41no—
24C2×C3×C3⋊Dic3C2×C411yescharacteristic
25C2×C3×C3⋊Dic3D411yescharacteristic
26#6296—41no—
27C3×#622—41no—
28#4467—91nomaximal
29C22×C3⋊Dic3—21no—
30C22×C3⋊Dic3C2×D311yescharacteristic
31C22×C3×Dic3—61no—
32C22×C3×Dic3—61no—
33C22×C3×Dic3—61no—
34C22×C3×Dic3—61no—
35C22×C3×Dic3—121no—
36C2×#627—31no—
37C2×#627—61no—
38C2×#622—21no—
39#3757—21no—
40C3×#217—61no—
41C3×#217—61no—
42C3×#217—121no—
43#3724—31no—
44#3724—61no—
45Dic3⋊Dic3—61no—
46Dic3⋊Dic3—61no—
47Dic3⋊Dic3—61no—
48C22×C33—21no—
49C22×C33—21no—
50C22×C33C22⋊C411yescharacteristic
51C22×C33SD1611yescharacteristic
52C22×C33Q1611yescharacteristic
53C3×C3⋊Dic3—21no—
54C3×C3⋊Dic3—21no—
55C3×C3⋊Dic3—42no—
56C3×C3⋊Dic3—41no—
57C3×C3⋊Dic3—41no—
58#924—91no—
59#878—181no—
60C3×C22⋊C8—91no—
61C3⋊#95—91no—
62C3⋊#95—181no—
63C23×C32—21no—
64C23×C32—21no—
65C23×C32—41no—
66C23×C32—41no—
67C23×C32D1211yescharacteristic
68C2×C3⋊Dic3—21no—
69C2×C3⋊Dic3—21no—
70C2×C3⋊Dic3—22no—
71C2×C3⋊Dic3—41no—
72C2×C3⋊Dic3—41no—
73C2×C3⋊Dic3—41no—
74C2×C3⋊Dic3C4×D311yescharacteristic
75C2×C3⋊Dic3#6711yescharacteristic
76C2×C3×Dic3—61no—
77C2×C3×Dic3—61no—
78C2×C3×Dic3—61no—
79C2×C3×Dic3—61no—
80C2×C3×Dic3—61no—
81C2×C3×Dic3—61no—
82C2×C3×Dic3—62no—
83C2×C3×Dic3—62no—
84C2×C3×Dic3—61no—
85C2×C3×Dic3—61no—
86C2×C3×Dic3—122no—
87C2×C3×Dic3—121no—
88C2×C3×Dic3—121no—
89C2×C3×Dic3—122no—
90C2×C3×Dic3—121no—
91C2×C3×Dic3—121no—
92C2×C3×Dic3—121no—
93C2×C3×Dic3—121no—
94C2×C3×Dic3—121no—
95C2×C3×Dic3—122no—
96C2×C3×Dic3—122no—
97C2×C3×Dic3—121no—
98#627—122no—
99#627—121no—
100#622—41no—
101#328—271noSylow
102C2×C33—21no—
103C2×C33—21no—
104C2×C33#9911yescharacteristic
105C2×C33#10311yescharacteristic
106C2×C33#10411yescharacteristic
107C22×C3×C4—91no—
108C22×C3×C4—181no—
109C22×Dic3—61no—
110C22×Dic3—61no—
111C22×Dic3—61no—
112C22×Dic3—61no—
113C22×Dic3—121no—
114C2×Dic6—91no—
115C2×Dic6—181no—
116C2×C3×C8—181no—
117C3×C22⋊C4—181no—
118#217—61no—
119#217—61no—
120#217—121no—
121C4⋊Dic3—91no—
122C4⋊Dic3—181no—
123Dic3⋊C4—181no—
124Dic3⋊C4—181no—
125Dic3⋊C4—181no—
126C22×C32—21no—
127C22×C32—21no—
128C22×C32—21no—
129C22×C32—21no—
130C22×C32—21no—
131C22×C32—21no—
132C22×C32—21no—
133C22×C32—21no—
134C22×C32—21no—
135C22×C32—21no—
136C22×C32—21no—
137C22×C32—21no—
138C22×C32—41no—
139C22×C32—41no—
140C22×C32—41no—
141C22×C32—41no—
142C22×C32—41no—
143C22×C32—41no—
144C22×C32—41no—
145C22×C32—41no—
146C22×C32—41no—
147C22×C32—41no—
148C22×C32—41no—
149C22×C32—41no—
150C22×C32—81no—
151C22×C32—81no—
152C22×C32#20411yescharacteristic
153C22×C32Dic1211yescharacteristic
154C22×C32(C2×D3)⋊C411yescharacteristic
155C3⋊Dic3—21no—
156C3⋊Dic3—21no—
157C3⋊Dic3—42no—
158C3⋊Dic3—41no—
159C3⋊Dic3—41no—
160C3×Dic3—62no—
161C3×Dic3—121no—
162C3×Dic3—121no—
163C3×Dic3—121no—
164C3×Dic3—121no—
165C3×Dic3—121no—
166C3×Dic3—122no—
167C3×Dic3—122no—
168C3×Dic3—124no—
169C3×Dic3—121no—
170C3×Dic3—121no—
171C3×Dic3—241no—
172C3×Dic3—241no—
173C22⋊Q8—271no—
174C22⋊C8—91no—
175#95—271no—
176C33#32811yescharacteristic, complemented, Sylow
177C23×C3—21no—
178C23×C3—21no—
179C23×C3—41no—
180C23×C3—41no—
181C23×C3GO+(4,𝔽2)11yescharacteristic
182C2×C3×C4—91no—
183C2×C3×C4—91no—
184C2×C3×C4—181no—
185C2×C3×C4—181no—
186C2×C3×C4—182no—
187C2×C3×C4—361no—
188C2×C3×C4—361no—
189C2×C3×C4—361no—
190C2×Dic3—61no—
191C2×Dic3—61no—
192C2×Dic3—61no—
193C2×Dic3—61no—
194C2×Dic3—61no—
195C2×Dic3—61no—
196C2×Dic3—62no—
197C2×Dic3—62no—
198C2×Dic3—61no—
199C2×Dic3—61no—
200C2×Dic3—122no—
201C2×Dic3—121no—
202C2×Dic3—121no—
203C2×Dic3—122no—
204C2×Dic3—121no—
205C2×Dic3—121no—
206C2×Dic3—121no—
207C2×Dic3—121no—
208C2×Dic3—121no—
209C2×Dic3—122no—
210C2×Dic3—122no—
211C2×Dic3—121no—
212Dic6—362no—
213Dic6—361no—
214C3×C8—361no—
215C2×C32—21no—
216C2×C32—21no—
217C2×C32—21no—
218C2×C32—21no—
219C2×C32—21no—
220C2×C32—21no—
221C2×C32—21no—
222C2×C32—21no—
223C2×C32—21no—
224C2×C32—21no—
225C2×C32—21no—
226C2×C32—21no—
227C2×C32—41no—
228C2×C32—41no—
229C2×C32—41no—
230C2×C32—41no—
231C2×C32—41no—
232C2×C32—41no—
233C2×C32—41no—
234C2×C32—41no—
235C2×C32—41no—
236C2×C32—41no—
237C2×C32—41no—
238C2×C32—41no—
239C2×C32—81no—
240C2×C32—81no—
241C2×C32#80611yescharacteristic
242C2×C32#81611yescharacteristic
243C2×C32#82511yescharacteristic
244C2×Q8—271no—
245C22×C4—91no—
246C22×C4—181no—
247C2×C8—181no—
248C4⋊C4—271no—
249C4⋊C4—541no—
250C22⋊C4—181no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—21no—
257C22×C3—21no—
258C22×C3—21no—
259C22×C3—21no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—41no—
264C22×C3—41no—
265C22×C3—41no—
266C22×C3—41no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3—81no—
276C22×C3—81no—
277C22×C3D32⋊C411yescharacteristic
278C22×C3#377511yescharacteristic
279C22×C3#377611yescharacteristic
280C3×C4—181no—
281C3×C4—181no—
282C3×C4—361no—
283C3×C4—362no—
284C3×C4—361no—
285Dic3—62no—
286Dic3—121no—
287Dic3—121no—
288Dic3—121no—
289Dic3—121no—
290Dic3—122no—
291Dic3—122no—
292Dic3—121no—
293Dic3—124no—
294Dic3—121no—
295Dic3—121no—
296Dic3—241no—
297Dic3—241no—
298C32—21no—
299C32—21no—
300C32—41no—
301C32—41no—
302C32#446711yescharacteristic, complemented
303C23#157⋊D311yescharacteristic
304Q8—1081no—
305C2×C4—91no—
306C2×C4—91no—
307C2×C4—182no—
308C2×C4—181no—
309C2×C4—181no—
310C2×C4—361no—
311C2×C4—361no—
312C2×C4—361no—
313C8—361no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—21no—
322C2×C3—21no—
323C2×C3—21no—
324C2×C3—21no—
325C2×C3—21no—
326C2×C3—41no—
327C2×C3—41no—
328C2×C3—41no—
329C2×C3—41no—
330C2×C3—41no—
331C2×C3—41no—
332C2×C3—41no—
333C2×C3—41no—
334C2×C3—41no—
335C2×C3—41no—
336C2×C3—41no—
337C2×C3—41no—
338C2×C3—81no—
339C2×C3—81no—
340C2×C3#6373811yescharacteristic
341C2×C3#6374011yescharacteristic
342C2×C3#6374111yescharacteristic
343C22—21no—
344C22—21no—
345C22#8893911yescharacteristic
346C22#8894211yescharacteristic
347C22#8894311yescharacteristic
348C4—181no—
349C4—181no—
350C4—362no—
351C4—361no—
352C4—361no—
353C3—21no—
354C3—21no—
355C3—41no—
356C3—41no—
357C3#1059007211yescharacteristic, complemented
358C2—21no—
359C2—21no—
360C2#1172279211yescharacteristic
361C2#1172279411yescharacteristic
362C2#1172279511yescharacteristic
subgroupG/H
centerC22#88939
commutatorC2×C3×C3⋊Dic3C2×C4
Frattini, hypercenterC23#157⋊D3
Fitting, prime radicalC23×C33D4
socleC22×C33C22⋊C4
cosocleC22×C3×C3⋊Dic3C22
hypocenterC33#328
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
7. C2×#6296C4semidirect
11. #88815C4semidirect
176. C33#328semidirect
302. C32#4467semidirect
357. C3#10590072semidirect

Maximal subgroups

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

subgroupindex
#117237212
C3×C22⋊#6222
#117220822
#105900723
#44679