Finite Group Databasebeta

← #13167280

Subgroups — #13167280

Order 1728.

522 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 59 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×#88815C211yesmaximal, characteristic, complemented
2C3×#63980C211yesmaximal, characteristic
3#11723112C211yesmaximal, characteristic, complemented
4#11722779C212yesmaximal, complemented
5#11722773C212yesmaximal, complemented
6#10593492—31nomaximal
7C22×C3×C3⋊Dic3—21no—
8C22×C3×C3⋊Dic3C2211yescharacteristic
9C2×C32×#67—21no—
10C2×C3×#625C2211yescharacteristic
11C2×C3⋊#622C2212yes—
12#88815—21no—
13#88815C2212yes—
14C3×#3724C2211yescharacteristic
15C3×(C2×D3)⋊Dic3—21no—
16C3×#939—61no—
17#63980D311yescharacteristic
18C2×#3724—31no—
19C2×Dic3⋊Dic3—61no—
20C22⋊#622—31no—
21#63743—32no—
22#63739—32no—
23C23×C33D411yescharacteristic
24C22×C32×D3—21no—
25C2×C3×C3⋊Dic3—22no—
26C2×C3×C3⋊Dic3—22no—
27C2×C3×C3⋊Dic3—42no—
28C2×C3×C3⋊Dic3D411yescharacteristic
29C2×C3×C3⋊Dic3C2311yescharacteristic
30C32×#67—42no—
31C2×C32×Dic3—21no—
32C3×#625—22no—
33C3⋊#622—22no—
34#5033—91nomaximal
35C2×C32×D4—61no—
36C22×C3⋊Dic3—21no—
37C22×C3⋊Dic3C2×D311yescharacteristic
38C2×C3×#67—21no—
39C2×C3×#67—21no—
40C2×C3×#67—41no—
41C2×C3×#67—61no—
42C22×C3×Dic3—61no—
43C22×C3×Dic3—61no—
44C22×C3×Dic3—61no—
45C22×C3×Dic3—61no—
46C22×C3×Dic3—121no—
47C2×#625C2×D311yescharacteristic
48C2×#622—32no—
49C3×#217—61no—
50C3×(C2×D3)⋊C4—61no—
51C3×Dic3⋊C4—61no—
52#3724—32no—
53#3724—61no—
54#3724C2×D311yescharacteristic
55Dic3⋊Dic3—62no—
56Dic3⋊Dic3—62no—
57(C2×D3)⋊Dic3—21no—
58C22×C33—21no—
59C22×C33—21no—
60C22×C33D812yes—
61C22×C33C2×D411yescharacteristic
62C2×C32×D3—41no—
63C2×C32×D3—41no—
64C2×C32×D3—41no—
65C3×C3⋊Dic3—21no—
66C3×C3⋊Dic3—42no—
67C3×C3⋊Dic3#4112yes—
68C32×Dic3—41no—
69C3×#121—91no—
70#939—61no—
71C2×C4⋊Dic3—91no—
72C2×Dic3⋊C4—181no—
73#832—92no—
74#830—91no—
75#807—92no—
76C23×C32—21no—
77C23×C32—21no—
78C23×C32—41no—
79C23×C32—41no—
80C23×C32—61no—
81C23×C32#6711yescharacteristic
82C22×C3×D3—21no—
83C22×C3×D3—21no—
84C22×C3×D3—41no—
85C32×D4—122no—
86C2×C32×C4—61no—
87C2×C3⋊Dic3—22no—
88C2×C3⋊Dic3—22no—
89C2×C3⋊Dic3—42no—
90C2×C3⋊Dic3#6711yescharacteristic
91C2×C3⋊Dic3C22×D311yescharacteristic
92C3×#67—42no—
93C3×#67—44no—
94C3×#67—42no—
95C3×#67—122no—
96C2×C3×Dic3—21no—
97C2×C3×Dic3—21no—
98C2×C3×Dic3—41no—
99C2×C3×Dic3—61no—
100C2×C3×Dic3—61no—
101C2×C3×Dic3—61no—
102C2×C3×Dic3—61no—
103C2×C3×Dic3—64no—
104C2×C3×Dic3—62no—
105C2×C3×Dic3—62no—
106C2×C3×Dic3—122no—
107C2×C3×Dic3—122no—
108C2×C3×Dic3—122no—
109C2×C3×Dic3—124no—
110C2×C3×Dic3—122no—
111#625—22no—
112#622—62no—
113#480—271noSylow
114C2×C33—22no—
115C2×C33#12311yescharacteristic
116C2×C33C2×D811yescharacteristic
117C2×C33#13711yescharacteristic
118C32×D3—81no—
119C2×C3×D4—61no—
120C2×C3×D4—61no—
121C2×C3×D4—91no—
122C2×C3×D4—121no—
123C22×C3×C4—91no—
124C22×C3×C4—181no—
125C2×#67—21no—
126C2×#67—61no—
127C22×Dic3—61no—
128C22×Dic3—61no—
129C22×Dic3—61no—
130C22×Dic3—61no—
131C22×Dic3—121no—
132C3×C4⋊C4—91no—
133C3×C22⋊C4—181no—
134#217—61no—
135(C2×D3)⋊C4—61no—
136C4⋊Dic3—92no—
137C4⋊Dic3—181no—
138Dic3⋊C4—61no—
139Dic3⋊C4—182no—
140Dic3⋊C4—182no—
141C2×C3⋊C8—92no—
142C22×C32—22no—
143C22×C32—22no—
144C22×C32—21no—
145C22×C32—21no—
146C22×C32—21no—
147C22×C32—21no—
148C22×C32—22no—
149C22×C32—22no—
150C22×C32—42no—
151C22×C32—41no—
152C22×C32—41no—
153C22×C32—42no—
154C22×C32—41no—
155C22×C32—41no—
156C22×C32—42no—
157C22×C32—42no—
158C22×C32—81no—
159C22×C32—81no—
160C22×C32—121no—
161C22×C32—121no—
162C22×C32—121no—
163C22×C32#21312yes—
164C22×C32C2×#6711yescharacteristic
165C2×C3×D3—41no—
166C2×C3×D3—41no—
167C2×C3×D3—41no—
168C2×C3×D3—42no—
169C2×C3×D3—41no—
170C2×C3×D3—41no—
171C2×C3×D3—41no—
172C2×C3×D3—81no—
173C2×C3×D3—81no—
174C32×C4—121no—
175C3⋊Dic3—21no—
176C3⋊Dic3—42no—
177C3⋊Dic3#23712yes—
178C3×Dic3—41no—
179C3×Dic3—42no—
180C3×Dic3—41no—
181C3×Dic3—62no—
182C3×Dic3—62no—
183C3×Dic3—62no—
184C3×Dic3—121no—
185C3×Dic3—122no—
186C3×Dic3—124no—
187C3×Dic3—122no—
188#121—91no—
189C2×C4⋊C4—271no—
190#107—272no—
191#102—272no—
192C22⋊C8—271no—
193C33#48011yescharacteristic, complemented, Sylow
194C23×C3—21no—
195C23×C3—21no—
196C23×C3—41no—
197C23×C3—41no—
198C23×C3—61no—
199C23×C3—61no—
200C23×C3—121no—
201C23×C3GO+(4,𝔽2)11yescharacteristic
202C22×D3—21no—
203C3×D4—122no—
204C3×D4—124no—
205C3×D4—122no—
206C3×D4—182no—
207C2×C3×C4—61no—
208C2×C3×C4—61no—
209C2×C3×C4—91no—
210C2×C3×C4—91no—
211C2×C3×C4—121no—
212C2×C3×C4—182no—
213C2×C3×C4—182no—
214C2×C3×C4—362no—
215#67—42no—
216#67—122no—
217C2×Dic3—21no—
218C2×Dic3—61no—
219C2×Dic3—61no—
220C2×Dic3—61no—
221C2×Dic3—61no—
222C2×Dic3—64no—
223C2×Dic3—62no—
224C2×Dic3—62no—
225C2×Dic3—122no—
226C2×Dic3—122no—
227C2×Dic3—122no—
228C2×Dic3—124no—
229C2×Dic3—122no—
230C3⋊C8—182no—
231C2×C32—24no—
232C2×C32—22no—
233C2×C32—21no—
234C2×C32—21no—
235C2×C32—21no—
236C2×C32—21no—
237C2×C32—21no—
238C2×C32—21no—
239C2×C32—41no—
240C2×C32—41no—
241C2×C32—41no—
242C2×C32—41no—
243C2×C32—41no—
244C2×C32—41no—
245C2×C32—42no—
246C2×C32—44no—
247C2×C32—82no—
248C2×C32—241no—
249C2×C32C2×#21311yescharacteristic
250C2×C32#93511yescharacteristic
251C2×C32#95111yescharacteristic
252C3×D3—81no—
253C3×D3—82no—
254C3×D3—81no—
255C2×D4—61no—
256C2×D4—91no—
257C22×C4—91no—
258C22×C4—181no—
259C2×C8—272no—
260C4⋊C4—91no—
261C4⋊C4—272no—
262C4⋊C4—541no—
263C22⋊C4—181no—
264C22×C3—22no—
265C22×C3—22no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—22no—
271C22×C3—22no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—42no—
275C22×C3—42no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—42no—
279C22×C3—42no—
280C22×C3—81no—
281C22×C3—81no—
282C22×C3—121no—
283C22×C3—121no—
284C22×C3—121no—
285C22×C3—122no—
286C22×C3—121no—
287C22×C3—121no—
288C22×C3—121no—
289C22×C3—241no—
290C22×C3—241no—
291C22×C3#377412yes—
292C22×C3C2×GO+(4,𝔽2)11yescharacteristic
293C2×D3—41no—
294C2×D3—41no—
295C2×D3—41no—
296C3×C4—92no—
297C3×C4—121no—
298C3×C4—122no—
299C3×C4—121no—
300C3×C4—181no—
301C3×C4—362no—
302Dic3—41no—
303Dic3—62no—
304Dic3—62no—
305Dic3—62no—
306Dic3—121no—
307Dic3—122no—
308Dic3—124no—
309Dic3—122no—
310C32—21no—
311C32—21no—
312C32—41no—
313C32—41no—
314C32#503311yescharacteristic, complemented
315C23—61no—
316C23D32⋊D311yescharacteristic
317D4—122no—
318D4—182no—
319C2×C4—61no—
320C2×C4—91no—
321C2×C4—91no—
322C2×C4—182no—
323C2×C4—182no—
324C2×C4—362no—
325C8—542no—
326C2×C3—24no—
327C2×C3—22no—
328C2×C3—21no—
329C2×C3—21no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3—21no—
334C2×C3—42no—
335C2×C3—44no—
336C2×C3—41no—
337C2×C3—41no—
338C2×C3—41no—
339C2×C3—41no—
340C2×C3—41no—
341C2×C3—41no—
342C2×C3—82no—
343C2×C3—241no—
344C2×C3—242no—
345C2×C3—241no—
346C2×C3#6423311yescharacteristic
347C2×C3C2×#377411yescharacteristic
348C2×C3#6423911yescharacteristic
349D3—81no—
350C22—21no—
351C22—21no—
352C22—121no—
353C22—121no—
354C22—121no—
355C22#8893512yes—
356C22C2×D32⋊D311yescharacteristic
357C4—92no—
358C4—121no—
359C4—181no—
360C4—362no—
361C3—21no—
362C3—21no—
363C3—41no—
364C3—41no—
365C3#1059349211yescharacteristic, complemented
366C2—22no—
367C2—241no—
368C2#1172431911yescharacteristic
369C2C2×#8893511yescharacteristic
370C2#1172432811yescharacteristic
subgroupG/H
center, FrattiniC22C2×D32⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC23×C33D4
socleC22×C33C2×D4
prime radicalC3×#63980C2
hypercenterC23D32⋊D3
hypocenterC33#480
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C2×#88815C2semidirect
3. #11723112C2semidirect
4. #11722779C2semidirect
5. #11722773C2semidirect
193. C33#480semidirect
314. C32#5033semidirect
365. C3#10593492semidirect

Maximal subgroups

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

subgroupindex
C2×#888152
C3×#639802
#117231122
#117227792
#117227732
#105934923
#50339