Finite Group Databasebeta

← #13155115

Subgroups — #13155115

Order 1728.

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

normal subgroups only

This group has 458 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#11723717C211yesmaximal, characteristic
2C3×C22⋊#622C211yesmaximal, characteristic, complemented
3#11722082C211yesmaximal, characteristic, complemented
4#10590071—31nomaximal
5C2×C3×#638—21no—
6C22×C3×C3⋊Dic3—21no—
7C22×C3×C3⋊Dic3C2211yescharacteristic
8C2×#6295C411yescharacteristic, complemented
9C2×C3×#622—21no—
10#88815C411yescharacteristic, complemented
11#88814—21no—
12#63980—31no—
13#63978—61no—
14C22⋊#622D311yescharacteristic, complemented
15#63579—31no—
16#63579—61no—
17C23×C33D411yescharacteristic
18C22×C3×C3⋊D3—21no—
19C3×#638—41no—
20C3×#638—41no—
21C2×C3×C3⋊Dic3—22no—
22C2×C3×C3⋊Dic3—21no—
23C2×C3×C3⋊Dic3—41no—
24C2×C3×C3⋊Dic3—41no—
25C2×C3×C3⋊Dic3—41no—
26C2×C3×C3⋊Dic3C2×C411yescharacteristic
27C2×C3×C3⋊Dic3D411yescharacteristic
28#6295—41no—
29C3×#622—41no—
30#4470—91nomaximal
31C2×#638—21no—
32C22×C3⋊Dic3—21no—
33C22×C3⋊Dic3C2×D311yescharacteristic
34C2×C3×#67—61no—
35C2×C3×#67—61no—
36C2×C3×#67—121no—
37C22×C3×Dic3—61no—
38C22×C3×Dic3—61no—
39C22×C3×Dic3—61no—
40C22×C3×Dic3—61no—
41C22×C3×Dic3—121no—
42C2×#626—61no—
43C2×#625—31no—
44C2×#622—21no—
45#3724—31no—
46Dic3⋊Dic3—61no—
47#3722—61no—
48(C2×D3)⋊Dic3—61no—
49(C2×D3)⋊Dic3—61no—
50C22×C33—21no—
51C22×C33—21no—
52C22×C33C22⋊C411yescharacteristic
53C22×C33D811yescharacteristic
54C22×C33SD1611yescharacteristic
55C2×C3×C3⋊D3—41no—
56C2×C3×C3⋊D3—41no—
57C2×C3×C3⋊D3—41no—
58C3×C3⋊Dic3—21no—
59C3×C3⋊Dic3—21no—
60C3×C3⋊Dic3—42no—
61C3×C3⋊Dic3—41no—
62#939—181no—
63#930—91no—
64C3×C22⋊C8—91no—
65C3⋊#95—91no—
66C3⋊#95—181no—
67C23×C32—21no—
68C23×C32—21no—
69C23×C32—41no—
70C23×C32—41no—
71C23×C32D1211yescharacteristic
72C22×C3⋊D3—21no—
73C22×C3×D3—61no—
74C22×C3×D3—61no—
75C22×C3×D3—121no—
76#638—41no—
77#638—41no—
78C2×C3⋊Dic3—21no—
79C2×C3⋊Dic3—22no—
80C2×C3⋊Dic3—41no—
81C2×C3⋊Dic3—41no—
82C2×C3⋊Dic3—41no—
83C2×C3⋊Dic3C4×D311yescharacteristic
84C2×C3⋊Dic3#6711yescharacteristic
85C3×#67—121no—
86C3×#67—121no—
87C3×#67—122no—
88C3×#67—122no—
89C3×#67—121no—
90C3×#67—121no—
91C2×C3×Dic3—61no—
92C2×C3×Dic3—61no—
93C2×C3×Dic3—61no—
94C2×C3×Dic3—61no—
95C2×C3×Dic3—61no—
96C2×C3×Dic3—62no—
97C2×C3×Dic3—62no—
98C2×C3×Dic3—61no—
99C2×C3×Dic3—122no—
100C2×C3×Dic3—121no—
101C2×C3×Dic3—122no—
102C2×C3×Dic3—121no—
103C2×C3×Dic3—121no—
104C2×C3×Dic3—121no—
105C2×C3×Dic3—121no—
106C2×C3×Dic3—121no—
107C2×C3×Dic3—122no—
108C2×C3×Dic3—122no—
109C2×C3×Dic3—121no—
110#626—122no—
111#625—121no—
112#622—41no—
113#327—271noSylow
114C2×C33—21no—
115C2×C33—21no—
116C2×C33#9911yescharacteristic
117C2×C33#10211yescharacteristic
118C2×C33#10411yescharacteristic
119C3×C3⋊D3—81no—
120C2×C3×D4—181no—
121C22×C3×C4—91no—
122C22×C3×C4—181no—
123C2×#67—61no—
124C2×#67—61no—
125C2×#67—121no—
126C2×#67—181no—
127C22×Dic3—61no—
128C22×Dic3—61no—
129C22×Dic3—61no—
130C22×Dic3—61no—
131C22×Dic3—121no—
132C2×D12—91no—
133C2×C3×C8—181no—
134#217—181no—
135(C2×D3)⋊C4—181no—
136(C2×D3)⋊C4—181no—
137C4⋊Dic3—91no—
138Dic3⋊C4—181no—
139C22×C32—21no—
140C22×C32—21no—
141C22×C32—21no—
142C22×C32—21no—
143C22×C32—21no—
144C22×C32—21no—
145C22×C32—21no—
146C22×C32—21no—
147C22×C32—21no—
148C22×C32—21no—
149C22×C32—21no—
150C22×C32—21no—
151C22×C32—41no—
152C22×C32—41no—
153C22×C32—41no—
154C22×C32—41no—
155C22×C32—41no—
156C22×C32—41no—
157C22×C32—41no—
158C22×C32—41no—
159C22×C32—41no—
160C22×C32—41no—
161C22×C32—41no—
162C22×C32—41no—
163C22×C32—81no—
164C22×C32—81no—
165C22×C32#20411yescharacteristic
166C22×C32D2411yescharacteristic
167C22×C32(C2×D3)⋊C411yescharacteristic
168C2×C3⋊D3—41no—
169C2×C3⋊D3—41no—
170C2×C3⋊D3—41no—
171C2×C3×D3—121no—
172C2×C3×D3—121no—
173C2×C3×D3—121no—
174C2×C3×D3—122no—
175C2×C3×D3—121no—
176C2×C3×D3—121no—
177C2×C3×D3—121no—
178C2×C3×D3—241no—
179C2×C3×D3—241no—
180C3⋊Dic3—21no—
181C3⋊Dic3—21no—
182C3⋊Dic3—42no—
183C3⋊Dic3—41no—
184C3×Dic3—62no—
185C3×Dic3—121no—
186C3×Dic3—121no—
187C3×Dic3—121no—
188C3×Dic3—122no—
189C3×Dic3—122no—
190C3×Dic3—121no—
191C3×Dic3—124no—
192C3×Dic3—122no—
193C3×Dic3—121no—
194#121—271no—
195C22⋊C8—91no—
196#95—271no—
197C33#32711yescharacteristic, complemented, Sylow
198C23×C3—21no—
199C23×C3—21no—
200C23×C3—41no—
201C23×C3—41no—
202C23×C3—181no—
203C23×C3GO+(4,𝔽2)11yescharacteristic
204C22×D3—61no—
205C22×D3—61no—
206C22×D3—121no—
207C3×D4—361no—
208C3×D4—361no—
209C2×C3×C4—91no—
210C2×C3×C4—91no—
211C2×C3×C4—181no—
212C2×C3×C4—182no—
213C2×C3×C4—361no—
214C2×C3×C4—361no—
215C2×C3×C4—361no—
216#67—121no—
217#67—121no—
218#67—122no—
219#67—122no—
220#67—121no—
221#67—121no—
222#67—362no—
223C2×Dic3—61no—
224C2×Dic3—61no—
225C2×Dic3—61no—
226C2×Dic3—61no—
227C2×Dic3—61no—
228C2×Dic3—62no—
229C2×Dic3—62no—
230C2×Dic3—61no—
231C2×Dic3—122no—
232C2×Dic3—121no—
233C2×Dic3—122no—
234C2×Dic3—121no—
235C2×Dic3—121no—
236C2×Dic3—121no—
237C2×Dic3—121no—
238C2×Dic3—121no—
239C2×Dic3—122no—
240C2×Dic3—122no—
241C2×Dic3—121no—
242D12—361no—
243C3×C8—361no—
244C2×C32—21no—
245C2×C32—21no—
246C2×C32—21no—
247C2×C32—21no—
248C2×C32—21no—
249C2×C32—21no—
250C2×C32—21no—
251C2×C32—21no—
252C2×C32—21no—
253C2×C32—21no—
254C2×C32—21no—
255C2×C32—21no—
256C2×C32—41no—
257C2×C32—41no—
258C2×C32—41no—
259C2×C32—41no—
260C2×C32—41no—
261C2×C32—41no—
262C2×C32—41no—
263C2×C32—41no—
264C2×C32—41no—
265C2×C32—41no—
266C2×C32—41no—
267C2×C32—41no—
268C2×C32—81no—
269C2×C32—81no—
270C2×C32#80611yescharacteristic
271C2×C32#82111yescharacteristic
272C2×C32#82511yescharacteristic
273C3⋊D3—81no—
274C3×D3—241no—
275C3×D3—242no—
276C3×D3—241no—
277C2×D4—181no—
278C2×D4—271no—
279C22×C4—91no—
280C22×C4—181no—
281C2×C8—181no—
282C4⋊C4—271no—
283C22⋊C4—541no—
284C22×C3—21no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—41no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—41no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3—41no—
308C22×C3—81no—
309C22×C3—81no—
310C22×C3—361no—
311C22×C3—361no—
312C22×C3—361no—
313C22×C3D32⋊C411yescharacteristic
314C22×C3#377411yescharacteristic
315C22×C3#377511yescharacteristic
316C2×D3—121no—
317C2×D3—121no—
318C2×D3—121no—
319C2×D3—122no—
320C2×D3—121no—
321C2×D3—121no—
322C2×D3—121no—
323C2×D3—241no—
324C2×D3—241no—
325C3×C4—181no—
326C3×C4—181no—
327C3×C4—362no—
328C3×C4—361no—
329Dic3—62no—
330Dic3—121no—
331Dic3—121no—
332Dic3—121no—
333Dic3—122no—
334Dic3—122no—
335Dic3—121no—
336Dic3—124no—
337Dic3—122no—
338Dic3—121no—
339C32—21no—
340C32—21no—
341C32—41no—
342C32—41no—
343C32#447011yescharacteristic, complemented
344C23—181no—
345C23#157⋊D311yescharacteristic
346D4—361no—
347D4—361no—
348D4—1081no—
349C2×C4—91no—
350C2×C4—91no—
351C2×C4—182no—
352C2×C4—181no—
353C2×C4—361no—
354C2×C4—361no—
355C2×C4—361no—
356C8—361no—
357C2×C3—21no—
358C2×C3—21no—
359C2×C3—21no—
360C2×C3—21no—
361C2×C3—21no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3—21no—
365C2×C3—21no—
366C2×C3—21no—
367C2×C3—21no—
368C2×C3—21no—
369C2×C3—41no—
370C2×C3—41no—
371C2×C3—41no—
372C2×C3—41no—
373C2×C3—41no—
374C2×C3—41no—
375C2×C3—41no—
376C2×C3—41no—
377C2×C3—41no—
378C2×C3—41no—
379C2×C3—41no—
380C2×C3—41no—
381C2×C3—81no—
382C2×C3—81no—
383C2×C3—721no—
384C2×C3#6373811yescharacteristic
385C2×C3#6373911yescharacteristic
386C2×C3#6374111yescharacteristic
387D3—241no—
388D3—242no—
389D3—241no—
390C22—21no—
391C22—21no—
392C22—361no—
393C22—361no—
394C22—361no—
395C22#8893911yescharacteristic
396C22#8894111yescharacteristic
397C22#8894211yescharacteristic
398C4—181no—
399C4—181no—
400C4—362no—
401C4—361no—
402C3—21no—
403C3—21no—
404C3—41no—
405C3—41no—
406C3#1059007111yescharacteristic, complemented
407C2—21no—
408C2—21no—
409C2—721no—
410C2#1172279211yescharacteristic
411C2#1172279311yescharacteristic
412C2#1172279511yescharacteristic
subgroupG/H
centerC22#88939
commutatorC2×C3×C3⋊Dic3C2×C4
Frattini, hypercenterC23#157⋊D3
FittingC23×C33D4
socleC22×C33C22⋊C4
prime radical#11723717C2
cosocleC22×C3×C3⋊Dic3C22
hypocenterC33#327
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
2. C3×C22⋊#622C2semidirect
3. #11722082C2semidirect
8. C2×#6295C4semidirect
10. #88815C4semidirect
14. C22⋊#622D3semidirect
197. C33#327semidirect
343. C32#4470semidirect
406. C3#10590071semidirect

Maximal subgroups

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

subgroupindex
#117237172
C3×C22⋊#6222
#117220822
#105900713
#44709