Finite Group Databasebeta

← #13155514

Subgroups — #13155514

Order 1728.

698 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 132 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
1C4×D3×C3⋊Dic3C211yesmaximal, characteristic
2#11722078C211yesmaximal, characteristic, complemented
3#11722018C212yesmaximal, complemented
4C3×#63572C211yesmaximal, characteristic, complemented
5C3×(C2×D3)⋊(C3⋊C8)C212yesmaximal
6#10591061—31nomaximal
7#10590656—32nomaximal
8C2×C32×C4×D3C2211yescharacteristic
9C2×D3×C3⋊Dic3C411yescharacteristic
10C2×D3×C3⋊Dic3C411yescharacteristic
11C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C2×C3×C3⋊(C3⋊C8)C2212yes—
14C2×C32×C3⋊C8C2212yes—
15Dic3×C3⋊Dic3C411yescharacteristic
16Dic3×C3⋊Dic3C411yescharacteristic
17C2×C4×C3⋊Dic3—31no—
18C4×D3×Dic3—32no—
19C4×D3×Dic3—61no—
20C8×C3⋊Dic3—32no—
21C3×#830—32no—
22C3×(C2×D3)⋊C8—32no—
23C3×Dic3⋊C8—32no—
24#63572—31no—
25#63572D311yescharacteristic, complemented
26Dic3⋊(C3⋊C8)—32no—
27(C2×D3)⋊(C3⋊C8)D312yes—
28#63553—32no—
29#63553—32no—
30#63553—62no—
31C22×C32×D3C2×C411yescharacteristic
32C2×C33×C4C2311yescharacteristic
33C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
34C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
35C2×C3×C3⋊Dic3C2×C411yescharacteristic
36C2×C3×C3⋊Dic3C2×C411yescharacteristic
37C2×C32×Dic3C2×C411yescharacteristic
38C32×C4×D3—21no—
39C32×C4×D3—21no—
40D3×C3⋊Dic3—22no—
41D3×C3⋊Dic3C814yescomplemented
42C3×C3⋊(C3⋊C8)—22no—
43C32×C3⋊C8—22no—
44#4833—91no—
45#4718—92no—
46C22×C32×C4—31no—
47C22×C3⋊Dic3—31no—
48C22×C3⋊Dic3—31no—
49C2×C3×C4×D3—21no—
50C2×C3×C4×D3C2×D312yes—
51C2×D3×Dic3—32no—
52C2×D3×Dic3—32no—
53C2×D3×Dic3—61no—
54C2×D3×Dic3—61no—
55C2×C32×C8—32no—
56C4×C3⋊Dic3—31no—
57C4×C3⋊Dic3—32no—
58C4×C3⋊Dic3—31no—
59C4×C3⋊Dic3—31no—
60C4×C3⋊Dic3—62no—
61C4×C3⋊Dic3—61no—
62C4×C3⋊Dic3—121no—
63C4×C3⋊Dic3C2×D311yescharacteristic
64C2×C3⋊(C3⋊C8)C2×D312yes—
65C3×C4×Dic3—32no—
66C3×C4×Dic3—61no—
67C2×C3×C3⋊C8—22no—
68C2×C3×C3⋊C8—32no—
69C2×C3×C3⋊C8—32no—
70C2×C3×C3⋊C8—62no—
71C2×C3×C3⋊C8C2×D312yes—
72C2×C3×C3⋊C8C2×D312yes—
73Dic32—32no—
74Dic32—32no—
75Dic32—61no—
76Dic32—61no—
77C22×C33C22×C411yescharacteristic
78C2×C32×D3—21no—
79C2×C32×D3—21no—
80C2×C32×D3C2×C812yes—
81C33×C4#4111yescharacteristic
82C33×C4#4111yescharacteristic
83C3⋊(C3⋊Dic3)C2×C812yes—
84C3⋊(C3⋊Dic3)M1612yes—
85C3×C3⋊Dic3—21no—
86C3×C3⋊Dic3C2×C812yes—
87C32×Dic3—21no—
88C2×C4×Dic3—92no—
89C2×C4×Dic3—181no—
90C42×D3—91no—
91C3×C4⋊C8—91no—
92C3×C22⋊C8—92no—
93#830—32no—
94(C2×D3)⋊C8—32no—
95Dic3⋊C8—32no—
96Dic3⋊C8—92no—
97C8×Dic3—92no—
98C8×Dic3—92no—
99C8×Dic3—92no—
100C8×Dic3—182no—
101C4⋊(C3⋊C8)—91no—
102C23×C32—31no—
103C22×C3×D3—21no—
104C22×C3×D3C4×D312yes—
105C2×C32×C4—22no—
106C2×C32×C4—21no—
107C2×C32×C4—31no—
108C2×C32×C4—41no—
109C2×C32×C4—61no—
110C2×C32×C4—61no—
111C2×C32×C4C22×D312yes—
112C2×C32×C4C22×D311yescharacteristic
113C2×C3⋊Dic3—32no—
114C2×C3⋊Dic3—32no—
115C2×C3⋊Dic3—31no—
116C2×C3⋊Dic3—31no—
117C2×C3⋊Dic3—34no—
118C2×C3⋊Dic3—61no—
119C2×C3⋊Dic3—61no—
120C2×C3⋊Dic3—62no—
121C2×C3⋊Dic3—62no—
122C2×C3⋊Dic3—62no—
123C2×C3⋊Dic3—121no—
124C2×C3⋊Dic3—121no—
125C2×C3⋊Dic3C4×D311yescharacteristic
126C2×C3⋊Dic3C4×D311yescharacteristic
127C2×C3×Dic3—21no—
128C2×C3×Dic3—32no—
129C2×C3×Dic3—32no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3C4×D312yes—
133C3×C4×D3—22no—
134C3×C4×D3—22no—
135C3×C4×D3—22no—
136C3×C4×D3—22no—
137D3×Dic3—38no—
138D3×Dic3—64no—
139D3×Dic3—64no—
140D3×Dic3—64no—
141C32×C8—62no—
142C3⋊(C3⋊C8)—22no—
143C3×C3⋊C8—24no—
144C3×C3⋊C8—22no—
145C3×C3⋊C8—22no—
146C3×C3⋊C8—64no—
147C3×C3⋊C8—62no—
148C3×C3⋊C8—62no—
149#431—271noSylow
150C2×C33#11711yescharacteristic
151C2×C33C22×C811yescharacteristic
152C2×C33C2×M1611yescharacteristic
153C32×D3—22no—
154C22×C3×C4—32no—
155C22×C3×C4—61no—
156C22×Dic3—92no—
157C22×Dic3—92no—
158C22×Dic3—181no—
159C22×Dic3—181no—
160C2×C4×D3—91no—
161C2×C4×D3—91no—
162C2×C4×D3D3211yescharacteristic
163C2×C3×C8—32no—
164C2×C3×C8—32no—
165C2×C3×C8—62no—
166C2×C3×C8—92no—
167C3×C42—91no—
168C4×Dic3—32no—
169C4×Dic3—61no—
170C4×Dic3—92no—
171C4×Dic3—92no—
172C4×Dic3—91no—
173C4×Dic3—92no—
174C4×Dic3—91no—
175C4×Dic3—91no—
176C4×Dic3—181no—
177C4×Dic3—181no—
178C4×Dic3—182no—
179C4×Dic3—181no—
180C4×Dic3—361no—
181C2×C3⋊C8—32no—
182C2×C3⋊C8—32no—
183C2×C3⋊C8—62no—
184C2×C3⋊C8D3212yes—
185C22×C32—22no—
186C22×C32—21no—
187C22×C32—32no—
188C22×C32—41no—
189C22×C32—61no—
190C22×C32—61no—
191C22×C32C2×C4×D311yescharacteristic
192C22×C32C2×C4×D312yes—
193C2×C3×D3—22no—
194C2×C3×D3—22no—
195C2×C3×D3—22no—
196C2×C3×D3—22no—
197C2×C3×D3—22no—
198C2×C3×D3D3×C814yes—
199C32×C4—22no—
200C32×C4—22no—
201C32×C4—21no—
202C32×C4—21no—
203C32×C4—41no—
204C32×C4—41no—
205C32×C4—61no—
206C32×C4#23712yes—
207C32×C4#23712yes—
208C32×C4#23711yescharacteristic
209C32×C4#23911yescharacteristic
210C3⋊Dic3—21no—
211C3⋊Dic3—34no—
212C3⋊Dic3—32no—
213C3⋊Dic3—34no—
214C3⋊Dic3—32no—
215C3⋊Dic3—62no—
216C3⋊Dic3—64no—
217C3⋊Dic3—62no—
218C3⋊Dic3—64no—
219C3⋊Dic3—122no—
220C3⋊Dic3—122no—
221C3⋊Dic3D3×C812yes—
222C3×Dic3—22no—
223C3×Dic3—22no—
224C3×Dic3—34no—
225C3×Dic3—62no—
226C3×Dic3—62no—
227C3×Dic3—62no—
228C2×C42—271no—
229C4⋊C8—91no—
230C4⋊C8—271no—
231C22⋊C8—92no—
232C4×C8—272no—
233C33#43111yescharacteristic, complemented, Sylow
234C23×C3—32no—
235C23×C3—61no—
236C22×D3#62411yescharacteristic
237C2×C3×C4—22no—
238C2×C3×C4—21no—
239C2×C3×C4—32no—
240C2×C3×C4—41no—
241C2×C3×C4—61no—
242C2×C3×C4—62no—
243C2×C3×C4—62no—
244C2×C3×C4—62no—
245C2×C3×C4—62no—
246C2×C3×C4—91no—
247C2×C3×C4—91no—
248C2×C3×C4C2×D3212yes—
249C2×C3×C4C2×D3211yescharacteristic
250C2×Dic3—32no—
251C2×Dic3—32no—
252C2×Dic3—61no—
253C2×Dic3—61no—
254C2×Dic3—98no—
255C2×Dic3—92no—
256C2×Dic3—92no—
257C2×Dic3—91no—
258C2×Dic3—91no—
259C2×Dic3—184no—
260C2×Dic3—182no—
261C2×Dic3—182no—
262C2×Dic3—181no—
263C2×Dic3—181no—
264C2×Dic3—184no—
265C2×Dic3—184no—
266C2×Dic3—361no—
267C2×Dic3—361no—
268C2×Dic3#62411yescharacteristic
269C4×D3—21no—
270C4×D3—21no—
271C4×D3—94no—
272C4×D3—182no—
273C3×C8—64no—
274C3×C8—62no—
275C3×C8—62no—
276C3×C8—182no—
277C3⋊C8—22no—
278C3⋊C8—64no—
279C3⋊C8—62no—
280C3⋊C8—62no—
281C2×C32—22no—
282C2×C32—22no—
283C2×C32—21no—
284C2×C32—21no—
285C2×C32—22no—
286C2×C32—21no—
287C2×C32—41no—
288C2×C32—41no—
289C2×C32—41no—
290C2×C32—62no—
291C2×C32#87712yes—
292C2×C32#89211yescharacteristic
293C2×C32C2×D3×C811yescharacteristic
294C2×C32C2×D3×C812yes—
295C2×C32D3×M1611yescharacteristic
296C2×C32D3×M1612yes—
297C3×D3—24no—
298C3×D3—24no—
299C22×C4—31no—
300C22×C4—271no—
301C22×C4—271no—
302C2×C8—32no—
303C2×C8—92no—
304C42—91no—
305C42—271no—
306C42—271no—
307C42—271no—
308C22×C3—21no—
309C22×C3—22no—
310C22×C3—34no—
311C22×C3—41no—
312C22×C3—62no—
313C22×C3—62no—
314C22×C3—62no—
315C22×C3—62no—
316C22×C3—62no—
317C22×C3C4×D3212yes—
318C22×C3C2×#62411yescharacteristic
319C2×D3—21no—
320C2×D3—21no—
321C2×D3#371012yes—
322C3×C4—22no—
323C3×C4—21no—
324C3×C4—22no—
325C3×C4—21no—
326C3×C4—41no—
327C3×C4—41no—
328C3×C4—62no—
329C3×C4—62no—
330C3×C4—92no—
331C3×C4—181no—
332C3×C4#379612yes—
333C3×C4#380512yes—
334C3×C4#380511yescharacteristic
335C3×C4#380511yescharacteristic
336Dic3—21no—
337Dic3—34no—
338Dic3—62no—
339Dic3—62no—
340Dic3—62no—
341Dic3—94no—
342Dic3—94no—
343Dic3—92no—
344Dic3—92no—
345Dic3—182no—
346Dic3—182no—
347Dic3—184no—
348Dic3—184no—
349Dic3—362no—
350Dic3—362no—
351C32—21no—
352C32—22no—
353C32—41no—
354C32#471812yescomplemented
355C32#483311yescharacteristic, complemented
356C23—31no—
357C2×C4—31no—
358C2×C4—61no—
359C2×C4—61no—
360C2×C4—91no—
361C2×C4—91no—
362C2×C4—274no—
363C2×C4—271no—
364C2×C4—271no—
365C2×C4—542no—
366C2×C4D3311yescharacteristic
367C8—62no—
368C8—182no—
369C2×C3—21no—
370C2×C3—22no—
371C2×C3—21no—
372C2×C3—22no—
373C2×C3—21no—
374C2×C3—22no—
375C2×C3—41no—
376C2×C3—41no—
377C2×C3—41no—
378C2×C3—64no—
379C2×C3—64no—
380C2×C3D32×C812yes—
381C2×C3#6379112yes—
382C2×C3C2×#371011yescharacteristic
383C2×C3#6381811yescharacteristic
384C2×C3#6387812yes—
385C2×C3#6395711yescharacteristic
386D3—22no—
387C22—32no—
388C22—61no—
389C22—61no—
390C22D3×#62411yescharacteristic
391C4—61no—
392C4—92no—
393C4—181no—
394C4—272no—
395C4—272no—
396C4#8896311yescharacteristic
397C4#8896411yescharacteristic
398C3—21no—
399C3—22no—
400C3—41no—
401C3#1059065612yescomplemented
402C3#1059106111yescharacteristic, complemented
403C2—62no—
404C2D3×#371011yescharacteristic
405C2#1172281911yescharacteristic
406C2#1172291711yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D33
commutatorC2×C33C22×C8
Fitting, cosocleC2×C33×C4C23
socleC22×C33C22×C4
prime radicalC22×C32×D3C2×C4
hypocenterC33#431
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11722078C2semidirect
3. #11722018C2semidirect
4. C3×#63572C2semidirect
25. #63572D3semidirect
41. D3×C3⋊Dic3C8semidirect
233. C33#431semidirect
354. C32#4718semidirect
355. C32#4833semidirect
401. C3#10590656semidirect
402. C3#10591061semidirect

Maximal subgroups

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

subgroupindex
C4×D3×C3⋊Dic32
#117220782
#117220182
C3×#635722
C3×(C2×D3)⋊(C3⋊C8)2
#105910613
#105906563