Finite Group Databasebeta

← #13165667

Subgroups — #13165667

Order 1728.

434 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 50 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#11723015C211yesmaximal, characteristic, complemented
2#11722938C211yesmaximal, characteristic
3#11722793C211yesmaximal, characteristic
4#11722778C211yesmaximal, characteristic, complemented
5#11722772C211yesmaximal, characteristic
6C3×#63743C211yesmaximal, characteristic, complemented
7Dic3⋊#622C211yesmaximal, characteristic, complemented
8#10593364—31nomaximal
9C2×#6295C2211yescharacteristic
10C2×Dic3×C3⋊D3—21no—
11C2×C3⋊#622C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13#88815C2211yescharacteristic
14#88807—21no—
15Dic3×C3⋊Dic3C2211yescharacteristic
16C3×#3724C2211yescharacteristic
17C3×#3724C2211yescharacteristic
18C3×Dic3⋊Dic3—21no—
19C3×#3722—21no—
20#63902—31no—
21#63881—61no—
22#63870—31no—
23#63846—61no—
24C4⋊#622—31no—
25#63743D311yescharacteristic, complemented
26#63742—31no—
27#63739—31no—
28#63739—31no—
29C22×C3×C3⋊D3—21no—
30C2×C3⋊(C3⋊Dic3)D411yescharacteristic
31C2×C3×C3⋊Dic3—21no—
32C2×C3×C3⋊Dic3C2311yescharacteristic
33C2×C32×Dic3—21no—
34C2×C32×Dic3—21no—
35C2×C32×Dic3D411yescharacteristic
36#6295—21no—
37#6295Q812yes—
38Dic3×C3⋊D3—42no—
39C3⋊#622—21no—
40C3×#622—21no—
41#4891—91nomaximal
42C2×C4×C3⋊D3—61no—
43C2×#626—61no—
44C2×#625—31no—
45C2×D3×Dic3—61no—
46C2×D3×Dic3—61no—
47C2×D3×Dic3—121no—
48C2×#622—31no—
49C2×#622C2×D311yescharacteristic
50#3750—61no—
51C4×C3⋊Dic3—31no—
52C3×(C2×D3)⋊C4—61no—
53C3×(C2×D3)⋊C4—61no—
54C3×C4⋊Dic3—61no—
55C3×Dic3⋊C4—61no—
56C3×Dic3⋊C4—61no—
57C3×Dic3⋊C4—61no—
58#3724—31no—
59#3724C2×D311yescharacteristic
60#3724C2×D311yescharacteristic
61Dic3⋊Dic3—21no—
62Dic3⋊Dic3—61no—
63Dic3⋊Dic3—61no—
64Dic3⋊Dic3—61no—
65Dic3⋊Dic3—121no—
66#3722—21no—
67Dic32—61no—
68Dic32—61no—
69C22×C33C2×D411yescharacteristic
70C2×C3×C3⋊D3—22no—
71C2×C3×C3⋊D3—41no—
72C2×C3×C3⋊D3—41no—
73C3⋊(C3⋊Dic3)—21no—
74C3×C3⋊Dic3—41no—
75C3×C3⋊Dic3C2×Q811yescharacteristic
76C3×C3⋊Dic3#4111yescharacteristic
77C32×Dic3—21no—
78C32×Dic3—41no—
79C32×Dic3—41no—
80#893—91no—
81#890—91no—
82#889—181no—
83#67⋊C4—181no—
84C3×#107—91no—
85#821—91no—
86Dic3⋊C8—91no—
87#809—91no—
88#808—91no—
89C22×C3⋊D3—21no—
90C22×C3×D3—61no—
91C22×C3×D3—61no—
92C22×C3×D3—121no—
93C2×C32×C4—31no—
94C2×C32×C4—61no—
95C2×C32×C4—61no—
96C2×C3⋊Dic3—21no—
97C2×C3⋊Dic3—31no—
98C2×C3⋊Dic3—61no—
99C2×C3⋊Dic3—61no—
100C2×C3⋊Dic3—121no—
101C2×C3⋊Dic3—121no—
102C2×C3⋊Dic3C22×D311yescharacteristic
103C4×C3⋊D3—122no—
104C2×C3×Dic3—21no—
105C2×C3×Dic3—21no—
106C2×C3×Dic3—21no—
107C2×C3×Dic3—21no—
108C2×C3×Dic3—21no—
109C2×C3×Dic3—21no—
110C2×C3×Dic3—41no—
111C2×C3×Dic3—41no—
112C2×C3×Dic3—61no—
113C2×C3×Dic3—61no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—121no—
117#626—62no—
118#626—62no—
119#625—32no—
120#625—61no—
121D3×Dic3—122no—
122D3×Dic3—122no—
123D3×Dic3—124no—
124#622—21no—
125#622—61no—
126#478—271noSylow
127C2×C33C22⋊Q811yescharacteristic
128C2×C33#13511yescharacteristic
129C2×C33AGL(1,ℤ/8)11yescharacteristic
130C3×C3⋊D3—42no—
131C2×#67—181no—
132C22×Dic3—181no—
133C2×D12—91no—
134C2×C4×D3—181no—
135C2×C4×D3—181no—
136C2×C4×D3—361no—
137C2×C3×C8—91no—
138C3×C4⋊C4—91no—
139C3×C4⋊C4—91no—
140C3×C4⋊C4—181no—
141C3×C22⋊C4—181no—
142(C2×D3)⋊C4—61no—
143(C2×D3)⋊C4—61no—
144C4⋊Dic3—61no—
145C4⋊Dic3—91no—
146C4⋊Dic3—181no—
147Dic3⋊C4—61no—
148Dic3⋊C4—61no—
149Dic3⋊C4—61no—
150Dic3⋊C4—181no—
151Dic3⋊C4—181no—
152Dic3⋊C4—181no—
153Dic3⋊C4—361no—
154C4×Dic3—91no—
155C4×Dic3—181no—
156C4×Dic3—181no—
157C2×C3⋊C8—91no—
158C22×C32—21no—
159C22×C32—21no—
160C22×C32—41no—
161C22×C32—41no—
162C22×C32D3×D411yescharacteristic
163C2×C3⋊D3—22no—
164C2×C3⋊D3—41no—
165C2×C3⋊D3—41no—
166C2×C3×D3—62no—
167C2×C3×D3—62no—
168C2×C3×D3—122no—
169C2×C3×D3—121no—
170C2×C3×D3—121no—
171C2×C3×D3—122no—
172C2×C3×D3—122no—
173C2×C3×D3—121no—
174C2×C3×D3—121no—
175C32×C4—61no—
176C32×C4—121no—
177C32×C4—121no—
178C3⋊Dic3—41no—
179C3⋊Dic3—62no—
180C3⋊Dic3—61no—
181C3⋊Dic3—121no—
182C3⋊Dic3—122no—
183C3⋊Dic3—241no—
184C3⋊Dic3#23511yescharacteristic
185C3⋊Dic3D3×Q811yescharacteristic
186C3×Dic3—41no—
187C3×Dic3—41no—
188C3×Dic3—41no—
189C3×Dic3—41no—
190C3×Dic3—42no—
191C3×Dic3—41no—
192C3×Dic3—41no—
193C3×Dic3—61no—
194C3×Dic3—61no—
195C3×Dic3—61no—
196C3×Dic3—61no—
197C3×Dic3—81no—
198C3×Dic3—121no—
199C3×Dic3—122no—
200C3×Dic3—121no—
201#125—271no—
202C4×D4—271no—
203#107—91no—
204#106—271no—
205C4⋊C8—271no—
206#102—271no—
207#102—271no—
208C33#47811yescharacteristic, complemented, Sylow
209C23×C3—181no—
210C22×D3—61no—
211C22×D3—61no—
212C22×D3—121no—
213C2×C3×C4—61no—
214C2×C3×C4—61no—
215C2×C3×C4—61no—
216C2×C3×C4—61no—
217C2×C3×C4—61no—
218C2×C3×C4—61no—
219C2×C3×C4—91no—
220C2×C3×C4—121no—
221C2×C3×C4—121no—
222C2×C3×C4—181no—
223#67—182no—
224#67—182no—
225C2×Dic3—21no—
226C2×Dic3—21no—
227C2×Dic3—61no—
228C2×Dic3—61no—
229C2×Dic3—61no—
230C2×Dic3—61no—
231C2×Dic3—91no—
232C2×Dic3—121no—
233C2×Dic3—181no—
234C2×Dic3—181no—
235C2×Dic3—361no—
236C2×Dic3—361no—
237C2×Dic3—362no—
238C2×Dic3GO+(4,𝔽2)11yescharacteristic
239D12—92no—
240D12—181no—
241C4×D3—362no—
242C4×D3—362no—
243C4×D3—364no—
244C3×C8—181no—
245C3⋊C8—181no—
246C2×C32—21no—
247C2×C32—21no—
248C2×C32—21no—
249C2×C32—21no—
250C2×C32—21no—
251C2×C32—21no—
252C2×C32—41no—
253C2×C32—41no—
254C2×C32—41no—
255C2×C32—41no—
256C2×C32—41no—
257C2×C32—41no—
258C2×C32#89611yescharacteristic
259C2×C32#91111yescharacteristic
260C2×C32#91911yescharacteristic
261C3⋊D3—42no—
262C3×D3—122no—
263C3×D3—124no—
264C3×D3—122no—
265C2×D4—271no—
266C22×C4—541no—
267C2×C8—91no—
268C2×C8—271no—
269C4⋊C4—91no—
270C4⋊C4—91no—
271C4⋊C4—181no—
272C4⋊C4—271no—
273C4⋊C4—541no—
274C22⋊C4—181no—
275C42—271no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—182no—
281C22×C3—361no—
282C22×C3—361no—
283C22×C3C2×GO+(4,𝔽2)11yescharacteristic
284C2×D3—62no—
285C2×D3—62no—
286C2×D3—122no—
287C2×D3—121no—
288C2×D3—121no—
289C2×D3—122no—
290C2×D3—122no—
291C2×D3—121no—
292C2×D3—121no—
293C3×C4—91no—
294C3×C4—91no—
295C3×C4—121no—
296C3×C4—121no—
297C3×C4—121no—
298C3×C4—121no—
299C3×C4—122no—
300C3×C4—121no—
301C3×C4—121no—
302C3×C4—241no—
303C3×C4—361no—
304Dic3—21no—
305Dic3—41no—
306Dic3—41no—
307Dic3—61no—
308Dic3—61no—
309Dic3—61no—
310Dic3—61no—
311Dic3—121no—
312Dic3—122no—
313Dic3—121no—
314Dic3—182no—
315Dic3—181no—
316Dic3—361no—
317Dic3—362no—
318Dic3—721no—
319C32—21no—
320C32—21no—
321C32—41no—
322C32—41no—
323C32#489111yescharacteristic, complemented
324C23—181no—
325D4—272no—
326D4—541no—
327C2×C4—31no—
328C2×C4—61no—
329C2×C4—61no—
330C2×C4—91no—
331C2×C4—181no—
332C2×C4—271no—
333C2×C4—1082no—
334C8—181no—
335C8—541no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—21no—
342C2×C3—41no—
343C2×C3—41no—
344C2×C3—41no—
345C2×C3—41no—
346C2×C3—41no—
347C2×C3—41no—
348C2×C3—362no—
349C2×C3D32⋊Q811yescharacteristic
350C2×C3#6422411yescharacteristic
351C2×C3#6423711yescharacteristic
352D3—122no—
353D3—124no—
354D3—122no—
355C22—182no—
356C22—361no—
357C22—361no—
358C22D3×GO+(4,𝔽2)11yescharacteristic
359C4—61no—
360C4—91no—
361C4—91no—
362C4—121no—
363C4—121no—
364C4—361no—
365C4—541no—
366C3—21no—
367C3—21no—
368C3—41no—
369C3—41no—
370C3#1059336411yescharacteristic, complemented
371C2—362no—
372C2#1172412011yescharacteristic
373C2#1172413411yescharacteristic
374C2#1172414011yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socleC22×C33C2×D4
prime radicalC2×#6295C22
hypocenterC33#478
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. #11723015C2semidirect
4. #11722778C2semidirect
6. C3×#63743C2semidirect
7. Dic3⋊#622C2semidirect
25. #63743D3semidirect
208. C33#478semidirect
323. C32#4891semidirect
370. C3#10593364semidirect

Maximal subgroups

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

subgroupindex
#117230152
#117229382
#117227932
#117227782
#117227722
C3×#637432
Dic3⋊#6222
#105933643
#48919