Finite Group Databasebeta

← #13165718

Subgroups — #13165718

Order 1728.

384 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 48 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#11723007C211yesmaximal, characteristic
3#11722793C211yesmaximal, characteristic
4#11722775C211yesmaximal, characteristic, complemented
5#11722772C211yesmaximal, characteristic
6C3×#63740C211yesmaximal, characteristic, complemented
7Dic3×#622C211yesmaximal, characteristic, complemented
8#10593353—31nomaximal
9C2×#6295C2211yescharacteristic
10C2×C3×#627C2211yescharacteristic
11C2×C3⋊#622C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13#88815C2211yescharacteristic
14#88807—21no—
15#88803—21no—
16Dic3×C3⋊Dic3C2211yescharacteristic
17C3×#3724C2211yescharacteristic
18C3×Dic3⋊Dic3—21no—
19C3×#3722—21no—
20#63915—31no—
21#63907—61no—
22#63870—31no—
23#63846—61no—
24C4×#622—31no—
25#63740—31no—
26#63740D311yescharacteristic, complemented
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#6295—21no—
38C3×#627—21no—
39C3×#627—21no—
40C3⋊#622—21no—
41C3×#622—21no—
42#4800—91nomaximal
43C2×C3×Dic6—61no—
44C2×#627C2×D311yescharacteristic
45C2×#626—61no—
46C2×#625—31no—
47C2×#622—31no—
48C2×#622C2×D311yescharacteristic
49#3752—61no—
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—
57#3724—31no—
58#3724C2×D311yescharacteristic
59Dic3⋊Dic3—21no—
60Dic3⋊Dic3—61no—
61Dic3⋊Dic3—61no—
62Dic3⋊Dic3—61no—
63Dic3⋊Dic3—121no—
64#3722—21no—
65(C2×D3)⋊Dic3—61no—
66(C2×D3)⋊Dic3—61no—
67(C2×D3)⋊Dic3—121no—
68Dic32—61no—
69Dic32—61no—
70C22×C33C2×D411yescharacteristic
71C2×C3×C3⋊D3—41no—
72C2×C3×C3⋊D3—41no—
73C2×C3×C3⋊D3—41no—
74C3⋊(C3⋊Dic3)—21no—
75C3×C3⋊Dic3—41no—
76C3×C3⋊Dic3#4111yescharacteristic
77C3×C3⋊Dic3#4111yescharacteristic
78C32×Dic3—21no—
79C32×Dic3—41no—
80C32×Dic3—41no—
81#947—91no—
82#890—91no—
83#889—181no—
84#885—181no—
85C3×#103—91no—
86#835—91no—
87#821—91no—
88C8×Dic3—91no—
89#809—91no—
90C22×C3⋊D3—21no—
91C22×C3×D3—61no—
92C22×C3×D3—61no—
93C22×C3×D3—121no—
94C2×C32×C4—31no—
95C2×C32×C4—61no—
96C2×C32×C4—61no—
97C2×C3⋊Dic3—21no—
98C2×C3⋊Dic3—31no—
99C2×C3⋊Dic3—61no—
100C2×C3⋊Dic3—61no—
101C2×C3⋊Dic3—121no—
102C2×C3⋊Dic3—121no—
103C2×C3⋊Dic3C22×D311yescharacteristic
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—
117C3×Dic6—121no—
118C3×Dic6—121no—
119#627—21no—
120#627—21no—
121#626—121no—
122#626—121no—
123#625—61no—
124#625—61no—
125#622—21no—
126#622—61no—
127#488—271noSylow
128C2×C33#12411yescharacteristic
129C2×C33#13511yescharacteristic
130C2×C33#13511yescharacteristic
131C3×C3⋊D3—81no—
132C2×C3×Q8—91no—
133C2×#67—181no—
134C2×D12—91no—
135C2×Dic6—61no—
136C2×C3×C8—91no—
137C3×C4⋊C4—91no—
138C3×C4⋊C4—181no—
139C3×C22⋊C4—181no—
140#217—181no—
141(C2×D3)⋊C4—61no—
142(C2×D3)⋊C4—61no—
143(C2×D3)⋊C4—181no—
144(C2×D3)⋊C4—181no—
145(C2×D3)⋊C4—361no—
146C4⋊Dic3—61no—
147C4⋊Dic3—91no—
148C4⋊Dic3—181no—
149Dic3⋊C4—61no—
150Dic3⋊C4—61no—
151Dic3⋊C4—181no—
152Dic3⋊C4—181no—
153Dic3⋊C4—181no—
154Dic3⋊C4—361no—
155C4×Dic3—91no—
156C4×Dic3—181no—
157C4×Dic3—181no—
158C2×C3⋊C8—91no—
159C22×C32—21no—
160C22×C32—21no—
161C22×C32—41no—
162C22×C32—41no—
163C22×C32D3×D411yescharacteristic
164C2×C3⋊D3—41no—
165C2×C3⋊D3—41no—
166C2×C3⋊D3—41no—
167C2×C3×D3—121no—
168C2×C3×D3—121no—
169C2×C3×D3—121no—
170C2×C3×D3—122no—
171C2×C3×D3—121no—
172C2×C3×D3—121no—
173C2×C3×D3—121no—
174C2×C3×D3—241no—
175C2×C3×D3—241no—
176C32×C4—61no—
177C32×C4—121no—
178C32×C4—121no—
179C3⋊Dic3—41no—
180C3⋊Dic3—61no—
181C3⋊Dic3—121no—
182C3⋊Dic3—121no—
183C3⋊Dic3—241no—
184C3⋊Dic3—241no—
185C3⋊Dic3#23511yescharacteristic
186C3⋊Dic3#23711yescharacteristic
187C3×Dic3—41no—
188C3×Dic3—41no—
189C3×Dic3—41no—
190C3×Dic3—41no—
191C3×Dic3—41no—
192C3×Dic3—41no—
193C3×Dic3—61no—
194C3×Dic3—61no—
195C3×Dic3—61no—
196C3×Dic3—61no—
197C3×Dic3—81no—
198C3×Dic3—81no—
199C3×Dic3—121no—
200C3×Dic3—122no—
201C3×Dic3—121no—
202#125—271no—
203#124—271no—
204#103—91no—
205#103—271no—
206#102—271no—
207#102—271no—
208C4×C8—271no—
209C33#48811yescharacteristic, complemented, Sylow
210C23×C3—181no—
211C22×D3—61no—
212C22×D3—61no—
213C22×D3—121no—
214C3×Q8—181no—
215C3×Q8—181no—
216C2×C3×C4—61no—
217C2×C3×C4—61no—
218C2×C3×C4—61no—
219C2×C3×C4—61no—
220C2×C3×C4—61no—
221C2×C3×C4—61no—
222C2×C3×C4—91no—
223C2×C3×C4—121no—
224C2×C3×C4—121no—
225C2×C3×C4—181no—
226#67—361no—
227#67—361no—
228C2×Dic3—21no—
229C2×Dic3—21no—
230C2×Dic3—61no—
231C2×Dic3—61no—
232C2×Dic3—61no—
233C2×Dic3—61no—
234C2×Dic3—91no—
235C2×Dic3—121no—
236C2×Dic3—181no—
237C2×Dic3—181no—
238C2×Dic3—361no—
239C2×Dic3—361no—
240C2×Dic3GO+(4,𝔽2)11yescharacteristic
241D12—181no—
242D12—181no—
243Dic6—121no—
244Dic6—121no—
245C3×C8—181no—
246C3⋊C8—181no—
247C2×C32—21no—
248C2×C32—21no—
249C2×C32—21no—
250C2×C32—21no—
251C2×C32—21no—
252C2×C32—21no—
253C2×C32—41no—
254C2×C32—41no—
255C2×C32—41no—
256C2×C32—41no—
257C2×C32—41no—
258C2×C32—41no—
259C2×C32#88511yescharacteristic
260C2×C32#91611yescharacteristic
261C2×C32#91911yescharacteristic
262C3⋊D3—81no—
263C3×D3—241no—
264C3×D3—242no—
265C3×D3—241no—
266C2×Q8—91no—
267C2×D4—271no—
268C2×C8—91no—
269C2×C8—271no—
270C4⋊C4—91no—
271C4⋊C4—181no—
272C4⋊C4—271no—
273C4⋊C4—541no—
274C22⋊C4—181no—
275C22⋊C4—541no—
276C42—271no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—361no—
282C22×C3—361no—
283C22×C3—361no—
284C22×C3C2×GO+(4,𝔽2)11yescharacteristic
285C2×D3—121no—
286C2×D3—121no—
287C2×D3—121no—
288C2×D3—122no—
289C2×D3—121no—
290C2×D3—121no—
291C2×D3—121no—
292C2×D3—241no—
293C2×D3—241no—
294C3×C4—91no—
295C3×C4—91no—
296C3×C4—121no—
297C3×C4—121no—
298C3×C4—121no—
299C3×C4—121no—
300C3×C4—121no—
301C3×C4—121no—
302C3×C4—241no—
303C3×C4—241no—
304C3×C4—361no—
305Dic3—21no—
306Dic3—41no—
307Dic3—41no—
308Dic3—61no—
309Dic3—61no—
310Dic3—61no—
311Dic3—61no—
312Dic3—121no—
313Dic3—122no—
314Dic3—121no—
315Dic3—181no—
316Dic3—361no—
317Dic3—361no—
318Dic3—721no—
319Dic3—721no—
320C32—21no—
321C32—21no—
322C32—41no—
323C32—41no—
324C32#480011yescharacteristic, complemented
325C23—181no—
326Q8—181no—
327Q8—181no—
328D4—541no—
329D4—541no—
330C2×C4—31no—
331C2×C4—61no—
332C2×C4—61no—
333C2×C4—91no—
334C2×C4—181no—
335C2×C4—271no—
336C8—181no—
337C8—541no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—21no—
342C2×C3—21no—
343C2×C3—21no—
344C2×C3—41no—
345C2×C3—41no—
346C2×C3—41no—
347C2×C3—41no—
348C2×C3—41no—
349C2×C3—41no—
350C2×C3—721no—
351C2×C3#6422111yescharacteristic
352C2×C3#6422311yescharacteristic
353C2×C3#6423711yescharacteristic
354D3—241no—
355D3—242no—
356D3—241no—
357C22—361no—
358C22—361no—
359C22—361no—
360C22D3×GO+(4,𝔽2)11yescharacteristic
361C4—61no—
362C4—91no—
363C4—91no—
364C4—121no—
365C4—121no—
366C4—361no—
367C4—541no—
368C3—21no—
369C3—21no—
370C3—41no—
371C3—41no—
372C3#1059335311yescharacteristic, complemented
373C2—721no—
374C2#1172412311yescharacteristic
375C2#1172413411yescharacteristic
376C2#1172413611yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socleC22×C33C2×D4
prime radicalC2×#6295C22
hypocenterC33#488
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. #11722775C2semidirect
6. C3×#63740C2semidirect
7. Dic3×#622C2semidirect
26. #63740D3semidirect
209. C33#488semidirect
324. C32#4800semidirect
372. C3#10593353semidirect

Maximal subgroups

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

subgroupindex
#117230152
#117230072
#117227932
#117227752
#117227722
C3×#637402
Dic3×#6222
#105933533
#48009