Finite Group Databasebeta

← #13167568

Subgroups — #13167568

Order 1728.

470 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 51 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
2#11723715C211yesmaximal, characteristic, complemented
3C3×C22⋊#622C211yesmaximal, characteristic, complemented
4#11722797C211yesmaximal, characteristic, complemented
5#11722796C211yesmaximal, characteristic, complemented
6#11722793C211yesmaximal, characteristic, complemented
7#11722793C211yesmaximal, characteristic, complemented
8#10593494—31nomaximal
9C2×C3×#638—21no—
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C2×#6295C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13C2×C3×#622C2211yescharacteristic
14C3×#3757—21no—
15#88815C2211yescharacteristic
16#88815C2211yescharacteristic
17#88815C2211yescharacteristic
18#88814—21no—
19#88813—21no—
20#63980—31no—
21#63978—61no—
22#63957—31no—
23#63955—61no—
24C22⋊#622D311yescharacteristic, complemented
25#63743—31no—
26#63742—31no—
27#63739—31no—
28#63739—31no—
29C23×C33D411yescharacteristic
30C22×C3×C3⋊D3—21no—
31C3×#638—42no—
32C2×C3×C3⋊Dic3—21no—
33C2×C3×C3⋊Dic3—21no—
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3—21no—
37C2×C3×C3⋊Dic3D411yescharacteristic
38C2×C3×C3⋊Dic3C2311yescharacteristic
39#6295—21no—
40#6295—21no—
41C3×#622—21no—
42C3×#622—21no—
43#4737—91nomaximal
44C2×#638—21no—
45C22×C3⋊Dic3C2×D311yescharacteristic
46C2×C3×#67—61no—
47C2×C3×#67—61no—
48C2×C3×#67—121no—
49C22×C3×Dic3—61no—
50C22×C3×Dic3—61no—
51C2×#626—61no—
52C2×#625—31no—
53C2×#622C2×D311yescharacteristic
54C2×#622C2×D311yescharacteristic
55#3757—21no—
56C3×#217—61no—
57C3×#217—61no—
58C3×#217—121no—
59#3724—31no—
60#3724—31no—
61#3724—31no—
62Dic3⋊Dic3—61no—
63Dic3⋊Dic3—61no—
64Dic3⋊Dic3—61no—
65#3722—61no—
66(C2×D3)⋊Dic3—61no—
67(C2×D3)⋊Dic3—61no—
68Dic32—61no—
69Dic32—61no—
70Dic32—61no—
71C22×C33—21no—
72C22×C33—21no—
73C22×C33—21no—
74C22×C33C2×D411yescharacteristic
75C2×C3×C3⋊D3—41no—
76C2×C3×C3⋊D3—41no—
77C2×C3×C3⋊D3—41no—
78C3×C3⋊Dic3—21no—
79C3×C3⋊Dic3—41no—
80C3×C3⋊Dic3—41no—
81C3×C3⋊Dic3—41no—
82C3×C3⋊Dic3#4111yescharacteristic
83C3×C3⋊Dic3#4111yescharacteristic
84#939—181no—
85#930—91no—
86#926—91no—
87#877—181no—
88C3×C22⋊C8—91no—
89#821—91no—
90#821—91no—
91#818—91no—
92#817—91no—
93C23×C32—21no—
94C23×C32—21no—
95C23×C32—41no—
96C23×C32—41no—
97C23×C32D1211yescharacteristic
98C22×C3⋊D3—21no—
99C22×C3×D3—61no—
100C22×C3×D3—61no—
101C22×C3×D3—121no—
102#638—42no—
103C2×C3⋊Dic3—21no—
104C2×C3⋊Dic3—21no—
105C2×C3⋊Dic3—21no—
106C2×C3⋊Dic3—21no—
107C2×C3⋊Dic3—21no—
108C2×C3⋊Dic3D1211yescharacteristic
109C2×C3⋊Dic3C22×D311yescharacteristic
110C3×#67—122no—
111C3×#67—122no—
112C3×#67—124no—
113C2×C3×Dic3—61no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3—62no—
118C2×C3×Dic3—62no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—121no—
126C2×C3×Dic3—121no—
127C2×C3×Dic3—121no—
128C2×C3×Dic3—121no—
129C2×C3×Dic3—121no—
130#626—121no—
131#626—121no—
132#625—61no—
133#625—61no—
134#622—21no—
135#622—21no—
136#482—271noSylow
137C2×C33—41no—
138C2×C33#12311yescharacteristic
139C2×C33#13511yescharacteristic
140C2×C33AGL(1,ℤ/8)11yescharacteristic
141C3×C3⋊D3—81no—
142C2×C3×D4—181no—
143C22×C3×C4—91no—
144C2×#67—61no—
145C2×#67—61no—
146C2×#67—121no—
147C2×#67—181no—
148C22×Dic3—61no—
149C22×Dic3—61no—
150C2×D12—91no—
151C2×C3×C8—91no—
152C2×C3×C8—91no—
153C3×C22⋊C4—181no—
154#217—61no—
155#217—61no—
156#217—121no—
157#217—181no—
158(C2×D3)⋊C4—181no—
159(C2×D3)⋊C4—181no—
160C4⋊Dic3—91no—
161C4⋊Dic3—91no—
162C4⋊Dic3—91no—
163Dic3⋊C4—181no—
164Dic3⋊C4—181no—
165Dic3⋊C4—181no—
166C4×Dic3—181no—
167C4×Dic3—181no—
168C4×Dic3—181no—
169C22×C32—22no—
170C22×C32—22no—
171C22×C32—21no—
172C22×C32—21no—
173C22×C32—21no—
174C22×C32—21no—
175C22×C32—21no—
176C22×C32—41no—
177C22×C32—41no—
178C22×C32—41no—
179C22×C32—41no—
180C22×C32—41no—
181C22×C32—41no—
182C22×C32—42no—
183C22×C32—42no—
184C22×C32—81no—
185C22×C32—81no—
186C22×C32—81no—
187C22×C32—81no—
188C22×C32C2×D1211yescharacteristic
189C2×C3⋊D3—41no—
190C2×C3⋊D3—41no—
191C2×C3⋊D3—41no—
192C2×C3×D3—121no—
193C2×C3×D3—121no—
194C2×C3×D3—121no—
195C2×C3×D3—122no—
196C2×C3×D3—121no—
197C2×C3×D3—121no—
198C2×C3×D3—121no—
199C2×C3×D3—241no—
200C2×C3×D3—241no—
201C3⋊Dic3—21no—
202C3⋊Dic3—41no—
203C3⋊Dic3—41no—
204C3⋊Dic3—41no—
205C3⋊Dic3#23711yescharacteristic
206C3⋊Dic3#23711yescharacteristic
207C3×Dic3—61no—
208C3×Dic3—61no—
209C3×Dic3—61no—
210C3×Dic3—61no—
211C3×Dic3—62no—
212C3×Dic3—121no—
213C3×Dic3—121no—
214C3×Dic3—121no—
215C3×Dic3—121no—
216C3×Dic3—122no—
217C3×Dic3—121no—
218C3×Dic3—121no—
219C3×Dic3—121no—
220C3×Dic3—241no—
221C3×Dic3—241no—
222#121—271no—
223#117—271no—
224#107—271no—
225#106—271no—
226#102—271no—
227#102—271no—
228C22⋊C8—91no—
229C33#48211yescharacteristic, complemented, Sylow
230C23×C3—21no—
231C23×C3—21no—
232C23×C3—41no—
233C23×C3—41no—
234C23×C3—181no—
235C23×C3GO+(4,𝔽2)11yescharacteristic
236C22×D3—61no—
237C22×D3—61no—
238C22×D3—121no—
239C3×D4—362no—
240C2×C3×C4—91no—
241C2×C3×C4—91no—
242C2×C3×C4—181no—
243C2×C3×C4—181no—
244C2×C3×C4—181no—
245C2×C3×C4—181no—
246C2×C3×C4—181no—
247#67—122no—
248#67—122no—
249#67—124no—
250#67—361no—
251#67—361no—
252C2×Dic3—61no—
253C2×Dic3—61no—
254C2×Dic3—61no—
255C2×Dic3—61no—
256C2×Dic3—62no—
257C2×Dic3—62no—
258C2×Dic3—61no—
259C2×Dic3—61no—
260C2×Dic3—61no—
261C2×Dic3—61no—
262C2×Dic3—61no—
263C2×Dic3—61no—
264C2×Dic3—121no—
265C2×Dic3—121no—
266C2×Dic3—121no—
267C2×Dic3—121no—
268C2×Dic3—121no—
269D12—181no—
270D12—181no—
271C3×C8—181no—
272C3×C8—181no—
273C2×C32—21no—
274C2×C32—21no—
275C2×C32—21no—
276C2×C32—21no—
277C2×C32—21no—
278C2×C32—21no—
279C2×C32—41no—
280C2×C32—41no—
281C2×C32—41no—
282C2×C32—41no—
283C2×C32—41no—
284C2×C32—41no—
285C2×C32—41no—
286C2×C32—42no—
287C2×C32—42no—
288C2×C32—82no—
289C2×C32—82no—
290C2×C32#88611yescharacteristic
291C2×C32#90411yescharacteristic
292C2×C32#90811yescharacteristic
293C3⋊D3—81no—
294C3×D3—241no—
295C3×D3—242no—
296C3×D3—241no—
297C2×D4—181no—
298C2×D4—271no—
299C22×C4—91no—
300C2×C8—91no—
301C2×C8—91no—
302C4⋊C4—271no—
303C4⋊C4—271no—
304C4⋊C4—271no—
305C22⋊C4—181no—
306C22⋊C4—541no—
307C42—541no—
308C22×C3—22no—
309C22×C3—22no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—21no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3—41no—
318C22×C3—41no—
319C22×C3—42no—
320C22×C3—42no—
321C22×C3—41no—
322C22×C3—41no—
323C22×C3—81no—
324C22×C3—81no—
325C22×C3—81no—
326C22×C3—81no—
327C22×C3—361no—
328C22×C3—361no—
329C22×C3—361no—
330C22×C3C2×GO+(4,𝔽2)11yescharacteristic
331C2×D3—121no—
332C2×D3—121no—
333C2×D3—121no—
334C2×D3—122no—
335C2×D3—121no—
336C2×D3—121no—
337C2×D3—121no—
338C2×D3—241no—
339C2×D3—241no—
340C3×C4—91no—
341C3×C4—91no—
342C3×C4—181no—
343C3×C4—361no—
344C3×C4—361no—
345C3×C4—361no—
346Dic3—61no—
347Dic3—61no—
348Dic3—61no—
349Dic3—61no—
350Dic3—62no—
351Dic3—121no—
352Dic3—121no—
353Dic3—121no—
354Dic3—121no—
355Dic3—122no—
356Dic3—121no—
357Dic3—121no—
358Dic3—121no—
359Dic3—241no—
360Dic3—241no—
361C32—21no—
362C32—21no—
363C32—41no—
364C32—41no—
365C32#473711yescharacteristic, complemented
366C23—181no—
367C23#157⋊D311yescharacteristic
368D4—362no—
369D4—541no—
370D4—541no—
371C2×C4—91no—
372C2×C4—91no—
373C2×C4—181no—
374C2×C4—181no—
375C2×C4—181no—
376C2×C4—181no—
377C2×C4—181no—
378C8—181no—
379C8—181no—
380C2×C3—21no—
381C2×C3—21no—
382C2×C3—21no—
383C2×C3—21no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3—42no—
387C2×C3—42no—
388C2×C3—41no—
389C2×C3—41no—
390C2×C3—41no—
391C2×C3—41no—
392C2×C3—41no—
393C2×C3—41no—
394C2×C3—41no—
395C2×C3—82no—
396C2×C3—82no—
397C2×C3—721no—
398C2×C3#6423311yescharacteristic
399C2×C3#6423611yescharacteristic
400C2×C3#6423711yescharacteristic
401D3—241no—
402D3—242no—
403D3—241no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—361no—
408C22—361no—
409C22—361no—
410C22C2×#157⋊D311yescharacteristic
411C4—91no—
412C4—91no—
413C4—181no—
414C4—361no—
415C4—361no—
416C4—361no—
417C3—21no—
418C3—21no—
419C3—41no—
420C3—41no—
421C3#1059349411yescharacteristic, complemented
422C2—41no—
423C2—721no—
424C2#1172434611yescharacteristic
425C2#1172435011yescharacteristic
426C2#1172435111yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC23×C33D4
socleC22×C33C2×D4
prime radical#11723717C2
hypercenterC23#157⋊D3
hypocenterC33#482
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
2. #11723715C2semidirect
3. C3×C22⋊#622C2semidirect
4. #11722797C2semidirect
5. #11722796C2semidirect
6. #11722793C2semidirect
7. #11722793C2semidirect
24. C22⋊#622D3semidirect
229. C33#482semidirect
365. C32#4737semidirect
421. C3#10593494semidirect

Maximal subgroups

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

subgroupindex
#117237172
#117237152
C3×C22⋊#6222
#117227972
#117227962
#117227932
#105934943
#47379