Finite Group Databasebeta

← #13167582

Subgroups — #13167582

Order 1728.

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

normal subgroups only

This group has 410 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#11723721C211yesmaximal, characteristic
2#11723715C211yesmaximal, characteristic
3C3×C22⋊#622C211yesmaximal, characteristic
4#11722797C211yesmaximal, characteristic, complemented
5#11722796C211yesmaximal, characteristic, complemented
6#11722794C211yesmaximal, characteristic, complemented
7#11722794C211yesmaximal, characteristic, complemented
8#10593501—31nomaximal
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C2×#6296C2211yescharacteristic
11C2×C3×#622C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13C3×#3757—21no—
14C3×#3757—21no—
15#88815—21no—
16#88815C2211yescharacteristic
17#88815C2211yescharacteristic
18#88815C2211yescharacteristic
19#88813—21no—
20#63982—31no—
21#63964—61no—
22#63957—31no—
23#63955—61no—
24C22⋊#622D311yescharacteristic
25#63743—31no—
26#63742—31no—
27#63740—31no—
28#63740—31no—
29C23×C33D411yescharacteristic
30C2×C3×C3⋊Dic3—21no—
31C2×C3×C3⋊Dic3—21no—
32C2×C3×C3⋊Dic3—21no—
33C2×C3×C3⋊Dic3—21no—
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3D411yescharacteristic
37C2×C3×C3⋊Dic3C2311yescharacteristic
38#6296—21no—
39#6296—21no—
40C3×#622—21no—
41C3×#622—21no—
42#4723—91nomaximal
43C22×C3⋊Dic3C2×D311yescharacteristic
44C22×C3×Dic3—61no—
45C22×C3×Dic3—61no—
46C2×#627—31no—
47C2×#627—61no—
48C2×#622C2×D311yescharacteristic
49C2×#622C2×D311yescharacteristic
50#3757—21no—
51#3757—21no—
52C3×#217—61no—
53C3×#217—61no—
54C3×#217—61no—
55C3×#217—61no—
56C3×#217—121no—
57C3×#217—121no—
58#3724—31no—
59#3724—31no—
60#3724—31no—
61#3724—61no—
62Dic3⋊Dic3—61no—
63Dic3⋊Dic3—61no—
64Dic3⋊Dic3—61no—
65Dic3⋊Dic3—61no—
66Dic3⋊Dic3—61no—
67Dic32—61no—
68Dic32—61no—
69Dic32—61no—
70C22×C33—21no—
71C22×C33—21no—
72C22×C33—21no—
73C22×C33C2×D411yescharacteristic
74C3×C3⋊Dic3—21no—
75C3×C3⋊Dic3—41no—
76C3×C3⋊Dic3—41no—
77C3×C3⋊Dic3—41no—
78C3×C3⋊Dic3—41no—
79C3×C3⋊Dic3#4111yescharacteristic
80C3×C3⋊Dic3#4111yescharacteristic
81#926—91no—
82#924—91no—
83#878—181no—
84#877—181no—
85C3×C22⋊C8—91no—
86#818—91no—
87#817—91no—
88#816—91no—
89#816—91no—
90C23×C32—21no—
91C23×C32—21no—
92C23×C32—41no—
93C23×C32—41no—
94C23×C32D1211yescharacteristic
95C2×C3⋊Dic3—21no—
96C2×C3⋊Dic3—21no—
97C2×C3⋊Dic3—21no—
98C2×C3⋊Dic3—21no—
99C2×C3⋊Dic3—21no—
100C2×C3⋊Dic3—21no—
101C2×C3⋊Dic3D1211yescharacteristic
102C2×C3⋊Dic3C22×D311yescharacteristic
103C2×C3×Dic3—61no—
104C2×C3×Dic3—61no—
105C2×C3×Dic3—61no—
106C2×C3×Dic3—61no—
107C2×C3×Dic3—62no—
108C2×C3×Dic3—62no—
109C2×C3×Dic3—61no—
110C2×C3×Dic3—61no—
111C2×C3×Dic3—61no—
112C2×C3×Dic3—61no—
113C2×C3×Dic3—61no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3—121no—
118C2×C3×Dic3—121no—
119C2×C3×Dic3—121no—
120C2×C3×Dic3—121no—
121C2×C3×Dic3—121no—
122C2×C3×Dic3—121no—
123#627—61no—
124#627—61no—
125#627—121no—
126#627—121no—
127#622—21no—
128#622—21no—
129#485—271noSylow
130C2×C33—41no—
131C2×C33#12311yescharacteristic
132C2×C33#13511yescharacteristic
133C2×C33#13711yescharacteristic
134C22×C3×C4—91no—
135C22×Dic3—61no—
136C22×Dic3—61no—
137C2×Dic6—91no—
138C2×Dic6—181no—
139C2×C3×C8—91no—
140C2×C3×C8—91no—
141C3×C22⋊C4—181no—
142C3×C22⋊C4—181no—
143#217—61no—
144#217—61no—
145#217—61no—
146#217—61no—
147#217—121no—
148#217—121no—
149C4⋊Dic3—91no—
150C4⋊Dic3—91no—
151C4⋊Dic3—91no—
152C4⋊Dic3—181no—
153Dic3⋊C4—181no—
154Dic3⋊C4—181no—
155Dic3⋊C4—181no—
156Dic3⋊C4—181no—
157Dic3⋊C4—181no—
158C4×Dic3—181no—
159C4×Dic3—181no—
160C4×Dic3—181no—
161C22×C32—22no—
162C22×C32—22no—
163C22×C32—21no—
164C22×C32—21no—
165C22×C32—21no—
166C22×C32—21no—
167C22×C32—21no—
168C22×C32—41no—
169C22×C32—41no—
170C22×C32—41no—
171C22×C32—41no—
172C22×C32—41no—
173C22×C32—41no—
174C22×C32—42no—
175C22×C32—42no—
176C22×C32—81no—
177C22×C32—81no—
178C22×C32—81no—
179C22×C32—81no—
180C22×C32C2×D1211yescharacteristic
181C3⋊Dic3—21no—
182C3⋊Dic3—41no—
183C3⋊Dic3—41no—
184C3⋊Dic3—41no—
185C3⋊Dic3—41no—
186C3⋊Dic3#23711yescharacteristic
187C3⋊Dic3#23711yescharacteristic
188C3×Dic3—61no—
189C3×Dic3—61no—
190C3×Dic3—61no—
191C3×Dic3—61no—
192C3×Dic3—62no—
193C3×Dic3—121no—
194C3×Dic3—121no—
195C3×Dic3—121no—
196C3×Dic3—121no—
197C3×Dic3—121no—
198C3×Dic3—121no—
199C3×Dic3—121no—
200C3×Dic3—121no—
201C3×Dic3—121no—
202C3×Dic3—241no—
203C3×Dic3—241no—
204C3×Dic3—241no—
205C3×Dic3—241no—
206C22⋊Q8—271no—
207#117—271no—
208#107—271no—
209#106—271no—
210#103—271no—
211#103—271no—
212C22⋊C8—91no—
213C33#48511yescharacteristic, complemented, Sylow
214C23×C3—21no—
215C23×C3—21no—
216C23×C3—41no—
217C23×C3—41no—
218C23×C3GO+(4,𝔽2)11yescharacteristic
219C2×C3×C4—91no—
220C2×C3×C4—91no—
221C2×C3×C4—181no—
222C2×C3×C4—181no—
223C2×C3×C4—181no—
224C2×C3×C4—181no—
225C2×C3×C4—181no—
226C2×C3×C4—181no—
227C2×Dic3—61no—
228C2×Dic3—61no—
229C2×Dic3—61no—
230C2×Dic3—61no—
231C2×Dic3—62no—
232C2×Dic3—62no—
233C2×Dic3—61no—
234C2×Dic3—61no—
235C2×Dic3—61no—
236C2×Dic3—61no—
237C2×Dic3—61no—
238C2×Dic3—61no—
239C2×Dic3—61no—
240C2×Dic3—61no—
241C2×Dic3—121no—
242C2×Dic3—121no—
243C2×Dic3—121no—
244C2×Dic3—121no—
245C2×Dic3—121no—
246C2×Dic3—121no—
247Dic6—181no—
248Dic6—181no—
249Dic6—361no—
250Dic6—361no—
251C3×C8—181no—
252C3×C8—181no—
253C2×C32—21no—
254C2×C32—21no—
255C2×C32—21no—
256C2×C32—21no—
257C2×C32—21no—
258C2×C32—21no—
259C2×C32—41no—
260C2×C32—41no—
261C2×C32—41no—
262C2×C32—41no—
263C2×C32—41no—
264C2×C32—41no—
265C2×C32—41no—
266C2×C32—42no—
267C2×C32—42no—
268C2×C32—82no—
269C2×C32—82no—
270C2×C32#88611yescharacteristic
271C2×C32#90411yescharacteristic
272C2×C32#90911yescharacteristic
273C2×Q8—271no—
274C22×C4—91no—
275C2×C8—91no—
276C2×C8—91no—
277C4⋊C4—271no—
278C4⋊C4—271no—
279C4⋊C4—271no—
280C4⋊C4—541no—
281C22⋊C4—181no—
282C22⋊C4—181no—
283C42—541no—
284C22×C3—22no—
285C22×C3—22no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—21no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3—41no—
295C22×C3—42no—
296C22×C3—42no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—81no—
300C22×C3—81no—
301C22×C3—81no—
302C22×C3—81no—
303C22×C3C2×GO+(4,𝔽2)11yescharacteristic
304C3×C4—91no—
305C3×C4—91no—
306C3×C4—181no—
307C3×C4—361no—
308C3×C4—361no—
309C3×C4—361no—
310C3×C4—361no—
311Dic3—61no—
312Dic3—61no—
313Dic3—61no—
314Dic3—61no—
315Dic3—62no—
316Dic3—121no—
317Dic3—121no—
318Dic3—121no—
319Dic3—121no—
320Dic3—121no—
321Dic3—121no—
322Dic3—121no—
323Dic3—121no—
324Dic3—121no—
325Dic3—241no—
326Dic3—241no—
327Dic3—241no—
328Dic3—241no—
329C32—21no—
330C32—21no—
331C32—41no—
332C32—41no—
333C32#472311yescharacteristic, complemented
334C23#157⋊D311yescharacteristic
335Q8—541no—
336Q8—541no—
337C2×C4—91no—
338C2×C4—91no—
339C2×C4—181no—
340C2×C4—181no—
341C2×C4—181no—
342C2×C4—181no—
343C2×C4—181no—
344C2×C4—181no—
345C8—181no—
346C8—181no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—42no—
354C2×C3—42no—
355C2×C3—41no—
356C2×C3—41no—
357C2×C3—41no—
358C2×C3—41no—
359C2×C3—41no—
360C2×C3—41no—
361C2×C3—41no—
362C2×C3—82no—
363C2×C3—82no—
364C2×C3#6423311yescharacteristic
365C2×C3#6423711yescharacteristic
366C2×C3#6423911yescharacteristic
367C22—21no—
368C22—21no—
369C22—21no—
370C22C2×#157⋊D311yescharacteristic
371C4—91no—
372C4—91no—
373C4—181no—
374C4—361no—
375C4—361no—
376C4—361no—
377C4—361no—
378C3—21no—
379C3—21no—
380C3—41no—
381C3—41no—
382C3#1059350111yescharacteristic, complemented
383C2—41no—
384C2#1172434611yescharacteristic
385C2#1172435111yescharacteristic
386C2#1172435211yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, prime radicalC23×C33D4
socleC22×C33C2×D4
hypercenterC23#157⋊D3
hypocenterC33#485
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
4. #11722797C2semidirect
5. #11722796C2semidirect
6. #11722794C2semidirect
7. #11722794C2semidirect
213. C33#485semidirect
333. C32#4723semidirect
382. C3#10593501semidirect

Maximal subgroups

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

subgroupindex
#117237212
#117237152
C3×C22⋊#6222
#117227972
#117227962
#117227942
#105935013
#47239