Finite Group Databasebeta

← #13165704

Subgroups — #13165704

Order 1728.

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

normal subgroups only

This group has 398 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#11722981C211yesmaximal, characteristic
2#11722917C211yesmaximal, characteristic
3#11722794C211yesmaximal, characteristic, complemented
4#11722778C211yesmaximal, characteristic, complemented
5#11722774C211yesmaximal, characteristic, complemented
6C3×#63743C211yesmaximal, characteristic, complemented
7(C2×D3)⋊#622C211yesmaximal, characteristic
8#10593501—31nomaximal
9C2×#6296C2211yescharacteristic
10C2×D3×C3⋊Dic3C2211yescharacteristic
11C2×C3⋊#622C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13#88815C2211yescharacteristic
14Dic3⋊(C3⋊Dic3)—21no—
15Dic3×C3⋊Dic3—21no—
16C3×#3724C2211yescharacteristic
17C3×#3724C2211yescharacteristic
18C3×(C2×D3)⋊Dic3—21no—
19C3×(C2×D3)⋊Dic3—21no—
20#63982—31no—
21#63957—31no—
22#63893—61no—
23#63878—61no—
24C22⋊#622—31no—
25#63743D311yescharacteristic, complemented
26#63742—31no—
27#63740—31no—
28#63740—31no—
29C22×C32×D3D411yescharacteristic
30C2×C3⋊(C3⋊Dic3)D411yescharacteristic
31C2×C3×C3⋊Dic3—21no—
32C2×C3×C3⋊Dic3—21no—
33C2×C3×C3⋊Dic3C2311yescharacteristic
34C2×C32×Dic3—21no—
35C2×C32×Dic3—21no—
36#6296—21no—
37#6296—21no—
38D3×C3⋊Dic3—21no—
39D3×C3⋊Dic3—21no—
40C3⋊#622—21no—
41C3×#622—21no—
42#4888—91nomaximal
43C22×C3⋊Dic3—31no—
44C2×#627—31no—
45C2×#627—61no—
46C2×D3×Dic3—61no—
47C2×D3×Dic3—61no—
48C2×#622—31no—
49C2×#622C2×D311yescharacteristic
50C4⋊(C3⋊Dic3)—61no—
51C4×C3⋊Dic3—61no—
52C3×#217—61no—
53C3×#217—61no—
54C3×(C2×D3)⋊C4—61no—
55C3×(C2×D3)⋊C4—61no—
56C3×Dic3⋊C4—61no—
57C3×Dic3⋊C4—61no—
58#3724—31no—
59#3724C2×D311yescharacteristic
60#3724C2×D311yescharacteristic
61Dic3⋊Dic3—61no—
62Dic3⋊Dic3—61no—
63Dic3⋊Dic3—61no—
64Dic3⋊Dic3—121no—
65(C2×D3)⋊Dic3—21no—
66(C2×D3)⋊Dic3—21no—
67Dic32—61no—
68Dic32—61no—
69Dic32—121no—
70C22×C33C2×D411yescharacteristic
71C2×C32×D3—21no—
72C2×C32×D3—21no—
73C2×C32×D3—21no—
74C3⋊(C3⋊Dic3)—21no—
75C3×C3⋊Dic3—41no—
76C3×C3⋊Dic3—41no—
77C3×C3⋊Dic3#4111yescharacteristic
78C3×C3⋊Dic3#4111yescharacteristic
79C32×Dic3—41no—
80C32×Dic3—41no—
81#897—91no—
82#892—91no—
83#878—181no—
84#877—181no—
85C3×#107—91no—
86(C2×D3)⋊C8—91no—
87#816—91no—
88#810—91no—
89#808—91no—
90C23×C32—31no—
91C22×C3×D3—21no—
92C22×C3×D3—21no—
93C2×C32×C4—61no—
94C2×C32×C4—61no—
95C2×C3⋊Dic3—21no—
96C2×C3⋊Dic3—21no—
97C2×C3⋊Dic3—31no—
98C2×C3⋊Dic3—61no—
99C2×C3⋊Dic3—61no—
100C2×C3⋊Dic3—61no—
101C2×C3⋊Dic3—61no—
102C2×C3⋊Dic3—121no—
103C2×C3⋊Dic3—121no—
104C2×C3⋊Dic3C22×D311yescharacteristic
105C2×C3×Dic3—21no—
106C2×C3×Dic3—21no—
107C2×C3×Dic3—21no—
108C2×C3×Dic3—21no—
109C2×C3×Dic3—41no—
110C2×C3×Dic3—41no—
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—
119#627—61no—
120#627—61no—
121#627—121no—
122#627—121no—
123D3×Dic3—62no—
124D3×Dic3—62no—
125D3×Dic3—121no—
126D3×Dic3—121no—
127#622—21no—
128#622—61no—
129#485—271noSylow
130C2×C33#12311yescharacteristic
131C2×C33#13511yescharacteristic
132C2×C33#13711yescharacteristic
133C32×D3—41no—
134C22×Dic3—181no—
135C22×Dic3—181no—
136C2×C4×D3—91no—
137C2×Dic6—91no—
138C2×Dic6—181no—
139C2×C3×C8—91no—
140C3×C4⋊C4—91no—
141C3×C4⋊C4—91no—
142C3×C22⋊C4—181no—
143C3×C22⋊C4—181no—
144#217—61no—
145#217—61no—
146(C2×D3)⋊C4—61no—
147(C2×D3)⋊C4—61no—
148C4⋊Dic3—91no—
149C4⋊Dic3—181no—
150C4⋊Dic3—181no—
151C4⋊Dic3—361no—
152Dic3⋊C4—61no—
153Dic3⋊C4—61no—
154Dic3⋊C4—181no—
155Dic3⋊C4—181no—
156C4×Dic3—181no—
157C4×Dic3—181no—
158C4×Dic3—181no—
159C4×Dic3—361no—
160C2×C3⋊C8—91no—
161C22×C32—21no—
162C22×C32—21no—
163C22×C32—41no—
164C22×C32—41no—
165C22×C32—61no—
166C22×C32—61no—
167C22×C32—61no—
168C22×C32D3×D411yescharacteristic
169C2×C3×D3—22no—
170C2×C3×D3—22no—
171C2×C3×D3—41no—
172C2×C3×D3—41no—
173C2×C3×D3—41no—
174C2×C3×D3—41no—
175C32×C4—121no—
176C32×C4—121no—
177C3⋊Dic3—41no—
178C3⋊Dic3—41no—
179C3⋊Dic3—62no—
180C3⋊Dic3—61no—
181C3⋊Dic3—121no—
182C3⋊Dic3—122no—
183C3⋊Dic3—241no—
184C3⋊Dic3#23511yescharacteristic
185C3⋊Dic3#23911yescharacteristic
186C3×Dic3—41no—
187C3×Dic3—41no—
188C3×Dic3—42no—
189C3×Dic3—42no—
190C3×Dic3—41no—
191C3×Dic3—41no—
192C3×Dic3—61no—
193C3×Dic3—61no—
194C3×Dic3—61no—
195C3×Dic3—61no—
196C3×Dic3—121no—
197C3×Dic3—121no—
198C3×Dic3—122no—
199C3×Dic3—121no—
200C3×Dic3—121no—
201C3×Dic3—241no—
202C22⋊Q8—271no—
203#117—271no—
204#107—91no—
205#106—271no—
206#103—271no—
207#103—271no—
208C22⋊C8—271no—
209C33#48511yescharacteristic, complemented, Sylow
210C23×C3—61no—
211C23×C3—61no—
212C22×D3GO+(4,𝔽2)11yescharacteristic
213C2×C3×C4—61no—
214C2×C3×C4—61no—
215C2×C3×C4—61no—
216C2×C3×C4—61no—
217C2×C3×C4—91no—
218C2×C3×C4—121no—
219C2×C3×C4—121no—
220C2×C3×C4—181no—
221C2×C3×C4—181no—
222C2×Dic3—21no—
223C2×Dic3—21no—
224C2×Dic3—61no—
225C2×Dic3—61no—
226C2×Dic3—61no—
227C2×Dic3—61no—
228C2×Dic3—61no—
229C2×Dic3—61no—
230C2×Dic3—91no—
231C2×Dic3—121no—
232C2×Dic3—121no—
233C2×Dic3—181no—
234C2×Dic3—181no—
235C2×Dic3—182no—
236C2×Dic3—182no—
237C2×Dic3—361no—
238C2×Dic3—361no—
239C2×Dic3—361no—
240C2×Dic3—361no—
241C4×D3—181no—
242C4×D3—181no—
243Dic6—181no—
244Dic6—181no—
245Dic6—361no—
246Dic6—361no—
247C3×C8—181no—
248C3⋊C8—181no—
249C2×C32—21no—
250C2×C32—21no—
251C2×C32—21no—
252C2×C32—21no—
253C2×C32—21no—
254C2×C32—21no—
255C2×C32—41no—
256C2×C32—41no—
257C2×C32—41no—
258C2×C32—41no—
259C2×C32—41no—
260C2×C32—41no—
261C2×C32—121no—
262C2×C32#89411yescharacteristic
263C2×C32#91211yescharacteristic
264C2×C32#91811yescharacteristic
265C3×D3—42no—
266C3×D3—42no—
267C2×Q8—271no—
268C22×C4—271no—
269C2×C8—91no—
270C2×C8—271no—
271C4⋊C4—91no—
272C4⋊C4—91no—
273C4⋊C4—271no—
274C4⋊C4—541no—
275C22⋊C4—181no—
276C22⋊C4—181no—
277C42—541no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—62no—
283C22×C3—62no—
284C22×C3—121no—
285C22×C3—121no—
286C22×C3—121no—
287C22×C3—121no—
288C22×C3C2×GO+(4,𝔽2)11yescharacteristic
289C2×D3—21no—
290C2×D3—21no—
291C2×D3—21no—
292C3×C4—91no—
293C3×C4—91no—
294C3×C4—121no—
295C3×C4—121no—
296C3×C4—122no—
297C3×C4—122no—
298C3×C4—121no—
299C3×C4—121no—
300C3×C4—361no—
301C3×C4—361no—
302Dic3—41no—
303Dic3—41no—
304Dic3—61no—
305Dic3—61no—
306Dic3—61no—
307Dic3—61no—
308Dic3—121no—
309Dic3—121no—
310Dic3—122no—
311Dic3—121no—
312Dic3—121no—
313Dic3—182no—
314Dic3—181no—
315Dic3—241no—
316Dic3—361no—
317Dic3—362no—
318Dic3—721no—
319C32—21no—
320C32—21no—
321C32—41no—
322C32—41no—
323C32#488811yescharacteristic, complemented
324C23—31no—
325Q8—541no—
326Q8—541no—
327C2×C4—61no—
328C2×C4—61no—
329C2×C4—91no—
330C2×C4—181no—
331C2×C4—181no—
332C2×C4—271no—
333C2×C4—541no—
334C2×C4—541no—
335C8—181no—
336C8—541no—
337C2×C3—21no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—21no—
342C2×C3—21no—
343C2×C3—41no—
344C2×C3—41no—
345C2×C3—41no—
346C2×C3—41no—
347C2×C3—41no—
348C2×C3—41no—
349C2×C3—122no—
350C2×C3—122no—
351C2×C3#6423311yescharacteristic
352C2×C3#6423711yescharacteristic
353C2×C3#6423911yescharacteristic
354D3—41no—
355C22—61no—
356C22—61no—
357C22—61no—
358C22D3×GO+(4,𝔽2)11yescharacteristic
359C4—91no—
360C4—91no—
361C4—121no—
362C4—121no—
363C4—361no—
364C4—361no—
365C4—541no—
366C3—21no—
367C3—21no—
368C3—41no—
369C3—41no—
370C3#1059350111yescharacteristic, complemented
371C2—121no—
372C2#1172412211yescharacteristic
373C2#1172413211yescharacteristic
374C2#1172414311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socleC22×C33C2×D4
prime radicalC22×C32×D3D4
hypocenterC33#485
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
3. #11722794C2semidirect
4. #11722778C2semidirect
5. #11722774C2semidirect
6. C3×#63743C2semidirect
25. #63743D3semidirect
209. C33#485semidirect
323. C32#4888semidirect
370. C3#10593501semidirect

Maximal subgroups

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

subgroupindex
#117229812
#117229172
#117227942
#117227782
#117227742
C3×#637432
(C2×D3)⋊#6222
#105935013
#48889