Finite Group Databasebeta

← #13165695

Subgroups — #13165695

Order 1728.

426 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 52 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#11722976C211yesmaximal, characteristic
2D3×#3724C211yesmaximal, characteristic
3#11722796C211yesmaximal, characteristic, complemented
4#11722778C211yesmaximal, characteristic, complemented
5#11722775C211yesmaximal, characteristic, complemented
6C3×#63740C211yesmaximal, characteristic, complemented
7(C2×D3)⋊#622C211yesmaximal, characteristic
8#10593504—31nomaximal
9C2×D3×C3⋊Dic3C2211yescharacteristic
10C2×C3×#627C2211yescharacteristic
11C2×C3×D3×Dic3—21no—
12C2×C3⋊#622C2211yescharacteristic
13C2×C3×#622C2211yescharacteristic
14#88815C2211yescharacteristic
15#88815C2211yescharacteristic
16#88807—21no—
17#88806—21no—
18C3×#3724C2211yescharacteristic
19C3×(C2×D3)⋊Dic3—21no—
20#63982—31no—
21C2×#3724—31no—
22#63887—61no—
23D3×Dic3⋊C4—61no—
24C22⋊#622—31no—
25#63742—31no—
26#63742—31no—
27#63740—31no—
28#63740D311yescharacteristic, complemented
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—
36D3×C3⋊Dic3—21no—
37D3×C3⋊Dic3—21no—
38C3×#627—21no—
39C3×#627—21no—
40C3×D3×Dic3—42no—
41C3⋊#622—21no—
42C3×#622—21no—
43#4805—91nomaximal
44C22×C3⋊Dic3—31no—
45C22×C3×Dic3—61no—
46C2×C3×C4×D3—61no—
47C2×C3×Dic6—61no—
48C2×#627C2×D311yescharacteristic
49C2×D3×Dic3—21no—
50C2×D3×Dic3—61no—
51C2×D3×Dic3—61no—
52C2×#622—31no—
53C2×#622C2×D311yescharacteristic
54#3750—61no—
55#3750—61no—
56C3×#217—61no—
57C3×(C2×D3)⋊C4—61no—
58C3×Dic3⋊C4—61no—
59#3724—31no—
60#3724—31no—
61#3724—61no—
62#3724—61no—
63#3724—121no—
64#3724C2×D311yescharacteristic
65Dic3⋊Dic3—61no—
66Dic3⋊Dic3—61no—
67Dic3⋊Dic3—61no—
68Dic3⋊Dic3—61no—
69Dic3⋊Dic3—121no—
70(C2×D3)⋊Dic3—21no—
71C22×C33C2×D411yescharacteristic
72C2×C32×D3—21no—
73C2×C32×D3—21no—
74C2×C32×D3SD1612yes—
75C3⋊(C3⋊Dic3)—21no—
76C3×C3⋊Dic3—41no—
77C3×C3⋊Dic3—41no—
78C3×C3⋊Dic3#4111yescharacteristic
79C3×C3⋊Dic3#4111yescharacteristic
80C32×Dic3—41no—
81C32×Dic3—41no—
82#946—91no—
83C2×Dic3⋊C4—181no—
84D3×C4⋊C4—91no—
85#878—181no—
86C3×#103—91no—
87#835—91no—
88(C2×D3)⋊C8—91no—
89#817—91no—
90#808—91no—
91C23×C32—31no—
92C22×C3×D3—21no—
93C22×C3×D3—21no—
94C2×C32×C4—61no—
95C2×C32×C4—61no—
96C2×C3⋊Dic3—21no—
97C2×C3⋊Dic3—21no—
98C2×C3⋊Dic3—31no—
99C2×C3⋊Dic3—61no—
100C2×C3⋊Dic3—61no—
101C2×C3⋊Dic3—61no—
102C2×C3⋊Dic3—61no—
103C2×C3⋊Dic3—121no—
104C2×C3⋊Dic3—121no—
105C2×C3⋊Dic3C22×D311yescharacteristic
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—61no—
117C2×C3×Dic3—61no—
118C2×C3×Dic3—121no—
119C2×C3×Dic3—121no—
120C2×C3×Dic3—122no—
121C3×C4×D3—122no—
122C3×Dic6—121no—
123C3×Dic6—121no—
124#627—21no—
125#627—21no—
126D3×Dic3—42no—
127D3×Dic3—62no—
128D3×Dic3—62no—
129D3×Dic3—121no—
130D3×Dic3—121no—
131#622—21no—
132#622—61no—
133#483—271noSylow
134C2×C33#12311yescharacteristic
135C2×C33C2×SD1611yescharacteristic
136C2×C33#13711yescharacteristic
137C32×D3—22no—
138C2×C3×Q8—91no—
139C22×C3×C4—181no—
140C22×Dic3—61no—
141C22×Dic3—181no—
142C22×Dic3—181no—
143C2×C4×D3—61no—
144C2×C4×D3—91no—
145C2×Dic6—61no—
146C2×C3×C8—91no—
147C3×C4⋊C4—91no—
148C3×C22⋊C4—181no—
149#217—61no—
150(C2×D3)⋊C4—61no—
151C4⋊Dic3—91no—
152C4⋊Dic3—91no—
153C4⋊Dic3—181no—
154Dic3⋊C4—61no—
155Dic3⋊C4—181no—
156Dic3⋊C4—181no—
157Dic3⋊C4—181no—
158Dic3⋊C4—181no—
159Dic3⋊C4—181no—
160Dic3⋊C4—181no—
161Dic3⋊C4—181no—
162Dic3⋊C4—361no—
163Dic3⋊C4—361no—
164C2×C3⋊C8—91no—
165C22×C32—21no—
166C22×C32—21no—
167C22×C32—32no—
168C22×C32—41no—
169C22×C32—41no—
170C22×C32—61no—
171C22×C32—61no—
172C22×C32D3×D411yescharacteristic
173C2×C3×D3—22no—
174C2×C3×D3—22no—
175C2×C3×D3—22no—
176C2×C3×D3—22no—
177C2×C3×D3—41no—
178C2×C3×D3—41no—
179C32×C4—121no—
180C32×C4—121no—
181C3⋊Dic3—41no—
182C3⋊Dic3—41no—
183C3⋊Dic3—62no—
184C3⋊Dic3—61no—
185C3⋊Dic3—121no—
186C3⋊Dic3—122no—
187C3⋊Dic3—241no—
188C3⋊Dic3#23511yescharacteristic
189C3⋊Dic3#23711yescharacteristic
190C3×Dic3—41no—
191C3×Dic3—41no—
192C3×Dic3—41no—
193C3×Dic3—41no—
194C3×Dic3—61no—
195C3×Dic3—61no—
196C3×Dic3—61no—
197C3×Dic3—61no—
198C3×Dic3—81no—
199C3×Dic3—81no—
200C3×Dic3—121no—
201C3×Dic3—121no—
202C3×Dic3—122no—
203C3×Dic3—121no—
204C3×Dic3—121no—
205C3×Dic3—241no—
206C22⋊Q8—271no—
207C2×C4⋊C4—271no—
208#106—271no—
209#106—271no—
210#103—91no—
211#103—271no—
212C22⋊C8—271no—
213C33#48311yescharacteristic, complemented, Sylow
214C23×C3—61no—
215C23×C3—61no—
216C22×D3GO+(4,𝔽2)11yescharacteristic
217C3×Q8—181no—
218C3×Q8—181no—
219C2×C3×C4—61no—
220C2×C3×C4—61no—
221C2×C3×C4—61no—
222C2×C3×C4—61no—
223C2×C3×C4—91no—
224C2×C3×C4—121no—
225C2×C3×C4—121no—
226C2×C3×C4—181no—
227C2×C3×C4—181no—
228C2×C3×C4—362no—
229C2×Dic3—21no—
230C2×Dic3—21no—
231C2×Dic3—61no—
232C2×Dic3—61no—
233C2×Dic3—61no—
234C2×Dic3—61no—
235C2×Dic3—61no—
236C2×Dic3—61no—
237C2×Dic3—91no—
238C2×Dic3—121no—
239C2×Dic3—121no—
240C2×Dic3—122no—
241C2×Dic3—181no—
242C2×Dic3—181no—
243C2×Dic3—182no—
244C2×Dic3—182no—
245C2×Dic3—361no—
246C2×Dic3—361no—
247C2×Dic3—361no—
248C2×Dic3—361no—
249C4×D3—122no—
250C4×D3—181no—
251C4×D3—181no—
252Dic6—121no—
253Dic6—121no—
254C3×C8—181no—
255C3⋊C8—181no—
256C2×C32—21no—
257C2×C32—21no—
258C2×C32—21no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32—21no—
262C2×C32—41no—
263C2×C32—41no—
264C2×C32—41no—
265C2×C32—41no—
266C2×C32—41no—
267C2×C32—41no—
268C2×C32—62no—
269C2×C32#88311yescharacteristic
270C2×C32D3×SD1611yescharacteristic
271C2×C32#91811yescharacteristic
272C3×D3—24no—
273C3×D3—42no—
274C2×Q8—91no—
275C22×C4—181no—
276C22×C4—271no—
277C2×C8—91no—
278C2×C8—271no—
279C4⋊C4—91no—
280C4⋊C4—271no—
281C4⋊C4—271no—
282C4⋊C4—541no—
283C4⋊C4—541no—
284C22⋊C4—181no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—41no—
288C22×C3—41no—
289C22×C3—62no—
290C22×C3—62no—
291C22×C3—62no—
292C22×C3—62no—
293C22×C3—121no—
294C22×C3—121no—
295C22×C3C2×GO+(4,𝔽2)11yescharacteristic
296C2×D3—21no—
297C2×D3—21no—
298C2×D3#377512yes—
299C3×C4—91no—
300C3×C4—91no—
301C3×C4—121no—
302C3×C4—121no—
303C3×C4—121no—
304C3×C4—121no—
305C3×C4—241no—
306C3×C4—241no—
307C3×C4—361no—
308C3×C4—361no—
309Dic3—41no—
310Dic3—41no—
311Dic3—61no—
312Dic3—61no—
313Dic3—61no—
314Dic3—61no—
315Dic3—121no—
316Dic3—121no—
317Dic3—122no—
318Dic3—121no—
319Dic3—121no—
320Dic3—182no—
321Dic3—181no—
322Dic3—241no—
323Dic3—361no—
324Dic3—362no—
325Dic3—721no—
326C32—21no—
327C32—21no—
328C32—41no—
329C32—41no—
330C32#480511yescharacteristic, complemented
331C23—31no—
332Q8—181no—
333Q8—181no—
334C2×C4—61no—
335C2×C4—61no—
336C2×C4—91no—
337C2×C4—181no—
338C2×C4—181no—
339C2×C4—271no—
340C2×C4—362no—
341C2×C4—541no—
342C2×C4—541no—
343C8—181no—
344C8—541no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3—41no—
352C2×C3—41no—
353C2×C3—41no—
354C2×C3—41no—
355C2×C3—41no—
356C2×C3—41no—
357C2×C3—64no—
358C2×C3—122no—
359C2×C3#6423311yescharacteristic
360C2×C3C2×#377511yescharacteristic
361C2×C3#6423911yescharacteristic
362D3—22no—
363C22—32no—
364C22—61no—
365C22—61no—
366C22D3×GO+(4,𝔽2)11yescharacteristic
367C4—91no—
368C4—91no—
369C4—121no—
370C4—121no—
371C4—361no—
372C4—361no—
373C4—541no—
374C3—21no—
375C3—21no—
376C3—41no—
377C3—41no—
378C3#1059350411yescharacteristic, complemented
379C2—62no—
380C2#1172412111yescharacteristic
381C2D3×#377511yescharacteristic
382C2#1172414311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socleC22×C33C2×D4
prime radicalC22×C32×D3D4
hypocenterC33#483
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. #11722796C2semidirect
4. #11722778C2semidirect
5. #11722775C2semidirect
6. C3×#63740C2semidirect
28. #63740D3semidirect
213. C33#483semidirect
330. C32#4805semidirect
378. C3#10593504semidirect

Maximal subgroups

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

subgroupindex
#117229762
D3×#37242
#117227962
#117227782
#117227752
C3×#637402
(C2×D3)⋊#6222
#105935043
#48059