Finite Group Databasebeta

← #13165644

Subgroups — #13165644

Order 1728.

420 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 50 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
1C2×#88938C211yesmaximal, characteristic, complemented
2C2×#88925C211yesmaximal, characteristic
3#11722981C211yesmaximal, characteristic
4#11722976C211yesmaximal, characteristic
5#11722794C211yesmaximal, characteristic, complemented
6#11722778C211yesmaximal, characteristic, complemented
7C3×#63740C211yesmaximal, characteristic, complemented
8#10593508—31nomaximal
9C2×#6296C2211yescharacteristic
10C2×D3×C3⋊Dic3C2211yescharacteristic
11C2×C3×#627C2211yescharacteristic
12C2×C3⋊#622C2211yescharacteristic
13C2×C3×#622C2211yescharacteristic
14#88938—22no—
15#88925—22no—
16#88815C2211yescharacteristic
17#88807—21no—
18Dic3⋊(C3⋊Dic3)—21no—
19C3×#3724C2211yescharacteristic
20C3×(C2×D3)⋊Dic3—21no—
21C3×(C2×D3)⋊Dic3—21no—
22C2×#3792—31no—
23C2×#3776—31no—
24#63982—31no—
25#63982—31no—
26#63893—61no—
27#63887—61no—
28#63742—31no—
29#63740—31no—
30#63740D311yescharacteristic, complemented
31C22×C32×D3D411yescharacteristic
32C2×C3⋊(C3⋊Dic3)D411yescharacteristic
33C2×C3×C3⋊Dic3—21no—
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3C2311yescharacteristic
36C2×C32×Dic3—21no—
37C2×C32×Dic3—21no—
38#6296—21no—
39#6296—21no—
40D3×C3⋊Dic3—21no—
41D3×C3⋊Dic3—21no—
42C3×#627—21no—
43C3×#627—21no—
44C3⋊#622D412yes—
45C3×#622—21no—
46#4816—91nomaximal
47C22×C3⋊Dic3—31no—
48C2×C3×Dic6—61no—
49C2×#627—31no—
50C2×#627—61no—
51C2×#627C2×D311yescharacteristic
52C2×D3×Dic3—61no—
53C2×D3×Dic3—61no—
54#3792—62no—
55C2×#622—31no—
56C2×#622C2×D311yescharacteristic
57#3776—62no—
58C4⋊(C3⋊Dic3)—61no—
59#3750—61no—
60C3×#217—61no—
61C3×#217—61no—
62C3×(C2×D3)⋊C4—61no—
63C3×(C2×D3)⋊C4—61no—
64C3×Dic3⋊C4—61no—
65#3724—31no—
66#3724C2×D311yescharacteristic
67Dic3⋊Dic3—61no—
68Dic3⋊Dic3—61no—
69Dic3⋊Dic3—61no—
70Dic3⋊Dic3—61no—
71Dic3⋊Dic3—61no—
72Dic3⋊Dic3—121no—
73Dic3⋊Dic3—121no—
74(C2×D3)⋊Dic3—21no—
75(C2×D3)⋊Dic3—21no—
76C22×C33C2×D411yescharacteristic
77C2×C32×D3—21no—
78C2×C32×D3—21no—
79C2×C32×D3—21no—
80C3⋊(C3⋊Dic3)—21no—
81C3×C3⋊Dic3—41no—
82C3×C3⋊Dic3—41no—
83C3×C3⋊Dic3C2×D411yescharacteristic
84C3×C3⋊Dic3#4111yescharacteristic
85C32×Dic3—41no—
86C32×Dic3—41no—
87#946—91no—
88C2×#216—91no—
89C2×ΓL(1,𝔽25)—91no—
90#897—91no—
91#878—181no—
92#878—181no—
93C3×#103—91no—
94#816—91no—
95#808—91no—
96C23×C32—31no—
97C22×C3×D3—21no—
98C22×C3×D3—21no—
99C2×C32×C4—61no—
100C2×C32×C4—61no—
101C2×C3⋊Dic3—21no—
102C2×C3⋊Dic3—21no—
103C2×C3⋊Dic3—31no—
104C2×C3⋊Dic3—61no—
105C2×C3⋊Dic3—61no—
106C2×C3⋊Dic3—61no—
107C2×C3⋊Dic3—61no—
108C2×C3⋊Dic3—121no—
109C2×C3⋊Dic3—121no—
110C2×C3⋊Dic3C22×D311yescharacteristic
111C2×C3×Dic3—21no—
112C2×C3×Dic3—21no—
113C2×C3×Dic3—21no—
114C2×C3×Dic3—21no—
115C2×C3×Dic3—41no—
116C2×C3×Dic3—41no—
117C2×C3×Dic3—61no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—121no—
124C2×C3×Dic3—121no—
125C3×Dic6—121no—
126C3×Dic6—121no—
127#627—21no—
128#627—21no—
129#627—61no—
130#627—61no—
131#627—121no—
132#627—121no—
133D3×Dic3—121no—
134D3×Dic3—121no—
135D3×Dic3—121no—
136D3×Dic3—121no—
137#622—21no—
138#622—32no—
139#470—271noSylow
140C2×C33#12111yescharacteristic
141C2×C33#13711yescharacteristic
142C2×C33#13711yescharacteristic
143C32×D3—41no—
144C2×C3×Q8—91no—
145C22×Dic3—181no—
146C22×Dic3—181no—
147C2×C4×D3—91no—
148C2×Dic6—61no—
149C2×Dic6—91no—
150C2×Dic6—181no—
151C2×C3×C8—91no—
152C3×C4⋊C4—91no—
153C3×C22⋊C4—181no—
154C3×C22⋊C4—181no—
155#217—61no—
156#217—61no—
157#216—182no—
158(C2×D3)⋊C4—61no—
159(C2×D3)⋊C4—61no—
160C4⋊Dic3—91no—
161C4⋊Dic3—181no—
162C4⋊Dic3—181no—
163C4⋊Dic3—181no—
164C4⋊Dic3—361no—
165Dic3⋊C4—61no—
166Dic3⋊C4—181no—
167Dic3⋊C4—181no—
168Dic3⋊C4—181no—
169Dic3⋊C4—181no—
170Dic3⋊C4—361no—
171C2×C3⋊C8—91no—
172ΓL(1,𝔽25)—182no—
173C22×C32—21no—
174C22×C32—21no—
175C22×C32—41no—
176C22×C32—41no—
177C22×C32—61no—
178C22×C32—61no—
179C22×C32—61no—
180C22×C32D3×D411yescharacteristic
181C2×C3×D3—22no—
182C2×C3×D3—22no—
183C2×C3×D3—41no—
184C2×C3×D3—41no—
185C2×C3×D3—41no—
186C2×C3×D3—41no—
187C32×C4—121no—
188C32×C4—121no—
189C3⋊Dic3—41no—
190C3⋊Dic3—41no—
191C3⋊Dic3—61no—
192C3⋊Dic3—121no—
193C3⋊Dic3—121no—
194C3⋊Dic3—241no—
195C3⋊Dic3—241no—
196C3⋊Dic3#23511yescharacteristic
197C3⋊Dic3D3×D411yescharacteristic
198C3×Dic3—41no—
199C3×Dic3—41no—
200C3×Dic3—42no—
201C3×Dic3—41no—
202C3×Dic3—41no—
203C3×Dic3—61no—
204C3×Dic3—61no—
205C3×Dic3—61no—
206C3×Dic3—61no—
207C3×Dic3—81no—
208C3×Dic3—121no—
209C3×Dic3—121no—
210C3×Dic3—122no—
211C3×Dic3—121no—
212C3×Dic3—121no—
213C3×Dic3—241no—
214C2×Q16—271no—
215C2×M16—271no—
216C22⋊Q8—271no—
217C22⋊Q8—271no—
218#106—271no—
219#103—91no—
220#103—271no—
221C33#47011yescharacteristic, complemented, Sylow
222C23×C3—61no—
223C23×C3—61no—
224C22×D3GO+(4,𝔽2)11yescharacteristic
225C3×Q8—181no—
226C3×Q8—181no—
227C2×C3×C4—61no—
228C2×C3×C4—61no—
229C2×C3×C4—61no—
230C2×C3×C4—61no—
231C2×C3×C4—91no—
232C2×C3×C4—121no—
233C2×C3×C4—121no—
234C2×C3×C4—181no—
235C2×C3×C4—181no—
236C2×Dic3—21no—
237C2×Dic3—21no—
238C2×Dic3—61no—
239C2×Dic3—61no—
240C2×Dic3—61no—
241C2×Dic3—61no—
242C2×Dic3—61no—
243C2×Dic3—61no—
244C2×Dic3—91no—
245C2×Dic3—121no—
246C2×Dic3—121no—
247C2×Dic3—181no—
248C2×Dic3—181no—
249C2×Dic3—361no—
250C2×Dic3—361no—
251C2×Dic3—361no—
252C2×Dic3—361no—
253C2×Dic3—361no—
254C2×Dic3—361no—
255C4×D3—181no—
256C4×D3—181no—
257Dic6—121no—
258Dic6—121no—
259Dic6—181no—
260Dic6—181no—
261Dic6—361no—
262Dic6—361no—
263C3×C8—181no—
264C3⋊C8—92no—
265C2×C32—21no—
266C2×C32—21no—
267C2×C32—21no—
268C2×C32—21no—
269C2×C32—21no—
270C2×C32—21no—
271C2×C32—41no—
272C2×C32—41no—
273C2×C32—41no—
274C2×C32—41no—
275C2×C32—41no—
276C2×C32—41no—
277C2×C32—121no—
278C2×C32#88411yescharacteristic
279C2×C32#91511yescharacteristic
280C2×C32#91811yescharacteristic
281C3×D3—42no—
282C3×D3—42no—
283C2×Q8—91no—
284C2×Q8—271no—
285C22×C4—271no—
286Q16—542no—
287M16—542no—
288C2×C8—91no—
289C2×C8—271no—
290C4⋊C4—91no—
291C4⋊C4—271no—
292C4⋊C4—541no—
293C4⋊C4—541no—
294C22⋊C4—181no—
295C22⋊C4—181no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—62no—
301C22×C3—62no—
302C22×C3—121no—
303C22×C3—121no—
304C22×C3—121no—
305C22×C3—121no—
306C22×C3C2×GO+(4,𝔽2)11yescharacteristic
307C2×D3—21no—
308C2×D3—21no—
309C2×D3—21no—
310C3×C4—91no—
311C3×C4—91no—
312C3×C4—121no—
313C3×C4—121no—
314C3×C4—122no—
315C3×C4—121no—
316C3×C4—121no—
317C3×C4—241no—
318C3×C4—361no—
319C3×C4—361no—
320Dic3—41no—
321Dic3—41no—
322Dic3—61no—
323Dic3—61no—
324Dic3—61no—
325Dic3—61no—
326Dic3—121no—
327Dic3—121no—
328Dic3—122no—
329Dic3—121no—
330Dic3—121no—
331Dic3—181no—
332Dic3—241no—
333Dic3—361no—
334Dic3—361no—
335Dic3—721no—
336Dic3—721no—
337C32—21no—
338C32—21no—
339C32—41no—
340C32—41no—
341C32#481611yescharacteristic, complemented
342C23—31no—
343Q8—181no—
344Q8—181no—
345Q8—541no—
346Q8—541no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—91no—
350C2×C4—181no—
351C2×C4—181no—
352C2×C4—271no—
353C2×C4—541no—
354C2×C4—541no—
355C8—181no—
356C8—272no—
357C2×C3—21no—
358C2×C3—21no—
359C2×C3—21no—
360C2×C3—21no—
361C2×C3—21no—
362C2×C3—21no—
363C2×C3—41no—
364C2×C3—41no—
365C2×C3—41no—
366C2×C3—41no—
367C2×C3—41no—
368C2×C3—41no—
369C2×C3—122no—
370C2×C3—122no—
371C2×C3#6423911yescharacteristic
372C2×C3#6423911yescharacteristic
373C2×C3#6424211yescharacteristic
374D3—41no—
375C22—61no—
376C22—61no—
377C22—61no—
378C22D3×GO+(4,𝔽2)11yescharacteristic
379C4—91no—
380C4—91no—
381C4—121no—
382C4—121no—
383C4—361no—
384C4—361no—
385C4—541no—
386C3—21no—
387C3—21no—
388C3—41no—
389C3—41no—
390C3#1059350811yescharacteristic, complemented
391C2—121no—
392C2#1172411511yescharacteristic
393C2#1172414211yescharacteristic
394C2#1172414311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socleC22×C33C2×D4
prime radicalC22×C32×D3D4
hypocenterC33#470
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. C2×#88938C2semidirect
5. #11722794C2semidirect
6. #11722778C2semidirect
7. C3×#63740C2semidirect
30. #63740D3semidirect
221. C33#470semidirect
341. C32#4816semidirect
390. C3#10593508semidirect

Maximal subgroups

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

subgroupindex
C2×#889382
C2×#889252
#117229812
#117229762
#117227942
#117227782
C3×#637402
#105935083
#48169