Finite Group Databasebeta

← #13165688

Subgroups — #13165688

Order 1728.

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

normal subgroups only

This group has 446 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#11722953C211yesmaximal, characteristic
2#11722917C211yesmaximal, characteristic, complemented
3#11722796C211yesmaximal, characteristic, complemented
4#11722779C211yesmaximal, characteristic, complemented
5#11722773C211yesmaximal, characteristic, complemented
6C3×#63739C211yesmaximal, characteristic, complemented
7(C2×D3)⋊#622C211yesmaximal, characteristic, complemented
8#10593494—31nomaximal
9C2×D3×C3⋊Dic3C2211yescharacteristic
10C2×C3×#625—21no—
11C2×C3×#625C2211yescharacteristic
12C2×C3⋊#622C2211yescharacteristic
13C2×C3×#622C2211yescharacteristic
14#88815C2211yescharacteristic
15#88815C2211yescharacteristic
16(C2×D3)⋊(C3⋊Dic3)—21no—
17Dic3×C3⋊Dic3—21no—
18C3×#3724C2211yescharacteristic
19C3×(C2×D3)⋊Dic3—21no—
20#63980—31no—
21#63970—61no—
22#63957—31no—
23#63878—61no—
24C22⋊#622—31no—
25#63743—31no—
26#63742—31no—
27#63739—31no—
28#63739D311yescharacteristic, complemented
29C22×C32×D3—21no—
30C22×C32×D3D411yescharacteristic
31C2×C3⋊(C3⋊Dic3)D411yescharacteristic
32C2×C3×C3⋊Dic3—21no—
33C2×C3×C3⋊Dic3—21no—
34C2×C3×C3⋊Dic3C2311yescharacteristic
35C2×C32×Dic3—21no—
36D3×C3⋊Dic3—21no—
37D3×C3⋊Dic3—21no—
38C3×#625—21no—
39C3×#625—21no—
40C3×#625—42no—
41C3⋊#622—21no—
42C3×#622—21no—
43#4779—91nomaximal
44C22×C3⋊Dic3—31no—
45C2×C3×#67—61no—
46C2×C3×#67—61no—
47C2×C3×#67—61no—
48C2×#625—21no—
49C2×#625C2×D311yescharacteristic
50C2×D3×Dic3—61no—
51C2×D3×Dic3—61no—
52C2×#622—31no—
53C2×#622C2×D311yescharacteristic
54#3757—61no—
55C4×C3⋊Dic3—61no—
56C3×#217—61no—
57C3×(C2×D3)⋊C4—61no—
58C3×Dic3⋊C4—61no—
59#3724—31no—
60#3724—31no—
61#3724C2×D311yescharacteristic
62Dic3⋊Dic3—61no—
63Dic3⋊Dic3—61no—
64(C2×D3)⋊Dic3—21no—
65(C2×D3)⋊Dic3—61no—
66(C2×D3)⋊Dic3—61no—
67(C2×D3)⋊Dic3—121no—
68Dic32—61no—
69Dic32—61no—
70Dic32—121no—
71C22×C33C2×D411yescharacteristic
72C2×C32×D3—21no—
73C2×C32×D3—21no—
74C2×C32×D3—21no—
75C2×C32×D3—41no—
76C2×C32×D3—41no—
77C2×C32×D3—41no—
78C3⋊(C3⋊Dic3)—21no—
79C3×C3⋊Dic3—41no—
80C3×C3⋊Dic3—41no—
81C3×C3⋊Dic3#4111yescharacteristic
82C3×C3⋊Dic3#4111yescharacteristic
83C32×Dic3—41no—
84#939—181no—
85#938—91no—
86#892—91no—
87#877—181no—
88C3×#102—91no—
89#832—91no—
90(C2×D3)⋊C8—91no—
91#817—91no—
92#807—91no—
93C23×C32—31no—
94C23×C32—61no—
95C22×C3×D3—21no—
96C22×C3×D3—21no—
97C22×C3×D3—21no—
98C22×C3×D3—21no—
99C22×C3×D3—41no—
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
111C3×#67—121no—
112C3×#67—121no—
113C3×#67—122no—
114C3×#67—122no—
115C2×C3×Dic3—21no—
116C2×C3×Dic3—21no—
117C2×C3×Dic3—41no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—121no—
125C2×C3×Dic3—121no—
126#625—21no—
127#625—21no—
128#625—42no—
129D3×Dic3—62no—
130D3×Dic3—62no—
131D3×Dic3—121no—
132D3×Dic3—121no—
133#622—21no—
134#622—61no—
135#482—271noSylow
136C2×C33#12311yescharacteristic
137C2×C33#13511yescharacteristic
138C2×C33AGL(1,ℤ/8)11yescharacteristic
139C32×D3—41no—
140C32×D3—81no—
141C2×C3×D4—91no—
142C2×C3×D4—181no—
143C2×#67—61no—
144C2×#67—61no—
145C2×#67—61no—
146C22×Dic3—181no—
147C22×Dic3—181no—
148C2×C4×D3—91no—
149C2×C3×C8—91no—
150C3×C4⋊C4—91no—
151C3×C22⋊C4—181no—
152#217—61no—
153#217—181no—
154#217—181no—
155#217—361no—
156(C2×D3)⋊C4—61no—
157(C2×D3)⋊C4—181no—
158C4⋊Dic3—91no—
159C4⋊Dic3—91no—
160Dic3⋊C4—61no—
161Dic3⋊C4—181no—
162Dic3⋊C4—181no—
163C4×Dic3—181no—
164C4×Dic3—181no—
165C4×Dic3—181no—
166C4×Dic3—361no—
167C2×C3⋊C8—91no—
168C22×C32—21no—
169C22×C32—21no—
170C22×C32—41no—
171C22×C32—41no—
172C22×C32—61no—
173C22×C32—61no—
174C22×C32—61no—
175C22×C32—121no—
176C22×C32—121no—
177C22×C32—121no—
178C22×C32D3×D411yescharacteristic
179C2×C3×D3—22no—
180C2×C3×D3—22no—
181C2×C3×D3—41no—
182C2×C3×D3—41no—
183C2×C3×D3—41no—
184C2×C3×D3—41no—
185C2×C3×D3—41no—
186C2×C3×D3—41no—
187C2×C3×D3—41no—
188C2×C3×D3—42no—
189C2×C3×D3—41no—
190C2×C3×D3—41no—
191C2×C3×D3—41no—
192C2×C3×D3—81no—
193C2×C3×D3—81no—
194C32×C4—121no—
195C3⋊Dic3—41no—
196C3⋊Dic3—41no—
197C3⋊Dic3—62no—
198C3⋊Dic3—61no—
199C3⋊Dic3—121no—
200C3⋊Dic3—122no—
201C3⋊Dic3—241no—
202C3⋊Dic3#23511yescharacteristic
203C3⋊Dic3#23711yescharacteristic
204C3×Dic3—41no—
205C3×Dic3—42no—
206C3×Dic3—41no—
207C3×Dic3—61no—
208C3×Dic3—61no—
209C3×Dic3—61no—
210C3×Dic3—61no—
211C3×Dic3—121no—
212C3×Dic3—121no—
213C3×Dic3—122no—
214C3×Dic3—121no—
215C3×Dic3—121no—
216C3×Dic3—241no—
217#121—271no—
218#117—271no—
219#107—271no—
220#106—271no—
221#102—91no—
222#102—271no—
223C22⋊C8—271no—
224C33#48211yescharacteristic, complemented, Sylow
225C23×C3—61no—
226C23×C3—61no—
227C23×C3—61no—
228C23×C3—61no—
229C23×C3—121no—
230C22×D3—21no—
231C22×D3GO+(4,𝔽2)11yescharacteristic
232C3×D4—181no—
233C3×D4—181no—
234C3×D4—362no—
235C2×C3×C4—61no—
236C2×C3×C4—61no—
237C2×C3×C4—91no—
238C2×C3×C4—121no—
239C2×C3×C4—181no—
240C2×C3×C4—181no—
241#67—121no—
242#67—121no—
243#67—122no—
244#67—122no—
245C2×Dic3—21no—
246C2×Dic3—61no—
247C2×Dic3—61no—
248C2×Dic3—61no—
249C2×Dic3—61no—
250C2×Dic3—61no—
251C2×Dic3—61no—
252C2×Dic3—91no—
253C2×Dic3—121no—
254C2×Dic3—121no—
255C2×Dic3—181no—
256C2×Dic3—181no—
257C2×Dic3—182no—
258C2×Dic3—182no—
259C2×Dic3—361no—
260C2×Dic3—361no—
261C2×Dic3—361no—
262C2×Dic3—361no—
263C4×D3—181no—
264C4×D3—181no—
265C3×C8—181no—
266C3⋊C8—181no—
267C2×C32—21no—
268C2×C32—21no—
269C2×C32—21no—
270C2×C32—21no—
271C2×C32—21no—
272C2×C32—21no—
273C2×C32—41no—
274C2×C32—41no—
275C2×C32—41no—
276C2×C32—41no—
277C2×C32—41no—
278C2×C32—41no—
279C2×C32—121no—
280C2×C32—241no—
281C2×C32#88311yescharacteristic
282C2×C32#91111yescharacteristic
283C2×C32#91611yescharacteristic
284C3×D3—42no—
285C3×D3—42no—
286C3×D3—81no—
287C3×D3—82no—
288C3×D3—81no—
289C2×D4—91no—
290C2×D4—181no—
291C22×C4—271no—
292C2×C8—91no—
293C2×C8—271no—
294C4⋊C4—91no—
295C4⋊C4—271no—
296C4⋊C4—271no—
297C22⋊C4—181no—
298C22⋊C4—541no—
299C42—541no—
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3—62no—
305C22×C3—62no—
306C22×C3—121no—
307C22×C3—121no—
308C22×C3—121no—
309C22×C3—121no—
310C22×C3—121no—
311C22×C3—121no—
312C22×C3—121no—
313C22×C3—122no—
314C22×C3—121no—
315C22×C3—121no—
316C22×C3—121no—
317C22×C3—241no—
318C22×C3—241no—
319C22×C3C2×GO+(4,𝔽2)11yescharacteristic
320C2×D3—21no—
321C2×D3—21no—
322C2×D3—21no—
323C2×D3—41no—
324C2×D3—41no—
325C2×D3—41no—
326C3×C4—91no—
327C3×C4—91no—
328C3×C4—121no—
329C3×C4—122no—
330C3×C4—121no—
331C3×C4—361no—
332C3×C4—361no—
333Dic3—41no—
334Dic3—61no—
335Dic3—61no—
336Dic3—61no—
337Dic3—61no—
338Dic3—121no—
339Dic3—121no—
340Dic3—122no—
341Dic3—121no—
342Dic3—121no—
343Dic3—182no—
344Dic3—181no—
345Dic3—241no—
346Dic3—361no—
347Dic3—362no—
348Dic3—721no—
349C32—21no—
350C32—21no—
351C32—41no—
352C32—41no—
353C32#477911yescharacteristic, complemented
354C23—31no—
355C23—61no—
356D4—181no—
357D4—181no—
358D4—362no—
359C2×C4—61no—
360C2×C4—91no—
361C2×C4—181no—
362C2×C4—181no—
363C2×C4—271no—
364C2×C4—541no—
365C2×C4—541no—
366C8—181no—
367C8—541no—
368C2×C3—21no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3—21no—
373C2×C3—21no—
374C2×C3—41no—
375C2×C3—41no—
376C2×C3—41no—
377C2×C3—41no—
378C2×C3—41no—
379C2×C3—41no—
380C2×C3—122no—
381C2×C3—122no—
382C2×C3—241no—
383C2×C3—242no—
384C2×C3—241no—
385C2×C3#6423311yescharacteristic
386C2×C3#6423611yescharacteristic
387C2×C3#6423711yescharacteristic
388D3—41no—
389D3—81no—
390C22—61no—
391C22—61no—
392C22—61no—
393C22—121no—
394C22—121no—
395C22—121no—
396C22D3×GO+(4,𝔽2)11yescharacteristic
397C4—91no—
398C4—91no—
399C4—121no—
400C4—361no—
401C4—361no—
402C4—541no—
403C3—21no—
404C3—21no—
405C3—41no—
406C3—41no—
407C3#1059349411yescharacteristic, complemented
408C2—121no—
409C2—241no—
410C2#1172412111yescharacteristic
411C2#1172413311yescharacteristic
412C2#1172413811yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socleC22×C33C2×D4
prime radical#11722953C2
hypocenterC33#482
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. #11722917C2semidirect
3. #11722796C2semidirect
4. #11722779C2semidirect
5. #11722773C2semidirect
6. C3×#63739C2semidirect
7. (C2×D3)⋊#622C2semidirect
28. #63739D3semidirect
224. C33#482semidirect
353. C32#4779semidirect
407. C3#10593494semidirect

Maximal subgroups

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

subgroupindex
#117229532
#117229172
#117227962
#117227792
#117227732
C3×#637392
(C2×D3)⋊#6222
#105934943
#47799