Finite Group Databasebeta

← #13167499

Subgroups — #13167499

Order 1728.

560 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 59 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×#88942C212yesmaximal, complemented
2C2×#88941C212yesmaximal, complemented
3#11723697C211yesmaximal, characteristic, complemented
4#11723672C211yesmaximal, characteristic, complemented
5C3×#63777C211yesmaximal, characteristic, complemented
6#10593427—31nomaximal
7C2×C3×C4⋊(C3⋊D3)—21no—
8C2×#6296C2211yescharacteristic
9C2×#6295—21no—
10C2×#6295C2211yescharacteristic
11C2×#6295C2212yes—
12C2×C3×#622C2212yes—
13#88942—24no—
14#88941—24no—
15C3×#3752—21no—
16C3×C4×C3⋊Dic3C2211yescharacteristic
17#88814—21no—
18C2×#3775—32no—
19C2×#3774—32no—
20#63915—31no—
21#63914—31no—
22#63911—61no—
23#63852—61no—
24#63777D311yescharacteristic, complemented
25C22×C3×C3⋊D3—21no—
26C22×C3×C3⋊D3—22no—
27C2×C33×C4D411yescharacteristic
28C2×C3×C3⋊Dic3—21no—
29C2×C3×C3⋊Dic3D411yescharacteristic
30C2×C3×C3⋊Dic3C2311yescharacteristic
31C3×C4⋊(C3⋊D3)—42no—
32#6296—22no—
33#6295—22no—
34#6295—22no—
35#6295—22no—
36#6295—42no—
37C3×#622D414yescomplemented
38#4712—91nomaximal
39C2×C4⋊(C3⋊D3)—21no—
40C2×C3×D12—61no—
41C2×C3×D12—61no—
42C2×C3×D12—121no—
43C2×#627—31no—
44C2×#627—61no—
45C2×#626—61no—
46C2×#626—62no—
47C2×#626—62no—
48C2×#625—32no—
49C2×#625—31no—
50C2×#625—61no—
51C2×#622C2×D312yes—
52#3775—64no—
53#3774—64no—
54#3752—21no—
55C4×C3⋊Dic3C2×D311yescharacteristic
56C3×(C2×D3)⋊C4—61no—
57C3×(C2×D3)⋊C4—61no—
58C3×(C2×D3)⋊C4—121no—
59C3×C4×Dic3—61no—
60C3×C4×Dic3—61no—
61#3722—61no—
62(C2×D3)⋊Dic3—61no—
63(C2×D3)⋊Dic3—61no—
64C22×C33C2×D411yescharacteristic
65C2×C3×C3⋊D3—41no—
66C2×C3×C3⋊D3—42no—
67C2×C3×C3⋊D3—42no—
68C2×C3×C3⋊D3—41no—
69C2×C3×C3⋊D3—42no—
70C2×C3×C3⋊D3—41no—
71C33×C4—21no—
72C3×C3⋊Dic3—21no—
73C3×C3⋊Dic3—41no—
74C3×C3⋊Dic3C2×D412yes—
75#940—181no—
76C2×D24—92no—
77C2×#204—92no—
78#885—181no—
79#875—91no—
80#874—91no—
81C3×#97—91no—
82C22×C3⋊D3—21no—
83C22×C3⋊D3—22no—
84C22×C3×D3—61no—
85C22×C3×D3—62no—
86C22×C3×D3—61no—
87C22×C3×D3—62no—
88C22×C3×D3—121no—
89C22×C3×D3—122no—
90C2×C32×C4—21no—
91C2×C32×C4—21no—
92C2×C32×C4—41no—
93C2×C32×C4—41no—
94C2×C32×C4D1211yescharacteristic
95C2×C3⋊Dic3—21no—
96C2×C3⋊Dic3D1211yescharacteristic
97C2×C3⋊Dic3C22×D311yescharacteristic
98C4⋊(C3⋊D3)—42no—
99C2×C3×Dic3—61no—
100C2×C3×Dic3—61no—
101C2×C3×Dic3—61no—
102C2×C3×Dic3—61no—
103C2×C3×Dic3—61no—
104C2×C3×Dic3—61no—
105C2×C3×Dic3—121no—
106C3×D12—122no—
107C3×D12—122no—
108C3×D12—124no—
109#627—62no—
110#627—122no—
111#626—122no—
112#626—122no—
113#626—122no—
114#626—124no—
115#625—62no—
116#625—62no—
117#625—62no—
118#625—122no—
119#622D1214yescomplemented
120#496—271noSylow
121C2×C33#12711yescharacteristic
122C2×C33AGL(1,ℤ/8)11yescharacteristic
123C2×C33AGL(1,ℤ/8)11yescharacteristic
124C3×C3⋊D3—82no—
125C3×C3⋊D3—81no—
126C2×C3×D4—181no—
127C2×#67—181no—
128C2×#67—182no—
129C2×#67—182no—
130C2×D12—61no—
131C2×D12—61no—
132C2×D12—92no—
133C2×D12—91no—
134C2×D12—121no—
135C2×D12—181no—
136C2×Dic6—91no—
137C2×Dic6—181no—
138C2×C3×C8—92no—
139C3×C22⋊C4—181no—
140C3×C42—91no—
141#217—181no—
142(C2×D3)⋊C4—61no—
143(C2×D3)⋊C4—61no—
144(C2×D3)⋊C4—121no—
145(C2×D3)⋊C4—181no—
146(C2×D3)⋊C4—181no—
147C4×Dic3—61no—
148C4×Dic3—61no—
149D24—184no—
150#204—184no—
151C22×C32—21no—
152C22×C32—21no—
153C22×C32—41no—
154C22×C32—41no—
155C22×C32C2×D1211yescharacteristic
156C2×C3⋊D3—42no—
157C2×C3⋊D3—41no—
158C2×C3⋊D3—42no—
159C2×C3⋊D3—41no—
160C2×C3⋊D3—42no—
161C2×C3⋊D3—41no—
162C2×C3×D3—122no—
163C2×C3×D3—121no—
164C2×C3×D3—122no—
165C2×C3×D3—122no—
166C2×C3×D3—121no—
167C2×C3×D3—121no—
168C2×C3×D3—124no—
169C2×C3×D3—122no—
170C2×C3×D3—121no—
171C2×C3×D3—122no—
172C2×C3×D3—122no—
173C2×C3×D3—122no—
174C2×C3×D3—121no—
175C2×C3×D3—121no—
176C2×C3×D3—241no—
177C2×C3×D3—242no—
178C2×C3×D3—242no—
179C2×C3×D3—241no—
180C32×C4—22no—
181C32×C4—21no—
182C32×C4—41no—
183C32×C4—42no—
184C32×C4—81no—
185C3⋊Dic3—21no—
186C3⋊Dic3—41no—
187C3⋊Dic3C2×D1212yes—
188C3×Dic3—62no—
189C3×Dic3—62no—
190C3×Dic3—62no—
191C3×Dic3—121no—
192C3×Dic3—121no—
193C3×Dic3—121no—
194C3×Dic3—241no—
195C2×SD16—272no—
196C2×D8—272no—
197#127—271no—
198#124—271no—
199#97—91no—
200C33#49611yescharacteristic, complemented, Sylow
201C23×C3—181no—
202C23×C3—182no—
203C22×D3—61no—
204C22×D3—62no—
205C22×D3—61no—
206C22×D3—62no—
207C22×D3—121no—
208C22×D3—122no—
209C3×D4—362no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—91no—
215C2×C3×C4—91no—
216C2×C3×C4—181no—
217C2×C3×C4GO+(4,𝔽2)11yescharacteristic
218#67—362no—
219#67—362no—
220#67—362no—
221#67—364no—
222C2×Dic3—61no—
223C2×Dic3—61no—
224C2×Dic3—61no—
225C2×Dic3—61no—
226C2×Dic3—61no—
227C2×Dic3—61no—
228C2×Dic3—121no—
229D12—122no—
230D12—122no—
231D12—124no—
232D12—182no—
233D12—182no—
234D12—182no—
235D12—362no—
236Dic6—182no—
237Dic6—362no—
238C3×C8—94no—
239C2×C32—21no—
240C2×C32—21no—
241C2×C32—21no—
242C2×C32—21no—
243C2×C32—21no—
244C2×C32—21no—
245C2×C32—41no—
246C2×C32—41no—
247C2×C32—41no—
248C2×C32—41no—
249C2×C32—41no—
250C2×C32—41no—
251C2×C32#87411yescharacteristic
252C2×C32#90811yescharacteristic
253C2×C32#90811yescharacteristic
254C3⋊D3—82no—
255C3⋊D3—81no—
256C3×D3—242no—
257C3×D3—241no—
258C3×D3—244no—
259C3×D3—242no—
260C3×D3—242no—
261C3×D3—241no—
262C2×Q8—271no—
263C2×D4—181no—
264C2×D4—271no—
265C2×D4—272no—
266C2×D4—541no—
267SD16—544no—
268D8—544no—
269C2×C8—92no—
270C22⋊C4—181no—
271C22⋊C4—541no—
272C42—91no—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—362no—
278C22×C3—361no—
279C22×C3—362no—
280C22×C3—362no—
281C22×C3—361no—
282C22×C3—361no—
283C22×C3C2×GO+(4,𝔽2)11yescharacteristic
284C2×D3—122no—
285C2×D3—122no—
286C2×D3—121no—
287C2×D3—122no—
288C2×D3—121no—
289C2×D3—121no—
290C2×D3—124no—
291C2×D3—122no—
292C2×D3—122no—
293C2×D3—121no—
294C2×D3—122no—
295C2×D3—121no—
296C2×D3—122no—
297C2×D3—121no—
298C2×D3—242no—
299C2×D3—241no—
300C2×D3—242no—
301C2×D3—241no—
302C3×C4—22no—
303C3×C4—21no—
304C3×C4—41no—
305C3×C4—42no—
306C3×C4—81no—
307C3×C4—92no—
308C3×C4—181no—
309C3×C4—361no—
310Dic3—62no—
311Dic3—62no—
312Dic3—62no—
313Dic3—121no—
314Dic3—121no—
315Dic3—121no—
316Dic3—241no—
317C32—21no—
318C32—21no—
319C32—41no—
320C32—41no—
321C32#471211yescharacteristic, complemented
322C23—181no—
323C23—182no—
324Q8—542no—
325D4—362no—
326D4—542no—
327D4—542no—
328D4—542no—
329D4—1082no—
330C2×C4—91no—
331C2×C4—91no—
332C2×C4—181no—
333C2×C4#157⋊D311yescharacteristic
334C8—94no—
335C2×C3—21no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—41no—
342C2×C3—41no—
343C2×C3—41no—
344C2×C3—41no—
345C2×C3—41no—
346C2×C3—41no—
347C2×C3—722no—
348C2×C3—721no—
349C2×C3#6422411yescharacteristic
350C2×C3#6423111yescharacteristic
351C2×C3#6423611yescharacteristic
352D3—242no—
353D3—241no—
354D3—244no—
355D3—242no—
356D3—242no—
357D3—241no—
358C22—362no—
359C22—361no—
360C22—361no—
361C22—362no—
362C22—362no—
363C22—361no—
364C22C2×#157⋊D311yescharacteristic
365C4—21no—
366C4—92no—
367C4—181no—
368C4—361no—
369C3—21no—
370C3—21no—
371C3—41no—
372C3—41no—
373C3#1059342711yescharacteristic, complemented
374C2—722no—
375C2—721no—
376C2#1172434011yescharacteristic
377C2#1172434411yescharacteristic
378C2#1172435011yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socleC22×C33C2×D4
hypercenterC2×C4#157⋊D3
hypocenterC33#496
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88942C2semidirect
2. C2×#88941C2semidirect
3. #11723697C2semidirect
4. #11723672C2semidirect
5. C3×#63777C2semidirect
24. #63777D3semidirect
37. C3×#622D4semidirect
119. #622D12semidirect
200. C33#496semidirect
321. C32#4712semidirect
373. C3#10593427semidirect

Maximal subgroups

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

subgroupindex
C2×#889422
C2×#889412
#117236972
#117236722
C3×#637772
#105934273
#47129