Finite Group Databasebeta

← #13156548

Subgroups — #13156548

Order 1728.

598 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 98 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#11723429C211yesmaximal, characteristic
2C3×#63919C211yesmaximal, characteristic, complemented
3#11722071C211yesmaximal, characteristic, complemented
4#11722041C212yesmaximal, complemented
5C3×#63562C212yesmaximal
6#10591496—31nomaximal
7#10590531—32nomaximal
8C2×#6308C2211yescharacteristic
9C2×C32×D12C2211yescharacteristic
10C2×C3×#627—21no—
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C2×C3×C3⋊(C3⋊C8)C2212yes—
13C32×C4⋊Dic3C2212yes—
14(C2×D3)⋊(C3⋊Dic3)—21no—
15(C2×D3)⋊(C3⋊Dic3)—21no—
16C3×#3724—21no—
17#64145—31no—
18C3×#888—32no—
19#63919D311yescharacteristic, complemented
20#63860—32no—
21#63860—61no—
22#63637—32no—
23C3×#832—32no—
24C3×#809—32no—
25#63567—32no—
26#63567—32no—
27#63567—62no—
28#63562D312yes—
29#63555—31no—
30#63554—32no—
31C22×C32×D3—21no—
32C2×C33×C4C2311yescharacteristic
33C2×C3⋊(C3⋊Dic3)—21no—
34#6308—22no—
35C2×C3×C3⋊Dic3D411yescharacteristic
36C2×C3×C3⋊Dic3D411yescharacteristic
37C2×C32×Dic3—22no—
38C32×D12—22no—
39C3×#627—42no—
40C3×C3⋊(C3⋊C8)—22no—
41#5089—91no—
42#4765—92no—
43C2×C32×D4—31no—
44C2×#634—31no—
45C2×#634—32no—
46C2×#634—62no—
47C2×#634—61no—
48C2×#634—121no—
49C2×C3×D12—21no—
50C2×C3×D12C2×D312yes—
51C2×C3×Dic6—62no—
52C2×#627—21no—
53C32×C4⋊C4—32no—
54#3757—61no—
55#3757—61no—
56C4×C3⋊Dic3C2×D311yescharacteristic
57C2×C3⋊(C3⋊C8)C2×D312yes—
58C3×C4⋊Dic3—22no—
59C3×C4⋊Dic3C2×D312yes—
60C3×C4⋊Dic3C2×D312yes—
61C3×Dic3⋊C4—62no—
62C3×C4×Dic3—32no—
63C3×C4×Dic3—61no—
64C2×C3×C3⋊C8—32no—
65C2×C3×C3⋊C8—32no—
66C2×C3×C3⋊C8—62no—
67#3724—21no—
68(C2×D3)⋊Dic3—62no—
69(C2×D3)⋊Dic3—62no—
70(C2×D3)⋊Dic3—62no—
71(C2×D3)⋊Dic3—62no—
72C22×C33C2×D411yescharacteristic
73C2×C32×D3—41no—
74C2×C32×D3—41no—
75C2×C32×D3—41no—
76C33×C4#4112yes—
77C3⋊(C3⋊Dic3)—41no—
78C3×C3⋊Dic3—21no—
79C3×C3⋊Dic3—21no—
80C32×Dic3—42no—
81C3×C4⋊Q8—91no—
82#936—92no—
83#936—181no—
84#888—32no—
85#875—91no—
86C3×#102—92no—
87#832—32no—
88#816—92no—
89#815—92no—
90#810—92no—
91#810—92no—
92#810—182no—
93#809—32no—
94#803—91no—
95C23×C32—61no—
96C22×C3×D3—22no—
97C22×C3×D3—22no—
98C32×D4—62no—
99C2×C32×C4—22no—
100C2×C32×C4—21no—
101C2×C32×C4—41no—
102C2×C32×C4—62no—
103C2×C32×C4C22×D312yes—
104C2×C32×C4C22×D311yescharacteristic
105C2×C3⋊Dic3—61no—
106C2×C3⋊Dic3—62no—
107C2×C3⋊Dic3—62no—
108C2×C3⋊Dic3—122no—
109C2×C3⋊Dic3—122no—
110C2×C3⋊Dic3#6711yescharacteristic
111C2×C3⋊Dic3#6711yescharacteristic
112#634—62no—
113#634—64no—
114#634—62no—
115#634—62no—
116#634—122no—
117#634—122no—
118#634—124no—
119C2×C3×Dic3—22no—
120C2×C3×Dic3—22no—
121C2×C3×Dic3—32no—
122C2×C3×Dic3—32no—
123C2×C3×Dic3—42no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C3×D12—22no—
127C3×D12—22no—
128C3×D12—42no—
129C3×Dic6—124no—
130#627—42no—
131C3⋊(C3⋊C8)—22no—
132C3×C3⋊C8—62no—
133C3×C3⋊C8—62no—
134C3×C3⋊C8—122no—
135#489—271noSylow
136C2×C33#12411yescharacteristic
137C2×C33AGL(1,ℤ/8)11yescharacteristic
138C2×C33#13711yescharacteristic
139C32×D3—81no—
140C2×C3×Q8—181no—
141C2×C3×D4—32no—
142C2×C3×D4—61no—
143C2×D12D3211yescharacteristic
144C2×Dic6—62no—
145C2×Dic6—91no—
146C2×Dic6—92no—
147C2×Dic6—182no—
148C2×Dic6—181no—
149C2×Dic6—361no—
150C2×C3×C8—92no—
151C3×C4⋊C4—32no—
152C3×C4⋊C4—32no—
153C3×C4⋊C4—62no—
154C3×C4⋊C4—181no—
155C3×C42—91no—
156#217—182no—
157#217—182no—
158#217—182no—
159#217—182no—
160(C2×D3)⋊C4—181no—
161(C2×D3)⋊C4—181no—
162C4⋊Dic3D3212yes—
163Dic3⋊C4—62no—
164C4×Dic3—32no—
165C4×Dic3—61no—
166C2×C3⋊C8—32no—
167C2×C3⋊C8—32no—
168C2×C3⋊C8—62no—
169C22×C32—22no—
170C22×C32—21no—
171C22×C32—41no—
172C22×C32—121no—
173C22×C32—121no—
174C22×C32—121no—
175C22×C32D3×D412yes—
176C22×C32C2×#6711yescharacteristic
177C2×C3×D3—42no—
178C2×C3×D3—42no—
179C2×C3×D3—42no—
180C2×C3×D3—42no—
181C2×C3×D3—42no—
182C2×C3×D3—42no—
183C32×C4—22no—
184C32×C4—22no—
185C32×C4—22no—
186C32×C4—42no—
187C32×C4—122no—
188C32×C4#23512yes—
189C32×C4#23712yes—
190C32×C4#23912yes—
191C3⋊Dic3—21no—
192C3⋊Dic3—21no—
193C3⋊Dic3—124no—
194C3⋊Dic3—122no—
195C3⋊Dic3—124no—
196C3⋊Dic3—122no—
197C3⋊Dic3—121no—
198C3×Dic3—42no—
199C3×Dic3—42no—
200C3×Dic3—62no—
201C3×Dic3—62no—
202C3×Dic3—82no—
203C3×Dic3—121no—
204C3×Dic3—121no—
205C4⋊Q8—91no—
206#124—271no—
207#103—272no—
208#102—92no—
209#97—271no—
210C33#48911yescharacteristic, complemented, Sylow
211C23×C3—62no—
212C23×C3—62no—
213C22×D3—21no—
214C3×Q8—362no—
215C3×D4—62no—
216C3×D4—62no—
217C3×D4—122no—
218C2×C3×C4—22no—
219C2×C3×C4—21no—
220C2×C3×C4—41no—
221C2×C3×C4—62no—
222C2×C3×C4—62no—
223C2×C3×C4—91no—
224C2×C3×C4—91no—
225C2×C3×C4—122no—
226C2×C3×C4C2×D3212yes—
227C2×C3×C4C2×D3211yescharacteristic
228C2×Dic3—22no—
229C2×Dic3—32no—
230C2×Dic3—32no—
231C2×Dic3—61no—
232C2×Dic3—61no—
233C2×Dic3—182no—
234C2×Dic3—182no—
235C2×Dic3—181no—
236C2×Dic3—362no—
237C2×Dic3—362no—
238D12—22no—
239Dic6—124no—
240Dic6—184no—
241Dic6—182no—
242Dic6—182no—
243Dic6—182no—
244Dic6—362no—
245Dic6—362no—
246Dic6—364no—
247C3×C8—182no—
248C3⋊C8—62no—
249C3⋊C8—62no—
250C3⋊C8—122no—
251C2×C32—22no—
252C2×C32—21no—
253C2×C32—21no—
254C2×C32—22no—
255C2×C32—22no—
256C2×C32—21no—
257C2×C32—41no—
258C2×C32—41no—
259C2×C32—41no—
260C2×C32—241no—
261C2×C32#88512yes—
262C2×C32#91112yes—
263C2×C32#91512yes—
264C2×C32#93211yescharacteristic
265C2×C32#94211yescharacteristic
266C2×C32#94711yescharacteristic
267C3×D3—82no—
268C3×D3—82no—
269C2×Q8—181no—
270C2×Q8—271no—
271C2×D4—31no—
272C2×C8—92no—
273C4⋊C4—32no—
274C4⋊C4—181no—
275C22⋊C4—541no—
276C22⋊C4—541no—
277C42—91no—
278C22×C3—21no—
279C22×C3—22no—
280C22×C3—41no—
281C22×C3—122no—
282C22×C3—122no—
283C22×C3—122no—
284C22×C3—122no—
285C22×C3—122no—
286C22×C3—122no—
287C22×C3D3×#6712yes—
288C22×C3#381111yescharacteristic
289C2×D3—41no—
290C2×D3—41no—
291C2×D3—41no—
292C3×C4—22no—
293C3×C4—22no—
294C3×C4—22no—
295C3×C4—42no—
296C3×C4—122no—
297C3×C4—122no—
298C3×C4—181no—
299C3×C4—181no—
300C3×C4—242no—
301C3×C4#379612yes—
302C3×C4#379912yes—
303C3×C4#380412yes—
304Dic3—42no—
305Dic3—62no—
306Dic3—62no—
307Dic3—121no—
308Dic3—121no—
309Dic3—364no—
310Dic3—362no—
311Dic3—364no—
312Dic3—362no—
313Dic3—361no—
314C32—21no—
315C32—22no—
316C32—41no—
317C32#476512yescomplemented
318C32#508911yescharacteristic, complemented
319C23—61no—
320Q8—362no—
321Q8—542no—
322D4—62no—
323C2×C4—62no—
324C2×C4—91no—
325C2×C4—91no—
326C2×C4—541no—
327C2×C4D3311yescharacteristic
328C8—182no—
329C2×C3—21no—
330C2×C3—22no—
331C2×C3—21no—
332C2×C3—22no—
333C2×C3—21no—
334C2×C3—22no—
335C2×C3—41no—
336C2×C3—41no—
337C2×C3—41no—
338C2×C3—242no—
339C2×C3—242no—
340C2×C3#6390712yes—
341C2×C3#6392711yescharacteristic
342C2×C3#6393111yescharacteristic
343C2×C3#6393212yes—
344C2×C3#6394612yes—
345C2×C3#6395311yescharacteristic
346D3—81no—
347C22—121no—
348C22—121no—
349C22—121no—
350C22#8895211yescharacteristic
351C4—122no—
352C4—181no—
353C4—181no—
354C4—1081no—
355C4#8896212yes—
356C3—21no—
357C3—22no—
358C3—41no—
359C3#1059053112yescomplemented
360C3#1059149611yescharacteristic, complemented
361C2—241no—
362C2#1172287211yescharacteristic
363C2#1172288911yescharacteristic
364C2#1172300511yescharacteristic
subgroupG/H
centerC22#88952
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypocenterC33#489
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C3×#63919C2semidirect
3. #11722071C2semidirect
4. #11722041C2semidirect
19. #63919D3semidirect
210. C33#489semidirect
317. C32#4765semidirect
318. C32#5089semidirect
359. C3#10590531semidirect
360. C3#10591496semidirect

Maximal subgroups

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

subgroupindex
#117234292
C3×#639192
#117220712
#117220412
C3×#635622
#105914963
#105905313