Finite Group Databasebeta

← #13156553

Subgroups — #13156553

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
1C4⋊#6296C211yesmaximal, characteristic, complemented
2#11723429C211yesmaximal, characteristic
3#11722042C212yesmaximal, complemented
4C3×#63563C212yesmaximal
5C3×#63555C211yesmaximal, characteristic, complemented
6#10591496—31nomaximal
7#10590895—32nomaximal
8C2×#6308C2211yescharacteristic
9C2×C32×D12C2211yescharacteristic
10C2×#6296—21no—
11C3×C4⋊(C3⋊Dic3)C2212yes—
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C2×C32×C3⋊C8C2212yes—
14#88815—21no—
15(C2×D3)⋊(C3⋊Dic3)—21no—
16(C2×D3)⋊(C3⋊Dic3)—21no—
17#64145—31no—
18#63919—31no—
19#63918—32no—
20#63860—32no—
21#63860—61no—
22#63643—32no—
23C3×#832—32no—
24C3×#821—32no—
25C3×#815—32no—
26#63567—32no—
27#63567—32no—
28#63567—62no—
29#63563D312yes—
30#63555D311yescharacteristic, complemented
31C22×C32×D3—21no—
32C2×C33×C4C2311yescharacteristic
33C2×C3⋊(C3⋊Dic3)—21no—
34#6308—22no—
35C2×C3×C3⋊Dic3—22no—
36C2×C3×C3⋊Dic3D411yescharacteristic
37C2×C3×C3⋊Dic3D411yescharacteristic
38C32×D12—22no—
39#6296—42no—
40C32×C3⋊C8—22no—
41#4765—92no—
42#4714—91no—
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×#627—62no—
52C2×#627—61no—
53C2×C32×C8—32no—
54#3757—61no—
55#3757—61no—
56C4⋊(C3⋊Dic3)C2×D312yes—
57C4×C3⋊Dic3C2×D311yescharacteristic
58C3×C4⋊Dic3—32no—
59C3×C4⋊Dic3—32no—
60C3×C4⋊Dic3—62no—
61C3×C4×Dic3—32no—
62C3×C4×Dic3—61no—
63C2×C3×C3⋊C8—22no—
64C2×C3×C3⋊C8C2×D312yes—
65C2×C3×C3⋊C8C2×D312yes—
66#3724—61no—
67Dic3⋊Dic3—62no—
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—
80C3×C3⋊Dic3—42no—
81#936—92no—
82#936—181no—
83#888—92no—
84#875—91no—
85#869—91no—
86C3×#102—92no—
87C3×#97—91no—
88#832—32no—
89#821—32no—
90#816—92no—
91#816—92no—
92#816—182no—
93#815—32no—
94#810—92no—
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×C4C22×D311yescharacteristic
103C2×C32×C4C22×D312yes—
104C2×C3⋊Dic3—22no—
105C2×C3⋊Dic3—61no—
106C2×C3⋊Dic3—62no—
107C2×C3⋊Dic3—62no—
108C2×C3⋊Dic3—122no—
109C2×C3⋊Dic3—122no—
110C2×C3⋊Dic3D1211yescharacteristic
111C2×C3⋊Dic3D1211yescharacteristic
112#634—62no—
113#634—64no—
114#634—62no—
115#634—62no—
116#634—122no—
117#634—122no—
118#634—124no—
119C2×C3×Dic3—32no—
120C2×C3×Dic3—32no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—62no—
124C2×C3×Dic3—62no—
125C2×C3×Dic3—122no—
126C3×D12—22no—
127C3×D12—22no—
128C3×D12—42no—
129#627—124no—
130#627—122no—
131C32×C8—62no—
132C3×C3⋊C8—22no—
133C3×C3⋊C8—22no—
134C3×C3⋊C8—42no—
135#489—271noSylow
136C2×C33#12411yescharacteristic
137C2×C33AGL(1,ℤ/8)11yescharacteristic
138C2×C33#13711yescharacteristic
139C32×D3—81no—
140C2×C3×D4—32no—
141C2×C3×D4—61no—
142C2×D12D3211yescharacteristic
143C2×Dic6—92no—
144C2×Dic6—91no—
145C2×Dic6—182no—
146C2×Dic6—181no—
147C2×Dic6—182no—
148C2×Dic6—181no—
149C2×Dic6—361no—
150C2×C3×C8—32no—
151C2×C3×C8—32no—
152C2×C3×C8—62no—
153C3×C4⋊C4—92no—
154C3×C42—91no—
155#217—182no—
156#217—182no—
157#217—182no—
158#217—182no—
159(C2×D3)⋊C4—181no—
160(C2×D3)⋊C4—181no—
161C4⋊Dic3—32no—
162C4⋊Dic3—32no—
163C4⋊Dic3—62no—
164C4⋊Dic3—181no—
165Dic3⋊C4—182no—
166C4×Dic3—32no—
167C4×Dic3—61no—
168C2×C3⋊C8D3212yes—
169C22×C32—22no—
170C22×C32—21no—
171C22×C32—41no—
172C22×C32—121no—
173C22×C32—121no—
174C22×C32—121no—
175C22×C32C2×D1211yescharacteristic
176C22×C32D3×D412yes—
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#23512yes—
188C32×C4#23512yes—
189C32×C4#23712yes—
190C3⋊Dic3—21no—
191C3⋊Dic3—21no—
192C3⋊Dic3—42no—
193C3⋊Dic3—124no—
194C3⋊Dic3—122no—
195C3⋊Dic3—124no—
196C3⋊Dic3—121no—
197C3⋊Dic3—122no—
198C3×Dic3—62no—
199C3×Dic3—62no—
200C3×Dic3—121no—
201C3×Dic3—121no—
202C3×Dic3—122no—
203C3×Dic3—122no—
204C3×Dic3—242no—
205C4⋊Q8—271no—
206#124—271no—
207#103—272no—
208#102—92no—
209#97—91no—
210C33#48911yescharacteristic, complemented, Sylow
211C23×C3—62no—
212C23×C3—62no—
213C22×D3—21no—
214C3×D4—62no—
215C3×D4—62no—
216C3×D4—122no—
217C2×C3×C4—21no—
218C2×C3×C4—22no—
219C2×C3×C4—41no—
220C2×C3×C4—91no—
221C2×C3×C4—91no—
222C2×C3×C4—182no—
223C2×C3×C4C2×D3212yes—
224C2×C3×C4C2×D3211yescharacteristic
225C2×Dic3—32no—
226C2×Dic3—32no—
227C2×Dic3—61no—
228C2×Dic3—61no—
229C2×Dic3—62no—
230C2×Dic3—62no—
231C2×Dic3—122no—
232C2×Dic3—182no—
233C2×Dic3—182no—
234C2×Dic3—181no—
235C2×Dic3—362no—
236C2×Dic3—362no—
237D12—22no—
238Dic6—184no—
239Dic6—182no—
240Dic6—182no—
241Dic6—182no—
242Dic6—362no—
243Dic6—362no—
244Dic6—364no—
245Dic6—364no—
246Dic6—362no—
247C3×C8—62no—
248C3×C8—62no—
249C3×C8—122no—
250C3⋊C8—22no—
251C2×C32—22no—
252C2×C32—22no—
253C2×C32—22no—
254C2×C32—21no—
255C2×C32—21no—
256C2×C32—21no—
257C2×C32—41no—
258C2×C32—41no—
259C2×C32—41no—
260C2×C32—241no—
261C2×C32#87511yescharacteristic
262C2×C32#88512yes—
263C2×C32#90811yescharacteristic
264C2×C32#90911yescharacteristic
265C2×C32#91112yes—
266C2×C32#91512yes—
267C3×D3—82no—
268C3×D3—82no—
269C2×Q8—271no—
270C2×Q8—541no—
271C2×D4—31no—
272C2×C8—32no—
273C4⋊C4—92no—
274C4⋊C4—541no—
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×D1212yes—
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—181no—
297C3×C4—181no—
298C3×C4—362no—
299C3×C4#379812yes—
300C3×C4#380412yes—
301C3×C4#380412yes—
302Dic3—62no—
303Dic3—62no—
304Dic3—121no—
305Dic3—121no—
306Dic3—122no—
307Dic3—122no—
308Dic3—242no—
309Dic3—364no—
310Dic3—362no—
311Dic3—364no—
312Dic3—362no—
313Dic3—361no—
314C32—21no—
315C32—22no—
316C32—41no—
317C32#471411yescharacteristic, complemented
318C32#476512yescomplemented
319C23—61no—
320Q8—542no—
321Q8—1082no—
322D4—62no—
323C2×C4—91no—
324C2×C4—91no—
325C2×C4—182no—
326C2×C4—541no—
327C2×C4D3311yescharacteristic
328C8—62no—
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#6379512yes—
341C2×C3#6380012yes—
342C2×C3#6385212yes—
343C2×C3#6392711yescharacteristic
344C2×C3#6393111yescharacteristic
345C2×C3#6395311yescharacteristic
346D3—81no—
347C22—121no—
348C22—121no—
349C22—121no—
350C22#8895511yescharacteristic
351C4—181no—
352C4—181no—
353C4—362no—
354C4—1081no—
355C4#8895912yes—
356C3—21no—
357C3—22no—
358C3—41no—
359C3#1059089512yescomplemented
360C3#1059149611yescharacteristic, complemented
361C2—241no—
362C2#1172288011yescharacteristic
363C2#1172289511yescharacteristic
364C2#1172300811yescharacteristic
subgroupG/H
centerC22#88955
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
1. C4⋊#6296C2semidirect
3. #11722042C2semidirect
5. C3×#63555C2semidirect
30. #63555D3semidirect
210. C33#489semidirect
317. C32#4714semidirect
318. C32#4765semidirect
359. C3#10590895semidirect
360. C3#10591496semidirect

Maximal subgroups

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

subgroupindex
C4⋊#62962
#117234292
#117220422
C3×#635632
C3×#635552
#105914963
#105908953