Finite Group Databasebeta

← #13155872

Subgroups — #13155872

Order 1728.

776 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 110 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×#89040C211yesmaximal, characteristic
2#11723435C211yesmaximal, characteristic
3C2×C3×#3719C212yesmaximal, complemented
4#11722040C212yesmaximal, complemented
5C3×#63557C211yesmaximal, characteristic
6#10591654—31nomaximal
7#10590949—32nomaximal
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C3×#634C2212yes—
10C2×C32×Dic6C2211yescharacteristic
11C2×#6296—21no—
12C2×#6294—21no—
13#89040—24no—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C2×C32×C3⋊C8C2212yes—
16#88806—21no—
17#88803—21no—
18C3×#3719—24no—
19C2×#3797—31no—
20C2×D3×Dic6—32no—
21#64155—31no—
22C2×C3×#216—32no—
23C2×C3×Dic12—32no—
24#63856—32no—
25#63856—61no—
26C2×#3719D312yescomplemented
27#63643—32no—
28C3×(C2×D3)⋊C8—32no—
29#63566—32no—
30#63566—32no—
31#63566—62no—
32#63557D311yescharacteristic
33C22×C3×C3⋊D3D411yescharacteristic
34C2×C33×C4C2311yescharacteristic
35C2×C3⋊(C3⋊Dic3)—21no—
36C2×C3×C3⋊Dic3—22no—
37C2×C3×C3⋊Dic3D411yescharacteristic
38C3×C4×C3⋊D3—22no—
39C3×#634—22no—
40C3×#634D414yes—
41C2×C32×Dic3—21no—
42C32×Dic6—22no—
43#6296—24no—
44#6294—24no—
45C32×C3⋊C8—22no—
46#5186—92no—
47#4739—91no—
48C2×C32×Q8—31no—
49C2×C4×C3⋊D3C2×D311yescharacteristic
50C2×#634C2×D312yes—
51C2×C3×C4×D3—32no—
52C2×C3×C4×D3—61no—
53C2×C3×Dic6—21no—
54C2×C3×Dic6—32no—
55C2×C3×Dic6—32no—
56C2×C3×Dic6—62no—
57C2×C3×Dic6C2×D312yes—
58C2×#627—62no—
59C2×#627—61no—
60C2×#624—61no—
61C2×D3×Dic3—62no—
62#3797—64no—
63D3×Dic6—64no—
64D3×Dic6—64no—
65C2×C32×C8—32no—
66#3752—61no—
67C4⋊(C3⋊Dic3)—31no—
68C4⋊(C3⋊Dic3)—32no—
69C4⋊(C3⋊Dic3)—61no—
70C4⋊(C3⋊Dic3)—62no—
71C4⋊(C3⋊Dic3)—121no—
72#3750—61no—
73C3×#216—64no—
74C2×C3×C3⋊C8—22no—
75C2×C3×C3⋊C8C2×D312yes—
76C2×C3×C3⋊C8C2×D312yes—
77C3×Dic12—64no—
78#3724—62no—
79#3724—62no—
80(C2×D3)⋊Dic3—62no—
81(C2×D3)⋊Dic3—62no—
82#3719—24no—
83C22×C33C2×D411yescharacteristic
84C2×C3×C3⋊D3—21no—
85C2×C3×C3⋊D3—21no—
86C2×C3×C3⋊D3Q1612yescomplemented
87C33×C4C2×D412yes—
88C3⋊(C3⋊Dic3)—41no—
89C3×C3⋊Dic3—24no—
90C3×C3⋊Dic3—21no—
91C32×Dic3—41no—
92C2×D3×Q8—92no—
93C22×Dic6—91no—
94C2×C3×Q16—92no—
95#946—92no—
96#946—181no—
97C2×#216—32no—
98#924—91no—
99C2×Dic12—32no—
100C3×C22⋊C8—91no—
101#835—92no—
102(C2×D3)⋊C8—32no—
103#816—92no—
104#816—92no—
105#816—182no—
106C22×C3⋊D3D1211yescharacteristic
107C22×C3×D3—32no—
108C22×C3×D3—61no—
109C32×Q8—62no—
110C2×C32×C4—22no—
111C2×C32×C4—21no—
112C2×C32×C4—41no—
113C2×C32×C4—61no—
114C2×C32×C4C22×D311yescharacteristic
115C2×C32×C4C22×D312yes—
116C2×C3⋊Dic3—22no—
117C2×C3⋊Dic3—61no—
118C2×C3⋊Dic3—62no—
119C2×C3⋊Dic3—62no—
120C2×C3⋊Dic3—122no—
121C2×C3⋊Dic3—122no—
122C2×C3⋊Dic3D1211yescharacteristic
123C4×C3⋊D3—22no—
124#634—22no—
125#634#6714yes—
126C2×C3×Dic3—22no—
127C2×C3×Dic3—22no—
128C2×C3×Dic3—32no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—62no—
131C2×C3×Dic3—62no—
132C2×C3×Dic3—122no—
133C3×C4×D3—62no—
134C3×C4×D3—62no—
135C3×C4×D3—122no—
136C3×Dic6—22no—
137C3×Dic6—22no—
138C3×Dic6—34no—
139C3×Dic6—34no—
140C3×Dic6—42no—
141C3×Dic6—64no—
142C3×Dic6—62no—
143C3×Dic6—62no—
144C3×Dic6—122no—
145#627—68no—
146#627—64no—
147#624—64no—
148D3×Dic3—68no—
149C32×C8—62no—
150C3×C3⋊C8—22no—
151C3×C3⋊C8—22no—
152C3×C3⋊C8—42no—
153#451—271noSylow
154C2×C33#12011yescharacteristic
155C2×C33C2×Q1611yescharacteristic
156C2×C33#13711yescharacteristic
157C3×C3⋊D3—22no—
158C2×C3×Q8—32no—
159C2×C3×Q8—61no—
160C2×C3×Q8—92no—
161C22×C3×C4—91no—
162C22×Dic3—181no—
163D3×Q8—184no—
164D3×Q8—184no—
165C2×C4×D3—32no—
166C2×C4×D3—61no—
167C2×C4×D3—182no—
168C2×Dic6—32no—
169C2×Dic6—32no—
170C2×Dic6—62no—
171C2×Dic6—184no—
172C2×Dic6—182no—
173C2×Dic6—181no—
174C2×Dic6D3211yescharacteristic
175C3×Q16—184no—
176C2×C3×C8—32no—
177C2×C3×C8—32no—
178C2×C3×C8—62no—
179#217—181no—
180#216—64no—
181(C2×D3)⋊C4—182no—
182(C2×D3)⋊C4—182no—
183C4⋊Dic3—91no—
184C4⋊Dic3—92no—
185C4⋊Dic3—181no—
186C4⋊Dic3—182no—
187C4⋊Dic3—361no—
188Dic3⋊C4—181no—
189Dic3⋊C4—182no—
190Dic3⋊C4—182no—
191C2×C3⋊C8D3212yes—
192Dic12—64no—
193C22×C32—22no—
194C22×C32—21no—
195C22×C32—41no—
196C22×C32C2×D1211yescharacteristic
197C22×C32D3×D412yes—
198C2×C3⋊D3—21no—
199C2×C3⋊D3—21no—
200C2×C3⋊D3Dic1212yescomplemented
201C2×C3×D3—34no—
202C2×C3×D3—62no—
203C2×C3×D3—62no—
204C2×C3×D3—62no—
205C2×C3×D3—121no—
206C2×C3×D3—121no—
207C32×C4—22no—
208C32×C4—22no—
209C32×C4—22no—
210C32×C4—42no—
211C32×C4—121no—
212C32×C4D3×D412yes—
213C32×C4D3×D412yes—
214C32×C4C2×#6712yes—
215C3⋊Dic3—21no—
216C3⋊Dic3—24no—
217C3⋊Dic3—121no—
218C3⋊Dic3—122no—
219C3⋊Dic3—122no—
220C3⋊Dic3—242no—
221C3⋊Dic3—242no—
222C3×Dic3—42no—
223C3×Dic3—42no—
224C3×Dic3—64no—
225C3×Dic3—64no—
226C3×Dic3—62no—
227C3×Dic3—124no—
228C3×Dic3—121no—
229C22×Q8—271no—
230C2×Q16—92no—
231C22⋊Q8—271no—
232#103—272no—
233C22⋊C8—91no—
234C33#45111yescharacteristic, complemented, Sylow
235C23×C3—91no—
236C22×D3—32no—
237C22×D3—61no—
238C3×Q8—62no—
239C3×Q8—62no—
240C3×Q8—94no—
241C3×Q8—122no—
242C3×Q8—182no—
243C2×C3×C4—22no—
244C2×C3×C4—21no—
245C2×C3×C4—41no—
246C2×C3×C4—62no—
247C2×C3×C4—62no—
248C2×C3×C4—91no—
249C2×C3×C4—182no—
250C2×C3×C4—182no—
251C2×C3×C4C2×D3212yes—
252C2×C3×C4C2×D3211yescharacteristic
253C2×Dic3—21no—
254C2×Dic3—32no—
255C2×Dic3—61no—
256C2×Dic3—62no—
257C2×Dic3—62no—
258C2×Dic3—122no—
259C2×Dic3—184no—
260C2×Dic3—182no—
261C2×Dic3—182no—
262C2×Dic3—181no—
263C2×Dic3—362no—
264C2×Dic3—362no—
265C4×D3—62no—
266C4×D3—62no—
267C4×D3—122no—
268C4×D3—188no—
269Dic6—22no—
270Dic6—34no—
271Dic6—34no—
272Dic6—64no—
273Dic6—62no—
274Dic6—62no—
275Dic6—122no—
276Dic6—184no—
277Dic6—188no—
278C3×C8—62no—
279C3×C8—62no—
280C3×C8—122no—
281C3⋊C8—22no—
282C2×C32—22no—
283C2×C32—22no—
284C2×C32—22no—
285C2×C32—21no—
286C2×C32—21no—
287C2×C32—21no—
288C2×C32—41no—
289C2×C32—41no—
290C2×C32—41no—
291C2×C32#88211yescharacteristic
292C2×C32C2×Dic1211yescharacteristic
293C2×C32#90911yescharacteristic
294C2×C32D3×Q1612yes—
295C2×C32#91812yes—
296C2×C32#93712yes—
297C3⋊D3—22no—
298C3×D3—64no—
299C3×D3—122no—
300C2×Q8—31no—
301C2×Q8—92no—
302C2×Q8—541no—
303C2×Q8—544no—
304C22×C4—91no—
305C22×C4—541no—
306Q16—184no—
307C2×C8—32no—
308C4⋊C4—271no—
309C4⋊C4—541no—
310C22⋊C4—541no—
311C22×C3—21no—
312C22×C3—22no—
313C22×C3—41no—
314C22×C3—92no—
315C22×C3—181no—
316C22×C3—181no—
317C22×C3D3×D1212yes—
318C22×C3#381111yescharacteristic
319C2×D3—34no—
320C2×D3—62no—
321C2×D3—62no—
322C2×D3—62no—
323C2×D3—121no—
324C2×D3—121no—
325C3×C4—22no—
326C3×C4—22no—
327C3×C4—22no—
328C3×C4—42no—
329C3×C4—122no—
330C3×C4—122no—
331C3×C4—184no—
332C3×C4—181no—
333C3×C4#380212yes—
334C3×C4D3×#6712yes—
335C3×C4D3×#6712yes—
336Dic3—41no—
337Dic3—64no—
338Dic3—62no—
339Dic3—64no—
340Dic3—124no—
341Dic3—121no—
342Dic3—362no—
343Dic3—362no—
344Dic3—361no—
345Dic3—722no—
346Dic3—722no—
347C32—21no—
348C32—22no—
349C32—41no—
350C32#473911yescharacteristic, complemented
351C32#518612yescomplemented
352C23—91no—
353Q8—62no—
354Q8—94no—
355Q8—182no—
356Q8—544no—
357C2×C4—61no—
358C2×C4—91no—
359C2×C4—182no—
360C2×C4—182no—
361C2×C4—544no—
362C2×C4—541no—
363C2×C4D3311yescharacteristic
364C8—62no—
365C2×C3—21no—
366C2×C3—22no—
367C2×C3—21no—
368C2×C3—22no—
369C2×C3—22no—
370C2×C3—21no—
371C2×C3—41no—
372C2×C3—41no—
373C2×C3—41no—
374C2×C3—182no—
375C2×C3D3×Dic1212yes—
376C2×C3#6380112yes—
377C2×C3#6392212yes—
378C2×C3#6394411yescharacteristic
379C2×C3#6394511yescharacteristic
380C2×C3#6398311yescharacteristic
381D3—64no—
382D3—122no—
383C22—92no—
384C22—181no—
385C22—181no—
386C22#8895511yescharacteristic
387C4—121no—
388C4—181no—
389C4—184no—
390C4—1081no—
391C4#8895312yes—
392C3—21no—
393C3—22no—
394C3—41no—
395C3#1059094912yescomplemented
396C3#1059165411yescharacteristic, complemented
397C2—182no—
398C2#1172284911yescharacteristic
399C2#1172289711yescharacteristic
400C2#1172295111yescharacteristic
subgroupG/H
centerC22#88955
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypocenterC33#451
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
3. C2×C3×#3719C2semidirect
4. #11722040C2semidirect
26. C2×#3719D3semidirect
86. C2×C3×C3⋊D3Q16semidirect
200. C2×C3⋊D3Dic12semidirect
234. C33#451semidirect
350. C32#4739semidirect
351. C32#5186semidirect
395. C3#10590949semidirect
396. C3#10591654semidirect

Maximal subgroups

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

subgroupindex
C2×#890402
#117234352
C2×C3×#37192
#117220402
C3×#635572
#105916543
#105909493