Finite Group Databasebeta

← #13151310

Subgroups — #13151310

Order 1728.

892 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 892 conjugacy classes of subgroups — too many to draw. Only its 181 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
1D4×#1084C211yesmaximal, characteristic
2D4×(C3⋊Dic3)⋊C3C212yesmaximal
3#11721309C212yesmaximal, complemented
4C2×#88449C212yesmaximal, complemented
5#11721299C214yesmaximal, complemented
6#11721117C211yesmaximal, characteristic, complemented
7C4×#6196C211yesmaximal, characteristic, complemented
8#11721093C212yesmaximal, complemented
9#10595272—32nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C22×#1084C2212yes—
12C22×(C3⋊Dic3)⋊C3C2214yes—
13C2×#6196C2211yescharacteristic
14C2×#6196C2212yes—
15#88552C2212yes—
16C4⋊#1084C2211yescharacteristic
17C4×#1084C2211yescharacteristic
18#3757⋊C3C2214yes—
19C4⋊((C3⋊Dic3)⋊C3)C2212yes—
20C4×(C3⋊Dic3)⋊C3C2212yes—
21#88449C2214yes—
22#88449C2211yescharacteristic
23#88449C2212yes—
24#88448C2212yes—
25#88448C2214yes—
26D4×C3⋊Dic3D312yes—
27C3×D4×Dic3—32no—
28C3×D4×Dic3—32no—
29C3×D4×Dic3—61no—
30#63982—34no—
31C2×Dic3⋊Dic3—34no—
32#63964—34no—
33#63964—34no—
34#63918—32no—
35C4×#627—32no—
36#63872—32no—
37#63869—32no—
38C23×H(3,𝔽3)C2312yes—
39D4×H(3,𝔽3)Q814yescomplemented
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084—24no—
42C2×#1084C2311yescharacteristic
43C2×#1084C2312yes—
44C2×#1084C2311yescharacteristic
45C2×(C3⋊Dic3)⋊C3—44no—
46C2×(C3⋊Dic3)⋊C3C2312yes—
47C2×(C3⋊Dic3)⋊C3C2314yes—
48C2×(C3⋊Dic3)⋊C3C2312yes—
49#6196—22no—
50#6196—42no—
51#5543—91no—
52#5539—92no—
53C2×C32×D4—21no—
54C2×C32×D4C2×D312yes—
55C22×C3⋊Dic3C2×D314yes—
56C22×C3×Dic3—34no—
57C22×C3×Dic3—34no—
58C22×C3×Dic3—62no—
59C2×#627—32no—
60C2×#627—34no—
61#3757C2×D314yes—
62C4⋊(C3⋊Dic3)C2×D312yes—
63C4×C3⋊Dic3C2×D312yes—
64C3×#217—34no—
65C3×#217—34no—
66C3×#217—62no—
67C3×C4⋊Dic3—32no—
68C3×C4⋊Dic3—32no—
69C3×C4⋊Dic3—61no—
70C3×C4×Dic3—32no—
71C3×C4×Dic3—32no—
72C3×C4×Dic3—61no—
73#3724—34no—
74#3724—32no—
75Dic3⋊Dic3—34no—
76Dic3⋊Dic3—34no—
77Dic3⋊Dic3—32no—
78Dic3⋊Dic3—34no—
79Dic3⋊Dic3—32no—
80Dic3⋊Dic3—34no—
81C22×H(3,𝔽3)—22no—
82C22×H(3,𝔽3)—22no—
83C22×H(3,𝔽3)C2×Q814yes—
84C22×H(3,𝔽3)C2411yescharacteristic
85C4×H(3,𝔽3)C2×Q812yes—
86#1084—22no—
87#1084—21no—
88#1084#4112yes—
89(C3⋊Dic3)⋊C3—22no—
90(C3⋊Dic3)⋊C3—24no—
91(C3⋊Dic3)⋊C3—22no—
92C3×C4×D4—92no—
93C3×C4×D4—91no—
94D4×Dic3—32no—
95D4×Dic3—32no—
96D4×Dic3—61no—
97D4×Dic3—92no—
98C2×C4⋊Dic3—92no—
99#924—92no—
100#924—94no—
101C2×Dic3⋊C4—94no—
102#890—92no—
103#890—92no—
104#888—92no—
105#878—94no—
106#878—94no—
107#878—94no—
108#870—92no—
109#869—91no—
110C4×Dic6—91no—
111C4×Dic6—92no—
112C23×C32—22no—
113C23×C32C22×D314yes—
114C32×D4—24no—
115C32×D4Dic618yescomplemented
116C2×C32×C4—21no—
117C2×C32×C4C22×D312yes—
118C2×C3⋊Dic3—44no—
119C2×C3⋊Dic3C22×D312yes—
120C2×C3⋊Dic3C22×D312yes—
121C2×C3⋊Dic3C22×D314yes—
122C2×C3×Dic3—32no—
123C2×C3×Dic3—32no—
124C2×C3×Dic3—34no—
125C2×C3×Dic3—32no—
126C2×C3×Dic3—34no—
127C2×C3×Dic3—32no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—62no—
131C2×C3×Dic3—68no—
132C2×C3×Dic3—124no—
133C2×C3×Dic3—124no—
134#627—64no—
135#627—124no—
136#554—271noSylow
137C2×H(3,𝔽3)—24no—
138C2×H(3,𝔽3)C22×Q811yescharacteristic
139C2×H(3,𝔽3)C2×#4111yescharacteristic
140C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
141C2×C3×D4—32no—
142C2×C3×D4—61no—
143C2×C3×D4D3211yescharacteristic
144C22×C3×C4—92no—
145C22×C3×C4—94no—
146C22×Dic3—34no—
147C22×Dic3—34no—
148C22×Dic3—62no—
149C22×Dic3—94no—
150C2×Dic6—91no—
151C2×Dic6—92no—
152C2×Dic6—92no—
153C2×Dic6—94no—
154C3×C4⋊C4—92no—
155C3×C4⋊C4—91no—
156C3×C22⋊C4—94no—
157C3×C22⋊C4—92no—
158C3×C42—92no—
159C3×C42—91no—
160#217—34no—
161#217—34no—
162#217—62no—
163#217—94no—
164C4⋊Dic3—32no—
165C4⋊Dic3—32no—
166C4⋊Dic3—61no—
167C4⋊Dic3—92no—
168C4⋊Dic3—91no—
169C4⋊Dic3—92no—
170C4⋊Dic3—94no—
171C4⋊Dic3—92no—
172C4⋊Dic3—94no—
173Dic3⋊C4—92no—
174Dic3⋊C4—94no—
175Dic3⋊C4—92no—
176Dic3⋊C4—92no—
177Dic3⋊C4—94no—
178Dic3⋊C4—94no—
179Dic3⋊C4—94no—
180Dic3⋊C4—94no—
181C4×Dic3—32no—
182C4×Dic3—32no—
183C4×Dic3—61no—
184C4×Dic3—92no—
185C22×C32—24no—
186C22×C32—24no—
187C22×C32—21no—
188C22×C32—24no—
189C22×C32—42no—
190C22×C32—42no—
191C22×C32C2×Dic618yes—
192C22×C32C23×D312yes—
193C32×C4—22no—
194C32×C4C2×Dic614yes—
195C3⋊Dic3—24no—
196C3⋊Dic3—22no—
197C3⋊Dic3—22no—
198C3×Dic3—34no—
199C3×Dic3—62no—
200C3×Dic3—64no—
201C3×Dic3—62no—
202C3×Dic3—62no—
203C3×Dic3—62no—
204C3×Dic3—64no—
205C3×Dic3—121no—
206C3×Dic3—122no—
207C4⋊Q8—271no—
208#125—272no—
209C22⋊Q8—274no—
210C22⋊Q8—272no—
211C4×Q8—271no—
212C4×D4—91no—
213C4×D4—92no—
214C2×C4⋊C4—272no—
215H(3,𝔽3)#55411yescharacteristic, complemented, Sylow
216C23×C3—34no—
217C23×C3—62no—
218C23×C3C2×D3212yes—
219C3×D4—38no—
220C3×D4—64no—
221C3×D4#62714yes—
222C2×C3×C4—32no—
223C2×C3×C4—61no—
224C2×C3×C4—91no—
225C2×C3×C4—92no—
226C2×C3×C4—92no—
227C2×C3×C4—94no—
228C2×C3×C4—92no—
229C2×C3×C4—91no—
230C2×C3×C4—184no—
231C2×C3×C4—364no—
232C2×C3×C4C2×D3211yescharacteristic
233C2×Dic3—32no—
234C2×Dic3—32no—
235C2×Dic3—34no—
236C2×Dic3—32no—
237C2×Dic3—32no—
238C2×Dic3—34no—
239C2×Dic3—61no—
240C2×Dic3—61no—
241C2×Dic3—62no—
242C2×Dic3—68no—
243C2×Dic3—92no—
244C2×Dic3—92no—
245C2×Dic3—94no—
246C2×Dic3—124no—
247C2×Dic3—124no—
248C2×Dic3—364no—
249Dic6—184no—
250Dic6—182no—
251Dic6—362no—
252Dic6—364no—
253C2×C32—28no—
254C2×C32—21no—
255C2×C32—21no—
256C2×C32—21no—
257C2×C32—44no—
258C2×C32C22×Dic612yes—
259C2×C32#100412yes—
260C2×C32D3×#4112yes—
261C2×Q8—271no—
262C2×Q8—272no—
263C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
264C22×C4—92no—
265C22×C4—94no—
266C4⋊C4—91no—
267C4⋊C4—92no—
268C4⋊C4—271no—
269C4⋊C4—274no—
270C4⋊C4—272no—
271C4⋊C4—272no—
272C4⋊C4—274no—
273C22⋊C4—92no—
274C22⋊C4—94no—
275C42—92no—
276C42—91no—
277C22×C3—22no—
278C22×C3—22no—
279C22×C3—32no—
280C22×C3—38no—
281C22×C3—64no—
282C22×C3—64no—
283C22×C3—61no—
284C22×C3—64no—
285C22×C3—122no—
286C22×C3—122no—
287C22×C3C2×#62714yes—
288C22×C3C22×D3211yescharacteristic
289C3×C4—34no—
290C3×C4—62no—
291C3×C4—92no—
292C3×C4—182no—
293C3×C4—184no—
294C3×C4—182no—
295C3×C4—182no—
296C3×C4—181no—
297C3×C4C2×#62712yes—
298Dic3—34no—
299Dic3—62no—
300Dic3—62no—
301Dic3—64no—
302Dic3—62no—
303Dic3—64no—
304Dic3—62no—
305Dic3—122no—
306Dic3—121no—
307Dic3—182no—
308Dic3—184no—
309Dic3—182no—
310C32—21no—
311C32#553912yescomplemented
312C23C2×((C3⋊D3)⋊C3)⋊C212yes—
313Q8—542no—
314Q8—1082no—
315D4#619614yescomplemented
316C2×C4—91no—
317C2×C4—92no—
318C2×C4—91no—
319C2×C4—92no—
320C2×C4—92no—
321C2×C4—94no—
322C2×C4—184no—
323C2×C4—364no—
324C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
325C2×C3—24no—
326C2×C3—32no—
327C2×C3—32no—
328C2×C3—32no—
329C2×C3—68no—
330C2×C3—61no—
331C2×C3—61no—
332C2×C3—61no—
333C2×C3—124no—
334C2×C3#6431211yescharacteristic
335C2×C3#6431311yescharacteristic
336C2×C3C22×#62711yescharacteristic
337C22—22no—
338C22—22no—
339C22C2×#619614yes—
340C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
341C4—92no—
342C4—182no—
343C4—182no—
344C4—182no—
345C4—184no—
346C4—181no—
347C4C2×#619612yes—
348C3—32no—
349C3—61no—
350C3#1059527411yescharacteristic
351C2—24no—
352C2#1172250511yescharacteristic
353C2#1172250611yescharacteristic
354C2C22×#619611yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
Fitting, prime radicalC2×D4×H(3,𝔽3)C22
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#554
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
3. #11721309C2semidirect
4. C2×#88449C2semidirect
5. #11721299C2semidirect
6. #11721117C2semidirect
7. C4×#6196C2semidirect
8. #11721093C2semidirect
39. D4×H(3,𝔽3)Q8semidirect
115. C32×D4Dic6semidirect
215. H(3,𝔽3)#554semidirect
311. C32#5539semidirect
315. D4#6196semidirect

Maximal subgroups

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

subgroupindex
D4×#10842
D4×(C3⋊Dic3)⋊C32
#117213092
C2×#884492
#117212992
#117211172
C4×#61962
#117210932
#105952723