Finite Group Databasebeta

← #13150669

Subgroups — #13150669

Order 1728.

797 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721826C211yesmaximal, characteristic
2#64135⋊C3C211yesmaximal, characteristic
3C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic
4#11721299C214yesmaximal
5#11721117C212yesmaximal, complemented
6#11721116C212yesmaximal, complemented
7#11721095C212yesmaximal, complemented
8#11721093C212yesmaximal, complemented
9#10595052—31nomaximal
10#10595051—31nomaximal
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C22×(C3⋊Dic3)⋊C3C2212yes—
13C2×#6196C2214yes—
14#88552C2212yes—
15C4⋊#1084C2211yescharacteristic
16C4⋊#1084C2211yescharacteristic
17C4×#1084C2212yes—
18#3757⋊C3C2212yes—
19C4⋊((C3⋊Dic3)⋊C3)C2212yes—
20C4⋊((C3⋊Dic3)⋊C3)C2212yes—
21C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
22C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
23C4×(C3⋊Dic3)⋊C3C2212yes—
24#88449C2214yes—
25#88448C2214yes—
26#88448C2214yes—
27#64135D311yescharacteristic
28C2×C4⋊(C3⋊Dic3)D311yescharacteristic
29C3×#926—31no—
30C3×#926—31no—
31C3×#926—31no—
32C3×#926—61no—
33C2×C3×C4⋊Dic3—31no—
34#63982—34no—
35#63964—34no—
36#63919—32no—
37#63918—32no—
38#63918—32no—
39#63918—32no—
40#63872—32no—
41#63869—32no—
42#63869—32no—
43#63869—32no—
44C23×H(3,𝔽3)C2311yescharacteristic
45C2×C4×H(3,𝔽3)Q814yescomplemented
46C2×C4×H(3,𝔽3)C2311yescharacteristic
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48C2×#1084C2312yes—
49C2×#1084C2312yes—
50C2×(C3⋊Dic3)⋊C3—42no—
51C2×(C3⋊Dic3)⋊C3C2312yes—
52C2×(C3⋊Dic3)⋊C3C2314yes—
53C2×(C3⋊Dic3)⋊C3C2312yes—
54#6196—44no—
55#5519—91no—
56#5519—91no—
57#5500—91no—
58C22×C32×C4—21no—
59C22×C32×C4C2×D311yescharacteristic
60C22×C32×C4C2×D311yescharacteristic
61C22×C3⋊Dic3C2×D312yes—
62C22×C3×Dic3—32no—
63C2×#627—34no—
64C2×#627—34no—
65#3757C2×D312yes—
66C4⋊(C3⋊Dic3)C2×D311yescharacteristic
67C4⋊(C3⋊Dic3)C2×D312yes—
68C4⋊(C3⋊Dic3)C2×D312yes—
69C4⋊(C3⋊Dic3)C2×D311yescharacteristic
70C4×C3⋊Dic3C2×D312yes—
71C3×#217—32no—
72C3×#217—32no—
73C3×#217—32no—
74C3×#217—62no—
75C3×C4⋊Dic3—32no—
76C3×C4⋊Dic3—32no—
77C3×C4⋊Dic3—31no—
78C3×C4⋊Dic3—31no—
79C3×C4⋊Dic3—31no—
80C3×C4⋊Dic3—31no—
81C3×C4⋊Dic3—31no—
82C3×C4⋊Dic3—31no—
83C3×C4⋊Dic3—61no—
84C3×C4⋊Dic3—61no—
85C3×C4×Dic3—32no—
86C3×C4×Dic3—32no—
87C3×C4×Dic3—32no—
88C3×C4×Dic3—62no—
89#3724—34no—
90#3724—34no—
91Dic3⋊Dic3—34no—
92Dic3⋊Dic3—34no—
93Dic3⋊Dic3—34no—
94Dic3⋊Dic3—34no—
95C22×H(3,𝔽3)—21no—
96C22×H(3,𝔽3)—21no—
97C22×H(3,𝔽3)C2×Q812yes—
98C22×H(3,𝔽3)C2411yescharacteristic
99C4×H(3,𝔽3)C2×Q812yes—
100C4×H(3,𝔽3)C2×Q812yes—
101#1084—22no—
102#1084—22no—
103(C3⋊Dic3)⋊C3—22no—
104(C3⋊Dic3)⋊C3—24no—
105(C3⋊Dic3)⋊C3—22no—
106C3×#117—91no—
107C3×#117—91no—
108C2×C3×C4⋊C4—91no—
109#926—31no—
110#926—31no—
111#926—31no—
112#926—61no—
113#926—91no—
114C2×C4⋊Dic3—31no—
115C2×C4⋊Dic3—91no—
116#924—94no—
117#890—92no—
118#890—92no—
119#890—92no—
120#890—92no—
121#888—92no—
122#888—92no—
123#888—92no—
124#888—92no—
125#878—94no—
126#878—94no—
127#870—92no—
128#870—92no—
129#869—92no—
130#869—92no—
131C23×C32—21no—
132C23×C32C22×D311yescharacteristic
133C23×C32C22×D311yescharacteristic
134C2×C32×C4—24no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4Dic614yescomplemented
138C2×C32×C4Dic614yescomplemented
139C2×C32×C4C22×D311yescharacteristic
140C2×C32×C4C22×D311yescharacteristic
141C2×C32×C4C22×D311yescharacteristic
142C2×C32×C4C22×D311yescharacteristic
143C2×C3⋊Dic3—42no—
144C2×C3⋊Dic3C22×D312yes—
145C2×C3⋊Dic3C22×D314yes—
146C2×C3⋊Dic3C22×D312yes—
147C2×C3×Dic3—32no—
148C2×C3×Dic3—34no—
149C2×C3×Dic3—32no—
150C2×C3×Dic3—32no—
151C2×C3×Dic3—32no—
152C2×C3×Dic3—32no—
153C2×C3×Dic3—32no—
154C2×C3×Dic3—62no—
155C2×C3×Dic3—62no—
156C2×C3×Dic3—122no—
157#627—124no—
158#627—124no—
159#544—271noSylow
160C2×H(3,𝔽3)—22no—
161C2×H(3,𝔽3)C22×Q811yescharacteristic
162C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
163C2×H(3,𝔽3)2−1+411yescharacteristic
164C22×C3×C4—31no—
165C22×C3×C4—31no—
166C22×C3×C4—61no—
167C22×C3×C4—92no—
168C22×C3×C4D3211yescharacteristic
169C22×Dic3—32no—
170C22×Dic3—92no—
171C2×Dic6—94no—
172C2×Dic6—94no—
173C2×Dic6—94no—
174C3×C4⋊C4—91no—
175C3×C4⋊C4—92no—
176C3×C4⋊C4—92no—
177C3×C4⋊C4—91no—
178C3×C4⋊C4—91no—
179C3×C4⋊C4—91no—
180C3×C22⋊C4—92no—
181C3×C22⋊C4—92no—
182C3×C42—92no—
183C3×C42—92no—
184#217—32no—
185#217—32no—
186#217—32no—
187#217—62no—
188#217—92no—
189C4⋊Dic3—31no—
190C4⋊Dic3—31no—
191C4⋊Dic3—31no—
192C4⋊Dic3—31no—
193C4⋊Dic3—31no—
194C4⋊Dic3—32no—
195C4⋊Dic3—31no—
196C4⋊Dic3—32no—
197C4⋊Dic3—61no—
198C4⋊Dic3—61no—
199C4⋊Dic3—92no—
200C4⋊Dic3—91no—
201C4⋊Dic3—92no—
202C4⋊Dic3—91no—
203C4⋊Dic3—94no—
204C4⋊Dic3—94no—
205C4⋊Dic3—94no—
206Dic3⋊C4—94no—
207Dic3⋊C4—94no—
208Dic3⋊C4—94no—
209Dic3⋊C4—94no—
210Dic3⋊C4—94no—
211Dic3⋊C4—94no—
212C4×Dic3—32no—
213C4×Dic3—32no—
214C4×Dic3—32no—
215C4×Dic3—62no—
216C4×Dic3—92no—
217C22×C32—21no—
218C22×C32—21no—
219C22×C32—21no—
220C22×C32—21no—
221C22×C32—21no—
222C22×C32—22no—
223C22×C32—41no—
224C22×C32—41no—
225C22×C32C2×Dic612yes—
226C22×C32C2×Dic612yes—
227C22×C32C23×D311yescharacteristic
228C22×C32C23×D311yescharacteristic
229C32×C4—22no—
230C32×C4—22no—
231C32×C4C2×Dic612yes—
232C32×C4C2×Dic612yes—
233C32×C4C2×Dic612yes—
234C32×C4C2×Dic612yes—
235C3⋊Dic3—24no—
236C3⋊Dic3—22no—
237C3⋊Dic3—22no—
238C3×Dic3—62no—
239C3×Dic3—62no—
240C3×Dic3—62no—
241C3×Dic3—62no—
242C3×Dic3—64no—
243C3×Dic3—62no—
244C3×Dic3—62no—
245C3×Dic3—122no—
246C3×Dic3—122no—
247C4⋊Q8—272no—
248C4⋊Q8—272no—
249#125—272no—
250#125—272no—
251C22⋊Q8—274no—
252#117—91no—
253#117—91no—
254C2×C4⋊C4—91no—
255H(3,𝔽3)#54411yescharacteristic, complemented, Sylow
256C23×C3—31no—
257C23×C3—31no—
258C23×C3—61no—
259C23×C3C2×D3211yescharacteristic
260C2×C3×C4—34no—
261C2×C3×C4—34no—
262C2×C3×C4—31no—
263C2×C3×C4—31no—
264C2×C3×C4—31no—
265C2×C3×C4—31no—
266C2×C3×C4—64no—
267C2×C3×C4—61no—
268C2×C3×C4—61no—
269C2×C3×C4—92no—
270C2×C3×C4—94no—
271C2×C3×C4—92no—
272C2×C3×C4—92no—
273C2×C3×C4—92no—
274C2×C3×C4—362no—
275C2×C3×C4#62714yes—
276C2×C3×C4C2×D3211yescharacteristic
277C2×C3×C4C2×D3211yescharacteristic
278C2×Dic3—32no—
279C2×Dic3—32no—
280C2×Dic3—32no—
281C2×Dic3—32no—
282C2×Dic3—32no—
283C2×Dic3—32no—
284C2×Dic3—34no—
285C2×Dic3—62no—
286C2×Dic3—62no—
287C2×Dic3—92no—
288C2×Dic3—92no—
289C2×Dic3—94no—
290C2×Dic3—122no—
291C2×Dic3—362no—
292Dic6—364no—
293Dic6—364no—
294Dic6—364no—
295C2×C32—22no—
296C2×C32—22no—
297C2×C32—21no—
298C2×C32—21no—
299C2×C32—21no—
300C2×C32—42no—
301C2×C32C22×Dic611yescharacteristic
302C2×C32C22×Dic611yescharacteristic
303C2×C32#100411yescharacteristic
304C2×C32#100711yescharacteristic
305C2×C32#100911yescharacteristic
306C2×C32#101011yescharacteristic
307C2×Q8—274no—
308C22×C4—92no—
309C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
310C4⋊C4—92no—
311C4⋊C4—92no—
312C4⋊C4—91no—
313C4⋊C4—91no—
314C4⋊C4—91no—
315C4⋊C4—91no—
316C4⋊C4—274no—
317C4⋊C4—274no—
318C4⋊C4—274no—
319C22⋊C4—92no—
320C22⋊C4—92no—
321C42—92no—
322C42—92no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—31no—
326C22×C3—31no—
327C22×C3—32no—
328C22×C3—32no—
329C22×C3—61no—
330C22×C3—61no—
331C22×C3—61no—
332C22×C3—61no—
333C22×C3—61no—
334C22×C3—62no—
335C22×C3—121no—
336C22×C3—121no—
337C22×C3C2×#62712yes—
338C22×C3C22×D3211yescharacteristic
339C3×C4—32no—
340C3×C4—32no—
341C3×C4—32no—
342C3×C4—32no—
343C3×C4—62no—
344C3×C4—62no—
345C3×C4—182no—
346C3×C4—184no—
347C3×C4—182no—
348C3×C4—182no—
349C3×C4—182no—
350C3×C4C2×#62712yes—
351C3×C4C2×#62712yes—
352Dic3—62no—
353Dic3—62no—
354Dic3—62no—
355Dic3—62no—
356Dic3—62no—
357Dic3—62no—
358Dic3—64no—
359Dic3—122no—
360Dic3—122no—
361Dic3—182no—
362Dic3—184no—
363Dic3—182no—
364C32—21no—
365C32#550011yescharacteristic, complemented
366C32#551911yescharacteristic, complemented
367C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
368Q8—1084no—
369C2×C4—92no—
370C2×C4—92no—
371C2×C4—92no—
372C2×C4—92no—
373C2×C4—94no—
374C2×C4—362no—
375C2×C4#619614yescomplemented
376C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
377C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
378C2×C3—22no—
379C2×C3—31no—
380C2×C3—31no—
381C2×C3—31no—
382C2×C3—31no—
383C2×C3—31no—
384C2×C3—31no—
385C2×C3—62no—
386C2×C3—62no—
387C2×C3—61no—
388C2×C3—61no—
389C2×C3—61no—
390C2×C3—122no—
391C2×C3#6429711yescharacteristic
392C2×C3#6430711yescharacteristic
393C2×C3C22×#62711yescharacteristic
394C22—21no—
395C22—21no—
396C22C2×#619612yes—
397C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
398C4—182no—
399C4—182no—
400C4—184no—
401C4—182no—
402C4—182no—
403C4C2×#619612yes—
404C4C2×#619612yes—
405C3—31no—
406C3—31no—
407C3—61no—
408C3#1059505111yescharacteristic
409C2—22no—
410C2#1172248211yescharacteristic
411C2#1172249211yescharacteristic
412C2C22×#619611yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC23×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#544
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
5. #11721117C2semidirect
6. #11721116C2semidirect
7. #11721095C2semidirect
8. #11721093C2semidirect
45. C2×C4×H(3,𝔽3)Q8semidirect
137. C2×C32×C4Dic6semidirect
138. C2×C32×C4Dic6semidirect
255. H(3,𝔽3)#544semidirect
365. C32#5500semidirect
366. C32#5519semidirect
375. C2×C4#6196semidirect

Maximal subgroups

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

subgroupindex
#117218262
#64135⋊C32
C2×C4⋊((C3⋊Dic3)⋊C3)2
#117212992
#117211172
#117211162
#117210952
#117210932
#105950523
#105950513