Finite Group Databasebeta

← #13140093

Subgroups — #13140093

Order 1728.

783 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 783 conjugacy classes of subgroups — too many to draw. Only its 125 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×#88542C211yesmaximal, characteristic
2C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
3C2×#88451C212yesmaximal
4C2×#88448C212yesmaximal, complemented
5#63658⋊C3C211yesmaximal, characteristic, complemented
6#10591324—31nomaximal
7#10591317—31nomaximal
8C23×#272C2211yescharacteristic
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×#1084C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3C2212yes—
12C22×(C3⋊Dic3)⋊C3C2212yes—
13#88542—22no—
14C4⋊((C3⋊Dic3)⋊C3)—22no—
15#88451—24no—
16#88448—24no—
17C2×C4⋊(C3⋊Dic3)D311yescharacteristic, complemented
18C2×C3×(C2×D3)⋊C4—31no—
19C2×C3×(C2×D3)⋊C4—31no—
20C2×C3×(C2×D3)⋊C4—61no—
21C2×C3×C4⋊Dic3—31no—
22C2×#3724—32no—
23C2×Dic3⋊Dic3—32no—
24C2×(C2×D3)⋊Dic3—32no—
25C2×(C2×D3)⋊Dic3—32no—
26#63658D311yescharacteristic, complemented
27C3×C3⋊#95—31no—
28C23×H(3,𝔽3)C2311yescharacteristic
29C22×#272—21no—
30C22×#272—22no—
31C22×#272—21no—
32C22×#272—22no—
33C22×#272Q812yes—
34C2×C4×H(3,𝔽3)—22no—
35C2×C4×H(3,𝔽3)D412yes—
36C2×#1084—21no—
37C2×#1084—22no—
38C2×(C3⋊Dic3)⋊C3—22no—
39C2×(C3⋊Dic3)⋊C3—22no—
40C2×(C3⋊Dic3)⋊C3—22no—
41C2×(C3⋊Dic3)⋊C3—22no—
42C2×(C3⋊Dic3)⋊C3—22no—
43C2×(C3⋊Dic3)⋊C3D414yes—
44#4993—91no—
45#4949—91no—
46#4677—91no—
47C23×C3×D3—31no—
48C23×C3×D3—31no—
49C23×C3×D3—61no—
50C22×C32×C4—21no—
51C22×C32×C4C2×D311yescharacteristic
52C22×C32×C4C2×D311yescharacteristic
53C22×C3⋊Dic3C2×D312yes—
54C22×C3⋊Dic3C2×D312yes—
55C22×C3×Dic3—32no—
56C22×C3×Dic3—32no—
57C22×C3×Dic3—31no—
58C22×C3×Dic3—31no—
59C22×C3×Dic3—61no—
60C4⋊(C3⋊Dic3)—22no—
61C3×(C2×D3)⋊C4—64no—
62C3×(C2×D3)⋊C4—62no—
63C3×(C2×D3)⋊C4—62no—
64C3×C4⋊Dic3—62no—
65#3724—64no—
66Dic3⋊Dic3—64no—
67(C2×D3)⋊Dic3—64no—
68(C2×D3)⋊Dic3—64no—
69C22×H(3,𝔽3)C2×D412yes—
70C22×H(3,𝔽3)C2×D411yescharacteristic
71C22×H(3,𝔽3)C2×Q811yescharacteristic
72C22×H(3,𝔽3)#4112yes—
73C22×H(3,𝔽3)#4111yescharacteristic
74C2×#272—22no—
75C2×#272—24no—
76C2×#272—42no—
77C2×#272—41no—
78C2×#272—41no—
79C4×H(3,𝔽3)—22no—
80#1084—41no—
81(C3⋊Dic3)⋊C3—24no—
82(C3⋊Dic3)⋊C3—42no—
83C2×C3×C4⋊C4—91no—
84C2×C3×C22⋊C4—91no—
85C2×#217—92no—
86C2×(C2×D3)⋊C4—31no—
87C2×(C2×D3)⋊C4—31no—
88C2×(C2×D3)⋊C4—61no—
89C2×(C2×D3)⋊C4—92no—
90C2×(C2×D3)⋊C4—92no—
91C2×C4⋊Dic3—31no—
92C2×C4⋊Dic3—91no—
93C2×C4⋊Dic3—92no—
94C2×Dic3⋊C4—92no—
95C2×Dic3⋊C4—92no—
96C3×#95—91no—
97C3⋊#95—31no—
98C3⋊#95—91no—
99C23×C32—21no—
100C23×C32C22×D311yescharacteristic
101C23×C32C22×D311yescharacteristic
102C22×C3×D3—32no—
103C22×C3×D3—32no—
104C22×C3×D3—62no—
105C22×C3×D3—62no—
106C22×C3×D3—62no—
107C22×C3×D3—62no—
108C22×C3×D3—61no—
109C22×C3×D3—62no—
110C22×C3×D3—62no—
111C22×C3×D3—61no—
112C22×C3×D3—62no—
113C22×C3×D3—61no—
114C22×C3×D3—61no—
115C22×C3×D3—122no—
116C22×C3×D3—122no—
117C2×C32×C4—22no—
118C2×C32×C4—22no—
119C2×C32×C4—22no—
120C2×C32×C4—42no—
121C2×C32×C4D1212yes—
122C2×C32×C4#6712yes—
123C2×C3⋊Dic3—22no—
124C2×C3⋊Dic3—22no—
125C2×C3⋊Dic3—22no—
126C2×C3⋊Dic3—22no—
127C2×C3⋊Dic3—22no—
128C2×C3⋊Dic3D1214yes—
129C2×C3×Dic3—34no—
130C2×C3×Dic3—62no—
131C2×C3×Dic3—62no—
132C2×C3×Dic3—62no—
133C2×C3×Dic3—62no—
134C2×C3×Dic3—62no—
135C2×C3×Dic3—62no—
136C2×C3×Dic3—62no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3—62no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—122no—
141#399—271noSylow
142C2×H(3,𝔽3)C22⋊Q812yes—
143C2×H(3,𝔽3)C22⋊Q811yescharacteristic
144C2×H(3,𝔽3)#12312yes—
145C2×H(3,𝔽3)#12311yescharacteristic
146C2×H(3,𝔽3)#12711yescharacteristic
147#272—42no—
148C24×C3—91no—
149C23×D3—31no—
150C23×D3—31no—
151C23×D3—61no—
152C22×C3×C4—31no—
153C22×C3×C4—31no—
154C22×C3×C4—61no—
155C22×C3×C4—92no—
156C22×C3×C4—92no—
157C22×C3×C4—91no—
158C22×C3×C4D3211yescharacteristic
159C22×Dic3—32no—
160C22×Dic3—31no—
161C22×Dic3—31no—
162C22×Dic3—32no—
163C22×Dic3—61no—
164C22×Dic3—92no—
165C22×Dic3—92no—
166C3×C4⋊C4—182no—
167C3×C22⋊C4—182no—
168#217—184no—
169(C2×D3)⋊C4—62no—
170(C2×D3)⋊C4—62no—
171(C2×D3)⋊C4—64no—
172(C2×D3)⋊C4—184no—
173(C2×D3)⋊C4—184no—
174C4⋊Dic3—62no—
175C4⋊Dic3—182no—
176C4⋊Dic3—184no—
177Dic3⋊C4—184no—
178Dic3⋊C4—184no—
179C22×C32—21no—
180C22×C32—21no—
181C22×C32—22no—
182C22×C32—22no—
183C22×C32—21no—
184C22×C32C2×D1212yes—
185C22×C32C2×D1211yescharacteristic
186C22×C32#23512yes—
187C22×C32D3×D412yes—
188C22×C32#23711yescharacteristic
189C22×C32#23712yes—
190C22×C32D3×Q811yescharacteristic
191C22×C32D3×Q811yescharacteristic
192C22×C32#23911yescharacteristic
193C22×C32C2×#6711yescharacteristic
194C2×C3×D3—64no—
195C2×C3×D3—64no—
196C2×C3×D3—62no—
197C2×C3×D3—64no—
198C2×C3×D3—62no—
199C2×C3×D3—124no—
200C2×C3×D3—124no—
201C2×C3×D3—122no—
202C2×C3×D3—122no—
203C2×C3×D3—121no—
204C2×C3×D3—122no—
205C2×C3×D3—121no—
206C2×C3×D3—122no—
207C2×C3×D3—121no—
208C2×C3×D3—121no—
209C32×C4—22no—
210C32×C4—22no—
211C32×C4—42no—
212C3⋊Dic3—24no—
213C3⋊Dic3—42no—
214C3×Dic3—64no—
215C3×Dic3—122no—
216C3×Dic3—122no—
217C3×Dic3—121no—
218C3×Dic3—121no—
219C2×C4⋊C4—91no—
220C2×C4⋊C4—272no—
221C2×C22⋊C4—91no—
222C2×C22⋊C4—272no—
223#95—91no—
224H(3,𝔽3)#39911yescharacteristic, complemented, Sylow
225C23×C3—31no—
226C23×C3—31no—
227C23×C3—61no—
228C23×C3—92no—
229C23×C3—181no—
230C23×C3—182no—
231C23×C3—182no—
232C23×C3—181no—
233C23×C3C2×D3211yescharacteristic
234C22×D3—32no—
235C22×D3—32no—
236C22×D3—62no—
237C22×D3—61no—
238C22×D3—61no—
239C22×D3—62no—
240C22×D3—62no—
241C22×D3—61no—
242C22×D3—61no—
243C22×D3—62no—
244C22×D3—62no—
245C22×D3—62no—
246C22×D3—62no—
247C22×D3—122no—
248C22×D3—122no—
249C2×C3×C4—22no—
250C2×C3×C4—32no—
251C2×C3×C4—32no—
252C2×C3×C4—62no—
253C2×C3×C4—62no—
254C2×C3×C4—62no—
255C2×C3×C4—94no—
256C2×C3×C4—122no—
257C2×C3×C4—182no—
258C2×C3×C4—182no—
259C2×C3×C4—182no—
260C2×C3×C4—182no—
261C2×C3×C4—182no—
262C2×C3×C4—182no—
263C2×C3×C4—181no—
264C2×C3×C4#62612yes—
265C2×Dic3—34no—
266C2×Dic3—62no—
267C2×Dic3—62no—
268C2×Dic3—62no—
269C2×Dic3—62no—
270C2×Dic3—61no—
271C2×Dic3—61no—
272C2×Dic3—62no—
273C2×Dic3—62no—
274C2×Dic3—62no—
275C2×Dic3—62no—
276C2×Dic3—94no—
277C2×Dic3—122no—
278C2×Dic3—182no—
279C2×Dic3—182no—
280C2×Dic3—182no—
281C2×Dic3—182no—
282C2×Dic3—182no—
283C2×C32—21no—
284C2×C32—22no—
285C2×C32—22no—
286C2×C32—21no—
287C2×C32—21no—
288C2×C32#87411yescharacteristic
289C2×C32#88612yes—
290C2×C32#88611yescharacteristic
291C2×C32#89412yes—
292C2×C32#89612yes—
293C2×C32#89711yescharacteristic
294C2×C32#89712yes—
295C2×C32#92911yescharacteristic
296C2×C32#94011yescharacteristic
297C2×C32#94611yescharacteristic
298C3×D3—124no—
299C3×D3—122no—
300C3×D3—122no—
301C24—91no—
302C22×C4—92no—
303C22×C4—91no—
304C22×C4—92no—
305C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
306C4⋊C4—182no—
307C4⋊C4—544no—
308C22⋊C4—182no—
309C22⋊C4—544no—
310C22×C3—31no—
311C22×C3—31no—
312C22×C3—31no—
313C22×C3—31no—
314C22×C3—32no—
315C22×C3—32no—
316C22×C3—32no—
317C22×C3—32no—
318C22×C3—31no—
319C22×C3—31no—
320C22×C3—61no—
321C22×C3—61no—
322C22×C3—62no—
323C22×C3—62no—
324C22×C3—61no—
325C22×C3—182no—
326C22×C3—184no—
327C22×C3—362no—
328C22×C3—361no—
329C22×C3—361no—
330C22×C3#379611yescharacteristic
331C22×C3#379711yescharacteristic
332C22×C3D3×D1212yes—
333C22×C3#380412yes—
334C22×C3C2×#62611yescharacteristic
335C2×D3—62no—
336C2×D3—62no—
337C2×D3—64no—
338C2×D3—64no—
339C2×D3—64no—
340C2×D3—124no—
341C2×D3—121no—
342C2×D3—121no—
343C2×D3—122no—
344C2×D3—122no—
345C2×D3—121no—
346C2×D3—121no—
347C2×D3—122no—
348C2×D3—124no—
349C2×D3—122no—
350C3×C4—22no—
351C3×C4—62no—
352C3×C4—62no—
353C3×C4—122no—
354C3×C4—184no—
355C3×C4—362no—
356C3×C4—361no—
357Dic3—64no—
358Dic3—121no—
359Dic3—121no—
360Dic3—122no—
361Dic3—122no—
362Dic3—184no—
363Dic3—362no—
364C32—21no—
365C32#467711yescharacteristic, complemented
366C32#499311yescharacteristic, complemented
367C23—92no—
368C23—181no—
369C23—182no—
370C23—182no—
371C23—181no—
372C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
373C2×C4—22no—
374C2×C4—94no—
375C2×C4—182no—
376C2×C4—182no—
377C2×C4—182no—
378C2×C4—182no—
379C2×C4—182no—
380C2×C4—181no—
381C2×C4—182no—
382C2×C4#620012yes—
383C2×C3—31no—
384C2×C3—31no—
385C2×C3—31no—
386C2×C3—31no—
387C2×C3—32no—
388C2×C3—32no—
389C2×C3—32no—
390C2×C3—32no—
391C2×C3—31no—
392C2×C3—31no—
393C2×C3—61no—
394C2×C3—61no—
395C2×C3—62no—
396C2×C3—62no—
397C2×C3—61no—
398C2×C3—362no—
399C2×C3#6387111yescharacteristic
400C2×C3#6389012yes—
401C2×C3#6389512yes—
402C2×C3#6391111yescharacteristic
403C2×C3#6396611yescharacteristic
404D3—122no—
405D3—122no—
406D3—124no—
407C22—182no—
408C22—184no—
409C22—361no—
410C22—362no—
411C22—361no—
412C22#8864911yescharacteristic
413C22#8865111yescharacteristic
414C22#8865412yes—
415C22#8867112yes—
416C22C2×#620011yescharacteristic
417C4—22no—
418C4—184no—
419C4—361no—
420C4—362no—
421C3—31no—
422C3—31no—
423C3—61no—
424C3#1059135011yescharacteristic
425C2—362no—
426C2#1172109411yescharacteristic
427C2#1172113012yes—
428C2#1172115012yes—
429C2#1172115811yescharacteristic
430C2#1172130111yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC23×H(3,𝔽3)C23
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC23×#272C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#399
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊((C3⋊Dic3)⋊C3)C2semidirect
4. C2×#88448C2semidirect
5. #63658⋊C3C2semidirect
17. C2×C4⋊(C3⋊Dic3)D3semidirect
26. #63658D3semidirect
224. H(3,𝔽3)#399semidirect
365. C32#4677semidirect
366. C32#4993semidirect

Maximal subgroups

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

subgroupindex
C2×#885422
C2×C4⋊((C3⋊Dic3)⋊C3)2
C2×#884512
C2×#884482
#63658⋊C32
#105913243
#105913173