Finite Group Databasebeta

← #13149582

Subgroups — #13149582

Order 1728.

795 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 795 conjugacy classes of subgroups — too many to draw. Only its 143 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
1#11721765C211yesmaximal, characteristic
2#64098⋊C3C212yesmaximal, complemented
3#11721150C212yesmaximal
4#11721147C211yesmaximal, characteristic
5#11721116C212yesmaximal, complemented
6#11721096C211yesmaximal, characteristic
7#11721085C212yesmaximal, complemented
8#11721080C212yesmaximal, complemented
9#11721076C212yesmaximal
10#10594389—32nomaximal
11C2×C4×#272C2211yescharacteristic
12C2×#6229C2211yescharacteristic
13C2×#6199C2211yescharacteristic
14C2×#6199C2211yescharacteristic
15C2×#6196C2212yes—
16C2×#6196C2211yescharacteristic
17C4⋊C4×H(3,𝔽3)C2211yescharacteristic
18#88542C2212yes—
19C4⋊#1084C2212yes—
20C4⋊((C3⋊Dic3)⋊C3)C2212yes—
21#3750⋊C3C2212yes—
22#3750⋊C3C2212yes—
23#3750⋊C3C2212yes—
24#3750⋊C3C2212yes—
25C4×(C3⋊Dic3)⋊C3C2212yes—
26#88451C2212yes—
27#88449C2211yescharacteristic
28#88448C2212yes—
29#88448C2212yes—
30#88448C2212yes—
31#88447C2212yes—
32#64098D312yescomplemented
33C3×#897—32no—
34C3×#897—61no—
35C3×#889—32no—
36#63919—32no—
37#63918—32no—
38#63899—32no—
39#63893—32no—
40#63878—32no—
41#63877—32no—
42D3×Dic3⋊C4—32no—
43#63851—32no—
44#63847—32no—
45#63846—32no—
46#63839—32no—
47#63839—32no—
48C22×#272C2311yescharacteristic
49C2×C4×H(3,𝔽3)C2311yescharacteristic
50C2×C4×H(3,𝔽3)C2312yes—
51C2×#1084C2312yes—
52C2×#1084C2311yescharacteristic
53C4×#272—41no—
54#6229—41no—
55C2×(C3⋊Dic3)⋊C3C2312yes—
56C2×(C3⋊Dic3)⋊C3C2312yes—
57C2×(C3⋊Dic3)⋊C3C2312yes—
58C2×(C3⋊Dic3)⋊C3C2312yes—
59#6199—41no—
60#6199Q814yescomplemented
61#6196—41no—
62#6196—42no—
63#5689—92no—
64#5623—91no—
65C2×C3×C4×D3—32no—
66C2×C3×C4×D3—61no—
67C2×C3×Dic6—32no—
68C2×C3×Dic6—61no—
69C2×#627—32no—
70C2×#627—32no—
71C2×#627—32no—
72C2×D3×Dic3—32no—
73C2×D3×Dic3—32no—
74C32×C4⋊C4—21no—
75C32×C4⋊C4C2×D312yes—
76C4⋊(C3⋊Dic3)C2×D312yes—
77#3750C2×D312yes—
78#3750C2×D312yes—
79#3750C2×D312yes—
80#3750C2×D312yes—
81C4×C3⋊Dic3C2×D312yes—
82C3×(C2×D3)⋊C4—32no—
83C3×(C2×D3)⋊C4—32no—
84C3×(C2×D3)⋊C4—62no—
85C3×C4⋊Dic3—32no—
86C3×C4⋊Dic3—32no—
87C3×C4⋊Dic3—32no—
88C3×C4⋊Dic3—62no—
89C3×Dic3⋊C4—32no—
90C3×Dic3⋊C4—32no—
91C3×Dic3⋊C4—32no—
92C3×Dic3⋊C4—32no—
93C3×C4×Dic3—32no—
94#3724—32no—
95#3724—32no—
96#3724—32no—
97Dic3⋊Dic3—32no—
98Dic3⋊Dic3—32no—
99Dic3⋊Dic3—32no—
100Dic3⋊Dic3—32no—
101(C2×D3)⋊Dic3—32no—
102(C2×D3)⋊Dic3—32no—
103Dic32—32no—
104Dic32—32no—
105C22×H(3,𝔽3)C2411yescharacteristic
106C2×#272—21no—
107C2×#272—21no—
108C2×#272C2×Q812yes—
109C4×H(3,𝔽3)—21no—
110C4×H(3,𝔽3)—22no—
111#1084—22no—
112#1084—21no—
113(C3⋊Dic3)⋊C3—22no—
114(C3⋊Dic3)⋊C3—22no—
115(C3⋊Dic3)⋊C3—22no—
116(C3⋊Dic3)⋊C3C2×Q814yes—
117C3×#125—92no—
118C3×C22⋊Q8—91no—
119#944—92no—
120#924—91no—
121C2×Dic3⋊C4—91no—
122#897—32no—
123#897—61no—
124#897—92no—
125#896—92no—
126#896—92no—
127#892—92no—
128D3×C4⋊C4—92no—
129#890—92no—
130#889—32no—
131#889—92no—
132#889—92no—
133#889—92no—
134#888—92no—
135#888—92no—
136#888—92no—
137#888—92no—
138#878—92no—
139#877—92no—
140#872—92no—
141#869—92no—
142C22×C3×D3—32no—
143C22×C3×D3—61no—
144C2×C32×C4—21no—
145C2×C32×C4—22no—
146C2×C32×C4C22×D312yes—
147C2×C32×C4C22×D312yes—
148C2×C32×C4C22×D312yes—
149C2×C3⋊Dic3C22×D312yes—
150C2×C3⋊Dic3C22×D312yes—
151C2×C3⋊Dic3C22×D312yes—
152C2×C3⋊Dic3C22×D312yes—
153C2×C3×Dic3—32no—
154C2×C3×Dic3—32no—
155C2×C3×Dic3—32no—
156C2×C3×Dic3—32no—
157C2×C3×Dic3—32no—
158C2×C3×Dic3—32no—
159C2×C3×Dic3—32no—
160C2×C3×Dic3—61no—
161C2×C3×Dic3—62no—
162C3×C4×D3—122no—
163C3×C4×D3—122no—
164C3×Dic6—122no—
165C3×Dic6—122no—
166#627—122no—
167#627—122no—
168#627—122no—
169D3×Dic3—38no—
170D3×Dic3—122no—
171#544—271noSylow
172C2×H(3,𝔽3)C22×Q811yescharacteristic
173C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
174C2×H(3,𝔽3)2−1+411yescharacteristic
175#272—22no—
176C2×C3×Q8—91no—
177C22×C3×C4—91no—
178C22×Dic3—91no—
179C22×Dic3—91no—
180C2×C4×D3—32no—
181C2×C4×D3—61no—
182C2×C4×D3—92no—
183C2×C4×D3—92no—
184C2×Dic6—32no—
185C2×Dic6—61no—
186C2×Dic6—92no—
187C2×Dic6—91no—
188C2×Dic6—92no—
189C2×Dic6—92no—
190C2×Dic6—92no—
191C3×C4⋊C4—32no—
192C3×C4⋊C4—61no—
193C3×C4⋊C4—92no—
194C3×C4⋊C4—92no—
195C3×C4⋊C4—92no—
196C3×C4⋊C4—92no—
197C3×C4⋊C4—92no—
198C3×C4⋊C4—92no—
199C3×C4⋊C4D3211yescharacteristic
200C3×C22⋊C4—92no—
201C3×C42—92no—
202#217—92no—
203(C2×D3)⋊C4—32no—
204(C2×D3)⋊C4—32no—
205(C2×D3)⋊C4—62no—
206(C2×D3)⋊C4—92no—
207(C2×D3)⋊C4—92no—
208C4⋊Dic3—32no—
209C4⋊Dic3—32no—
210C4⋊Dic3—32no—
211C4⋊Dic3—62no—
212C4⋊Dic3—92no—
213C4⋊Dic3—91no—
214C4⋊Dic3—92no—
215C4⋊Dic3—92no—
216C4⋊Dic3—92no—
217Dic3⋊C4—32no—
218Dic3⋊C4—32no—
219Dic3⋊C4—32no—
220Dic3⋊C4—32no—
221Dic3⋊C4—92no—
222Dic3⋊C4—92no—
223Dic3⋊C4—92no—
224Dic3⋊C4—92no—
225Dic3⋊C4—92no—
226Dic3⋊C4—92no—
227Dic3⋊C4—92no—
228Dic3⋊C4—92no—
229Dic3⋊C4—92no—
230Dic3⋊C4—92no—
231Dic3⋊C4—92no—
232C4×Dic3—32no—
233C4×Dic3—92no—
234C4×Dic3—92no—
235C4×Dic3—92no—
236C4×Dic3—92no—
237C22×C32—21no—
238C22×C32C23×D312yes—
239C2×C3×D3—34no—
240C2×C3×D3—62no—
241C2×C3×D3—62no—
242C2×C3×D3—62no—
243C2×C3×D3—121no—
244C2×C3×D3—121no—
245C32×C4—22no—
246C32×C4—22no—
247C32×C4—22no—
248C32×C4—22no—
249C32×C4—42no—
250C3⋊Dic3—22no—
251C3⋊Dic3—22no—
252C3⋊Dic3—22no—
253C3⋊Dic3D3×Q814yes—
254C3×Dic3—34no—
255C3×Dic3—62no—
256C3×Dic3—62no—
257C3×Dic3—62no—
258C3×Dic3—62no—
259C3×Dic3—62no—
260C3×Dic3—62no—
261C3×Dic3—122no—
262C3×Dic3—121no—
263C4⋊Q8—272no—
264C4⋊Q8—272no—
265#125—92no—
266#125—272no—
267C22⋊Q8—91no—
268C22⋊Q8—272no—
269C22⋊Q8—271no—
270#117—272no—
271C2×C4⋊C4—271no—
272H(3,𝔽3)#54411yescharacteristic, complemented, Sylow
273C23×C3—91no—
274C22×D3—32no—
275C22×D3—61no—
276C3×Q8—361no—
277C2×C3×C4—32no—
278C2×C3×C4—32no—
279C2×C3×C4—32no—
280C2×C3×C4—61no—
281C2×C3×C4—62no—
282C2×C3×C4—92no—
283C2×C3×C4—92no—
284C2×C3×C4—92no—
285C2×C3×C4—92no—
286C2×C3×C4—92no—
287C2×C3×C4—91no—
288C2×C3×C4—361no—
289C2×C3×C4C2×D3212yes—
290C2×C3×C4C2×D3211yescharacteristic
291C2×Dic3—32no—
292C2×Dic3—32no—
293C2×Dic3—32no—
294C2×Dic3—32no—
295C2×Dic3—32no—
296C2×Dic3—32no—
297C2×Dic3—32no—
298C2×Dic3—61no—
299C2×Dic3—62no—
300C2×Dic3—92no—
301C2×Dic3—92no—
302C2×Dic3—92no—
303C2×Dic3—92no—
304C2×Dic3—94no—
305C2×Dic3—361no—
306C4×D3—98no—
307C4×D3—122no—
308C4×D3—122no—
309C4×D3—362no—
310Dic6—122no—
311Dic6—122no—
312Dic6—361no—
313Dic6—362no—
314Dic6—362no—
315Dic6—362no—
316Dic6—362no—
317C2×C32—21no—
318C2×C32—21no—
319C2×C32—21no—
320C2×C32C2×D3×Q812yes—
321C2×C32#100912yes—
322C2×C32#101012yes—
323C3×D3—64no—
324C3×D3—122no—
325C2×Q8—91no—
326C2×Q8—271no—
327C2×Q8—272no—
328C22×C4—91no—
329C22×C4—271no—
330C22×C4—271no—
331C4⋊C4—92no—
332C4⋊C4—92no—
333C4⋊C4—92no—
334C4⋊C4—92no—
335C4⋊C4—92no—
336C4⋊C4—92no—
337C4⋊C4—272no—
338C4⋊C4—272no—
339C4⋊C4—271no—
340C4⋊C4—272no—
341C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
342C22⋊C4—92no—
343C22⋊C4—272no—
344C42—92no—
345C42—272no—
346C22×C3—32no—
347C22×C3—61no—
348C22×C3—92no—
349C22×C3—181no—
350C22×C3—181no—
351C22×C3C22×D3211yescharacteristic
352C2×D3—34no—
353C2×D3—62no—
354C2×D3—62no—
355C2×D3—62no—
356C2×D3—121no—
357C2×D3—121no—
358C3×C4—21no—
359C3×C4—22no—
360C3×C4—62no—
361C3×C4—62no—
362C3×C4—62no—
363C3×C4—62no—
364C3×C4—94no—
365C3×C4—122no—
366C3×C4—182no—
367C3×C4—182no—
368C3×C4—182no—
369C3×C4—181no—
370C3×C4—182no—
371Dic3—34no—
372Dic3—62no—
373Dic3—62no—
374Dic3—62no—
375Dic3—62no—
376Dic3—62no—
377Dic3—62no—
378Dic3—94no—
379Dic3—122no—
380Dic3—121no—
381Dic3—182no—
382Dic3—182no—
383Dic3—182no—
384C32—21no—
385C32#568912yescomplemented
386C23—91no—
387Q8—361no—
388Q8—1081no—
389Q8—1082no—
390C2×C4—92no—
391C2×C4—92no—
392C2×C4—92no—
393C2×C4—92no—
394C2×C4—92no—
395C2×C4—91no—
396C2×C4—274no—
397C2×C4—361no—
398C2×C4—1081no—
399C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
400C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
401C2×C3—32no—
402C2×C3—32no—
403C2×C3—32no—
404C2×C3—61no—
405C2×C3—61no—
406C2×C3—61no—
407C2×C3—182no—
408C2×C3#6430811yescharacteristic
409C2×C3#6430911yescharacteristic
410C2×C3D32×Q811yescharacteristic
411D3—64no—
412D3—122no—
413C22—92no—
414C22—181no—
415C22—181no—
416C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
417C4—22no—
418C4—21no—
419C4—94no—
420C4—182no—
421C4—181no—
422C4—182no—
423C4—182no—
424C4—182no—
425C3—32no—
426C3—61no—
427C3#1059440111yescharacteristic
428C2—182no—
429C2#1172249311yescharacteristic
430C2#1172250211yescharacteristic
431C2Q8×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC4⋊C4×H(3,𝔽3)C22
prime radicalC22×#272C23
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#544
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #64098⋊C3C2semidirect
5. #11721116C2semidirect
7. #11721085C2semidirect
8. #11721080C2semidirect
32. #64098D3semidirect
60. #6199Q8semidirect
272. H(3,𝔽3)#544semidirect
385. C32#5689semidirect

Maximal subgroups

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

subgroupindex
#117217652
#64098⋊C32
#117211502
#117211472
#117211162
#117210962
#117210852
#117210802
#117210762
#105943893