Finite Group Databasebeta

← #13149396

Subgroups — #13149396

Order 1728.

770 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 770 conjugacy classes of subgroups — too many to draw. Only its 145 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#11721755C211yesmaximal, characteristic
2#64098⋊C3C212yesmaximal
3#11721117C211yesmaximal, characteristic
4C4×#6196C211yesmaximal, characteristic
5#11721098C211yesmaximal, characteristic
6#11721098C211yesmaximal, characteristic
7#11721087C212yesmaximal
8#11721080C212yesmaximal
9#11721078C211yesmaximal, characteristic
10#11721078C211yesmaximal, characteristic
11#11721077C212yesmaximal
12#10594596—32nomaximal
13C2×#6229C2211yescharacteristic
14C2×#6196C2212yes—
15C2×#6196C2211yescharacteristic
16C4⋊C4×H(3,𝔽3)C2211yescharacteristic
17#88540C2211yescharacteristic
18#88540C2211yescharacteristic
19C4×#1084C2211yescharacteristic
20C4×#1084C2211yescharacteristic
21C4×#1084C2211yescharacteristic
22C4⋊((C3⋊Dic3)⋊C3)C2212yes—
23#3750⋊C3C2212yes—
24#3750⋊C3C2212yes—
25#3750⋊C3C2212yes—
26#3750⋊C3C2212yes—
27C4×(C3⋊Dic3)⋊C3C2212yes—
28#88449C2211yescharacteristic
29#88449C2211yescharacteristic
30#88449C2211yescharacteristic
31#88448C2212yes—
32#88448C2212yes—
33#88448C2212yes—
34#88447C2212yes—
35#88447C2212yes—
36#64098D312yes—
37C3×#889—32no—
38C3×#887—32no—
39C3×#887—61no—
40#63918—32no—
41C4×#627—32no—
42#63867—32no—
43#63867—32no—
44#63847—32no—
45#63845—32no—
46#63839—32no—
47#63839—32no—
48#63838—32no—
49#63837—32no—
50#63837—32no—
51#63837—32no—
52C2×C4×H(3,𝔽3)C2311yescharacteristic
53C2×C4×H(3,𝔽3)C2311yescharacteristic
54C2×C4×H(3,𝔽3)C2311yescharacteristic
55C2×#1084C2311yescharacteristic
56C2×#1084C2311yescharacteristic
57C2×#1084C2311yescharacteristic
58C2×#1084C2311yescharacteristic
59#6229—22no—
60C2×(C3⋊Dic3)⋊C3C2312yes—
61C2×(C3⋊Dic3)⋊C3C2312yes—
62C2×(C3⋊Dic3)⋊C3C2312yes—
63C2×(C3⋊Dic3)⋊C3C2312yes—
64#6196—24no—
65#6196Q814yescomplemented
66#5685—92no—
67#5567—91no—
68C2×C3×Dic6—32no—
69C2×C3×Dic6—61no—
70C2×#627—32no—
71C2×#627—32no—
72C2×#627—32no—
73C32×C4⋊C4—21no—
74C32×C4⋊C4C2×D312yes—
75C4⋊(C3⋊Dic3)C2×D312yes—
76#3750C2×D312yes—
77#3750C2×D312yes—
78#3750C2×D312yes—
79#3750C2×D312yes—
80C4×C3⋊Dic3C2×D312yes—
81C3×C4⋊Dic3—32no—
82C3×Dic3⋊C4—32no—
83C3×Dic3⋊C4—32no—
84C3×Dic3⋊C4—32no—
85C3×Dic3⋊C4—32no—
86C3×Dic3⋊C4—32no—
87C3×Dic3⋊C4—32no—
88C3×Dic3⋊C4—61no—
89C3×Dic3⋊C4—61no—
90C3×C4×Dic3—32no—
91C3×C4×Dic3—32no—
92C3×C4×Dic3—32no—
93C3×C4×Dic3—32no—
94C3×C4×Dic3—61no—
95C3×C4×Dic3—61no—
96C3×C4×Dic3—61no—
97#3724—32no—
98#3724—32no—
99#3724—32no—
100Dic3⋊Dic3—32no—
101Dic3⋊Dic3—32no—
102Dic3⋊Dic3—32no—
103Dic3⋊Dic3—32no—
104Dic3⋊Dic3—32no—
105Dic3⋊Dic3—32no—
106Dic32—32no—
107Dic32—32no—
108Dic32—32no—
109Dic32—32no—
110C22×H(3,𝔽3)C2411yescharacteristic
111C4×H(3,𝔽3)—21no—
112C4×H(3,𝔽3)—21no—
113C4×H(3,𝔽3)—21no—
114#1084—21no—
115#1084—21no—
116#1084C2×Q812yes—
117#1084#4112yes—
118(C3⋊Dic3)⋊C3—22no—
119(C3⋊Dic3)⋊C3—22no—
120(C3⋊Dic3)⋊C3—22no—
121(C3⋊Dic3)⋊C3C2×Q814yes—
122C3×#125—92no—
123C3×C4×Q8—91no—
124#944—92no—
125#890—92no—
126#889—32no—
127#889—92no—
128#889—92no—
129#889—92no—
130#889—92no—
131#889—92no—
132#888—92no—
133#888—92no—
134#888—92no—
135#887—32no—
136#887—61no—
137#887—92no—
138#887—92no—
139#887—92no—
140#887—92no—
141#870—91no—
142#870—91no—
143#869—91no—
144C4×Dic6—91no—
145C4×Dic6—92no—
146C4×Dic6—91no—
147C4×Dic6—91no—
148C4×Dic6—92no—
149C2×C32×C4—21no—
150C2×C32×C4—21no—
151C2×C32×C4—21no—
152C2×C32×C4C22×D312yes—
153C2×C32×C4C22×D312yes—
154C2×C32×C4C22×D312yes—
155C2×C3⋊Dic3C22×D312yes—
156C2×C3⋊Dic3C22×D312yes—
157C2×C3⋊Dic3C22×D312yes—
158C2×C3⋊Dic3C22×D312yes—
159C2×C3×Dic3—32no—
160C2×C3×Dic3—32no—
161C2×C3×Dic3—32no—
162C2×C3×Dic3—32no—
163C2×C3×Dic3—32no—
164C2×C3×Dic3—32no—
165C2×C3×Dic3—32no—
166C2×C3×Dic3—32no—
167C2×C3×Dic3—61no—
168C2×C3×Dic3—61no—
169C2×C3×Dic3—61no—
170C2×C3×Dic3—61no—
171C3×Dic6—64no—
172C3×Dic6—64no—
173#627—38no—
174#627—64no—
175#627—64no—
176#557—271noSylow
177C2×H(3,𝔽3)C22×Q811yescharacteristic
178C2×H(3,𝔽3)C2×#4111yescharacteristic
179C2×H(3,𝔽3)2−1+411yescharacteristic
180C2×C3×Q8—91no—
181C2×Dic6—32no—
182C2×Dic6—61no—
183C2×Dic6—91no—
184C2×Dic6—92no—
185C2×Dic6—92no—
186C2×Dic6—92no—
187C2×Dic6—92no—
188C3×C4⋊C4—32no—
189C3×C4⋊C4—61no—
190C3×C4⋊C4—92no—
191C3×C4⋊C4—92no—
192C3×C4⋊C4—92no—
193C3×C4⋊C4—92no—
194C3×C4⋊C4—92no—
195C3×C4⋊C4—91no—
196C3×C4⋊C4—91no—
197C3×C4⋊C4D3211yescharacteristic
198C3×C42—92no—
199C3×C42—91no—
200C3×C42—91no—
201C3×C42—91no—
202C4⋊Dic3—32no—
203C4⋊Dic3—92no—
204C4⋊Dic3—91no—
205C4⋊Dic3—91no—
206C4⋊Dic3—91no—
207C4⋊Dic3—92no—
208C4⋊Dic3—92no—
209C4⋊Dic3—92no—
210Dic3⋊C4—32no—
211Dic3⋊C4—32no—
212Dic3⋊C4—32no—
213Dic3⋊C4—32no—
214Dic3⋊C4—32no—
215Dic3⋊C4—32no—
216Dic3⋊C4—61no—
217Dic3⋊C4—61no—
218Dic3⋊C4—92no—
219Dic3⋊C4—92no—
220Dic3⋊C4—92no—
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—
231C4×Dic3—32no—
232C4×Dic3—32no—
233C4×Dic3—32no—
234C4×Dic3—32no—
235C4×Dic3—61no—
236C4×Dic3—61no—
237C4×Dic3—61no—
238C4×Dic3—92no—
239C4×Dic3—92no—
240C4×Dic3—92no—
241C4×Dic3—92no—
242C4×Dic3—92no—
243C4×Dic3—92no—
244C4×Dic3—92no—
245C22×C32—21no—
246C22×C32C23×D312yes—
247C32×C4—22no—
248C32×C4—22no—
249C32×C4—22no—
250C32×C4—22no—
251C32×C4—41no—
252C32×C4—41no—
253C3⋊Dic3—22no—
254C3⋊Dic3—22no—
255C3⋊Dic3—22no—
256C3⋊Dic3D3×Q814yes—
257C3×Dic3—34no—
258C3×Dic3—34no—
259C3×Dic3—34no—
260C3×Dic3—62no—
261C3×Dic3—62no—
262C3×Dic3—62no—
263C3×Dic3—62no—
264C3×Dic3—62no—
265C3×Dic3—62no—
266C3×Dic3—62no—
267C3×Dic3—62no—
268C3×Dic3—121no—
269C4⋊Q8—271no—
270C4⋊Q8—272no—
271#125—92no—
272#125—271no—
273#125—271no—
274#125—272no—
275C4×Q8—91no—
276C4×Q8—271no—
277C4×Q8—271no—
278C4×Q8—271no—
279C4×Q8—272no—
280H(3,𝔽3)#55711yescharacteristic, complemented, Sylow
281C3×Q8—182no—
282C2×C3×C4—32no—
283C2×C3×C4—32no—
284C2×C3×C4—32no—
285C2×C3×C4—61no—
286C2×C3×C4—61no—
287C2×C3×C4—61no—
288C2×C3×C4—92no—
289C2×C3×C4—91no—
290C2×C3×C4—91no—
291C2×C3×C4—92no—
292C2×C3×C4—92no—
293C2×C3×C4—92no—
294C2×C3×C4—91no—
295C2×C3×C4—91no—
296C2×C3×C4C2×D3211yescharacteristic
297C2×C3×C4C2×D3211yescharacteristic
298C2×C3×C4C2×D3211yescharacteristic
299C2×Dic3—32no—
300C2×Dic3—32no—
301C2×Dic3—32no—
302C2×Dic3—32no—
303C2×Dic3—32no—
304C2×Dic3—32no—
305C2×Dic3—32no—
306C2×Dic3—32no—
307C2×Dic3—61no—
308C2×Dic3—61no—
309C2×Dic3—61no—
310C2×Dic3—61no—
311C2×Dic3—92no—
312C2×Dic3—92no—
313C2×Dic3—92no—
314C2×Dic3—92no—
315Dic6—64no—
316Dic6—64no—
317Dic6—94no—
318Dic6—98no—
319Dic6—184no—
320Dic6—184no—
321Dic6—184no—
322C2×C32—21no—
323C2×C32—21no—
324C2×C32—21no—
325C2×C32C2×D3×Q812yes—
326C2×C32D3×#4112yes—
327C2×C32#101012yes—
328C2×Q8—91no—
329C2×Q8—271no—
330C2×Q8—272no—
331C4⋊C4—91no—
332C4⋊C4—91no—
333C4⋊C4—92no—
334C4⋊C4—92no—
335C4⋊C4—92no—
336C4⋊C4—92no—
337C4⋊C4—92no—
338C4⋊C4—271no—
339C4⋊C4—272no—
340C4⋊C4—271no—
341C4⋊C4—271no—
342C4⋊C4—272no—
343C4⋊C4—272no—
344C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
345C42—91no—
346C42—91no—
347C42—91no—
348C42—92no—
349C42—272no—
350C42—272no—
351C22×C3—32no—
352C22×C3—61no—
353C22×C3C22×D3211yescharacteristic
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—62no—
358C3×C4—62no—
359C3×C4—62no—
360C3×C4—62no—
361C3×C4—94no—
362C3×C4—92no—
363C3×C4—92no—
364C3×C4—121no—
365C3×C4—121no—
366C3×C4—182no—
367C3×C4—182no—
368C3×C4—182no—
369C3×C4—181no—
370C3×C4—181no—
371Dic3—34no—
372Dic3—34no—
373Dic3—34no—
374Dic3—62no—
375Dic3—62no—
376Dic3—62no—
377Dic3—62no—
378Dic3—62no—
379Dic3—62no—
380Dic3—62no—
381Dic3—62no—
382Dic3—94no—
383Dic3—121no—
384Dic3—182no—
385Dic3—182no—
386Dic3—182no—
387C32—21no—
388C32#568512yescomplemented
389Q8—182no—
390Q8—274no—
391Q8—544no—
392C2×C4—91no—
393C2×C4—92no—
394C2×C4—91no—
395C2×C4—91no—
396C2×C4—92no—
397C2×C4—91no—
398C2×C4—92no—
399C2×C4—92no—
400C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
401C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
402C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
403C2×C3—32no—
404C2×C3—32no—
405C2×C3—32no—
406C2×C3—61no—
407C2×C3—61no—
408C2×C3—61no—
409C2×C3#6430611yescharacteristic
410C2×C3#6430911yescharacteristic
411C2×C3D32×Q811yescharacteristic
412C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—92no—
417C4—94no—
418C4—92no—
419C4—181no—
420C4—182no—
421C4—181no—
422C4—182no—
423C4—182no—
424C3—32no—
425C3—61no—
426C3#1059459711yescharacteristic
427C2#1172249111yescharacteristic
428C2#1172250211yescharacteristic
429C2Q8×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, prime radical, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC4⋊C4×H(3,𝔽3)C22
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#557
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
65. #6196Q8semidirect
280. H(3,𝔽3)#557semidirect
388. C32#5685semidirect

Maximal subgroups

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

subgroupindex
#117217552
#64098⋊C32
#117211172
C4×#61962
#117210982
#117210872
#117210802
#117210782
#117210772
#105945963