Finite Group Databasebeta

← #13150639

Subgroups — #13150639

Order 1728.

1021 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 1021 conjugacy classes of subgroups — too many to draw. Only its 179 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×#3828⋊C3C211yesmaximal, characteristic
2C22×#634⋊C3C211yesmaximal, characteristic, complemented
3#11721826C211yesmaximal, characteristic, complemented
4#11721306C214yesmaximal, complemented
5#11721113C212yesmaximal, complemented
6#11721106C214yesmaximal, complemented
7#11721099C212yesmaximal, complemented
8#10594866—31nomaximal
9#10594856—31nomaximal
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3C2212yes—
12C22×(C3⋊Dic3)⋊C3C2212yes—
13#3828⋊C3—24no—
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15C2×C4×(C3⋊D3)⋊C3C2212yes—
16C2×#634⋊C3—24no—
17C2×#634⋊C3C2212yes—
18C2×#634⋊C3C2212yes—
19C2×#634⋊C3C2211yescharacteristic
20#88552C2212yes—
21C4⋊#1084C2212yes—
22C4×#1084C2212yes—
23#88450C2218yes—
24#88449C2218yes—
25C2×#3828D311yescharacteristic
26C22×#634D311yescharacteristic, complemented
27C2×C3×#235—31no—
28C22×C3×Dic6—31no—
29C3×#926—31no—
30C3×#926—31no—
31C3×#926—61no—
32#63966—34no—
33#63962—34no—
34#63871—34no—
35#63866—32no—
36#63866—32no—
37#63864—32no—
38#63860—32no—
39#63857—34no—
40C23×H(3,𝔽3)C2311yescharacteristic
41C22×(C3⋊D3)⋊C3C2312yes—
42C2×C4×H(3,𝔽3)D414yes—
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45C2×#1084C2314yes—
46#638⋊C3—24no—
47C2×(C3⋊Dic3)⋊C3—24no—
48C2×(C3⋊Dic3)⋊C3C2314yes—
49C2×(C3⋊Dic3)⋊C3C2312yes—
50C4⋊((C3⋊D3)⋊C3)—22no—
51C4×(C3⋊D3)⋊C3—24no—
52#634⋊C3—24no—
53#634⋊C3—24no—
54#634⋊C3—22no—
55#5825—91no—
56#5816—91no—
57#5526—91no—
58C22×C32×C4—21no—
59C22×C32×C4C2×D311yescharacteristic
60C22×C32×C4C2×D311yescharacteristic
61C2×#638C2×D312yes—
62C22×C3⋊Dic3C2×D312yes—
63#3828—24no—
64C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
65C2×C4×C3⋊D3C2×D312yes—
66C2×#634—24no—
67C2×#634C2×D312yes—
68C2×#634C2×D312yes—
69C2×#634C2×D311yescharacteristic
70C2×C3×#67—32no—
71C22×C3×Dic3—32no—
72C3×#235—64no—
73C2×C3×D12—31no—
74C2×C3×C4×D3—32no—
75C2×C3×Dic6—32no—
76C2×C3×Dic6—32no—
77C2×C3×Dic6—31no—
78C2×C3×Dic6—64no—
79C3×#217—32no—
80C3×#217—32no—
81C3×#217—62no—
82C3×C4⋊Dic3—32no—
83C3×C4⋊Dic3—32no—
84C3×C4⋊Dic3—62no—
85C3×C4×Dic3—32no—
86C3×C4×Dic3—32no—
87C3×C4×Dic3—62no—
88Dic3⋊Dic3—38no—
89Dic3⋊Dic3—38no—
90#3722—38no—
91(C2×D3)⋊Dic3—38no—
92C22×H(3,𝔽3)—22no—
93C22×H(3,𝔽3)C2×D412yes—
94C22×H(3,𝔽3)C2411yescharacteristic
95C2×(C3⋊D3)⋊C3—22no—
96C2×(C3⋊D3)⋊C3—24no—
97C4×H(3,𝔽3)C2×D412yes—
98C4×H(3,𝔽3)C2×D412yes—
99#1084—24no—
100(C3⋊Dic3)⋊C3—24no—
101(C3⋊Dic3)⋊C3—22no—
102C2×C3×#41—91no—
103C22×C3×Q8—91no—
104C2×#235—31no—
105C2×#235—91no—
106C22×Dic6—31no—
107C22×Dic6—91no—
108C3×#117—91no—
109#947—92no—
110#946—94no—
111#944—92no—
112#944—92no—
113#936—92no—
114#935—94no—
115#929—94no—
116#926—31no—
117#926—31no—
118#926—61no—
119#924—94no—
120#897—94no—
121#886—94no—
122#875—92no—
123#869—92no—
124C23×C32—21no—
125C23×C32C22×D311yescharacteristic
126C23×C32C22×D311yescharacteristic
127C22×C3⋊D3C22×D312yes—
128C22×C3×D3—32no—
129C2×C32×C4—24no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4#6714yes—
133C2×C32×C4#6714yes—
134C2×C32×C4C22×D311yescharacteristic
135C2×C32×C4C22×D311yescharacteristic
136C2×C32×C4C22×D311yescharacteristic
137C2×C32×C4C22×D311yescharacteristic
138#638—24no—
139C2×C3⋊Dic3—24no—
140C2×C3⋊Dic3C22×D314yes—
141C2×C3⋊Dic3C22×D312yes—
142C4⋊(C3⋊D3)—22no—
143C4×C3⋊D3—24no—
144#634—24no—
145#634—24no—
146#634—22no—
147C3×#67—64no—
148C2×C3×Dic3—34no—
149C2×C3×Dic3—32no—
150C2×C3×Dic3—34no—
151C2×C3×Dic3—34no—
152C2×C3×Dic3—64no—
153C2×C3×Dic3—64no—
154C3×D12—62no—
155C3×C4×D3—64no—
156C3×Dic6—64no—
157C3×Dic6—64no—
158C3×Dic6—62no—
159#536—271noSylow
160C2×H(3,𝔽3)—22no—
161C2×H(3,𝔽3)C22×D411yescharacteristic
162C2×H(3,𝔽3)2−1+412yes—
163(C3⋊D3)⋊C3—42no—
164C3×#41—184no—
165C2×C3×Q8—91no—
166C2×C3×Q8—92no—
167C2×C3×Q8—92no—
168C2×C3×Q8—184no—
169C2×C3×D4—91no—
170C2×C3×D4—92no—
171C22×C3×C4—31no—
172C22×C3×C4—31no—
173C22×C3×C4—61no—
174C22×C3×C4—92no—
175C22×C3×C4—92no—
176C22×C3×C4D3211yescharacteristic
177C2×#67—32no—
178C2×#67—92no—
179C22×Dic3—32no—
180C22×Dic3—92no—
181#235—64no—
182#235—184no—
183C2×D12—31no—
184C2×D12—91no—
185C2×C4×D3—32no—
186C2×C4×D3—92no—
187C2×Dic6—31no—
188C2×Dic6—32no—
189C2×Dic6—32no—
190C2×Dic6—64no—
191C2×Dic6—92no—
192C2×Dic6—92no—
193C2×Dic6—91no—
194C2×Dic6—184no—
195C3×C4⋊C4—92no—
196C3×C22⋊C4—92no—
197C3×C42—92no—
198#217—32no—
199#217—32no—
200#217—62no—
201#217—98no—
202(C2×D3)⋊C4—98no—
203(C2×D3)⋊C4—98no—
204C4⋊Dic3—32no—
205C4⋊Dic3—32no—
206C4⋊Dic3—62no—
207C4⋊Dic3—98no—
208Dic3⋊C4—98no—
209Dic3⋊C4—98no—
210C4×Dic3—32no—
211C4×Dic3—32no—
212C4×Dic3—62no—
213C22×C32—22no—
214C22×C32—22no—
215C22×C32—21no—
216C22×C32—22no—
217C22×C32—42no—
218C22×C32C2×#6712yes—
219C22×C32C2×#6712yes—
220C22×C32C23×D311yescharacteristic
221C22×C32C23×D311yescharacteristic
222C2×C3⋊D3—24no—
223C2×C3⋊D3—22no—
224C2×C3×D3—64no—
225C2×C3×D3—62no—
226C32×C4—22no—
227C32×C4—22no—
228C32×C4C2×#6712yes—
229C32×C4C2×#6712yes—
230C32×C4C2×#6712yes—
231C32×C4C2×#6712yes—
232C3⋊Dic3—24no—
233C3⋊Dic3—22no—
234C3×Dic3—64no—
235C3×Dic3—64no—
236C3×Dic3—64no—
237C3×Dic3—62no—
238C3×Dic3—124no—
239C2×#41—91no—
240C22×Q8—91no—
241C4⋊Q8—272no—
242#124—272no—
243#123—274no—
244C22⋊Q8—274no—
245#117—91no—
246H(3,𝔽3)#53611yescharacteristic, complemented, Sylow
247C23×C3—31no—
248C23×C3—31no—
249C23×C3—61no—
250C23×C3—92no—
251C23×C3C2×D3211yescharacteristic
252C22×D3—32no—
253C22×D3—92no—
254C3×Q8—184no—
255C3×Q8—184no—
256C3×Q8—182no—
257C3×D4—182no—
258C3×D4—184no—
259C2×C3×C4—34no—
260C2×C3×C4—34no—
261C2×C3×C4—31no—
262C2×C3×C4—31no—
263C2×C3×C4—31no—
264C2×C3×C4—31no—
265C2×C3×C4—64no—
266C2×C3×C4—61no—
267C2×C3×C4—61no—
268C2×C3×C4—92no—
269C2×C3×C4—94no—
270C2×C3×C4—94no—
271C2×C3×C4—184no—
272C2×C3×C4—184no—
273C2×C3×C4#62514yes—
274C2×C3×C4C2×D3211yescharacteristic
275C2×C3×C4C2×D3211yescharacteristic
276#67—64no—
277#67—184no—
278C2×Dic3—34no—
279C2×Dic3—34no—
280C2×Dic3—32no—
281C2×Dic3—34no—
282C2×Dic3—64no—
283C2×Dic3—64no—
284C2×Dic3—92no—
285C2×Dic3—94no—
286C2×Dic3—184no—
287D12—62no—
288D12—182no—
289C4×D3—64no—
290C4×D3—184no—
291Dic6—64no—
292Dic6—64no—
293Dic6—62no—
294Dic6—184no—
295Dic6—184no—
296Dic6—182no—
297C2×C32—22no—
298C2×C32—22no—
299C2×C32—22no—
300C2×C32—21no—
301C2×C32—42no—
302C2×C32#100712yes—
303C2×C32#101012yes—
304C2×C32C22×#6711yescharacteristic
305C2×C32C22×#6711yescharacteristic
306C3⋊D3—42no—
307C3×D3—122no—
308#41—184no—
309C2×Q8—92no—
310C2×Q8—92no—
311C2×Q8—91no—
312C2×Q8—184no—
313C2×D4—92no—
314C2×D4—91no—
315C22×C4—92no—
316C22×C4—92no—
317C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
318C4⋊C4—92no—
319C4⋊C4—278no—
320C22⋊C4—92no—
321C22⋊C4—278no—
322C42—92no—
323C22×C3—22no—
324C22×C3—31no—
325C22×C3—31no—
326C22×C3—32no—
327C22×C3—32no—
328C22×C3—62no—
329C22×C3—62no—
330C22×C3—61no—
331C22×C3—62no—
332C22×C3—122no—
333C22×C3—182no—
334C22×C3—184no—
335C22×C3C2×#62512yes—
336C22×C3C22×D3211yescharacteristic
337C2×D3—64no—
338C2×D3—62no—
339C2×D3—182no—
340C2×D3—184no—
341C3×C4—32no—
342C3×C4—32no—
343C3×C4—32no—
344C3×C4—32no—
345C3×C4—62no—
346C3×C4—62no—
347C3×C4—182no—
348C3×C4—184no—
349C3×C4—184no—
350C3×C4C2×#62512yes—
351C3×C4C2×#62512yes—
352Dic3—64no—
353Dic3—64no—
354Dic3—64no—
355Dic3—62no—
356Dic3—124no—
357Dic3—184no—
358Dic3—182no—
359C32—21no—
360C32#581611yescharacteristic, complemented
361C32#582511yescharacteristic, complemented
362C23—92no—
363C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
364Q8—184no—
365Q8—184no—
366Q8—182no—
367D4—184no—
368D4—182no—
369C2×C4—94no—
370C2×C4—94no—
371C2×C4—92no—
372C2×C4—184no—
373C2×C4—184no—
374C2×C4#619814yes—
375C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
376C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
377C2×C3—22no—
378C2×C3—31no—
379C2×C3—31no—
380C2×C3—32no—
381C2×C3—32no—
382C2×C3—62no—
383C2×C3—62no—
384C2×C3—61no—
385C2×C3—62no—
386C2×C3—122no—
387C2×C3—362no—
388C2×C3#6429712yes—
389C2×C3C22×#62511yescharacteristic
390D3—122no—
391D3—362no—
392C22—22no—
393C22—182no—
394C22—184no—
395C22C2×#619812yes—
396C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
397C4—182no—
398C4—184no—
399C4—184no—
400C4C2×#619812yes—
401C4C2×#619812yes—
402C3—31no—
403C3—31no—
404C3—61no—
405C3#1059485511yescharacteristic
406C2—22no—
407C2—362no—
408C2#1172248212yes—
409C2C22×#619811yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC2×#3828⋊C3C2
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#536
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
2. C22×#634⋊C3C2semidirect
3. #11721826C2semidirect
4. #11721306C2semidirect
5. #11721113C2semidirect
6. #11721106C2semidirect
7. #11721099C2semidirect
26. C22×#634D3semidirect
246. H(3,𝔽3)#536semidirect
360. C32#5816semidirect
361. C32#5825semidirect

Maximal subgroups

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

subgroupindex
C2×#3828⋊C32
C22×#634⋊C32
#117218262
#117213062
#117211132
#117211062
#117210992
#105948663
#105948563