Finite Group Databasebeta

← #13150623

Subgroups — #13150623

Order 1728.

1109 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 1109 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×#88740C211yesmaximal, characteristic
2C22×#634⋊C3C211yesmaximal, characteristic, complemented
3#64135⋊C3C211yesmaximal, characteristic, complemented
4#11721301C214yesmaximal, complemented
5#11721114C212yesmaximal, complemented
6#11721094C214yesmaximal, complemented
7#11721090C212yesmaximal, complemented
8#10594856—31nomaximal
9#10594855—31nomaximal
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#6235C2212yes—
12#88740—24no—
13C2×C4⋊#272C2211yescharacteristic
14C2×C4×#272C2212yes—
15C2×#6229C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2212yes—
17C2×#634⋊C3—24no—
18C2×#634⋊C3C2212yes—
19C2×#634⋊C3C2212yes—
20#3757⋊C3C2212yes—
21C4⋊((C3⋊Dic3)⋊C3)C2212yes—
22C4×(C3⋊Dic3)⋊C3C2212yes—
23#88451C2218yes—
24#88448C2218yes—
25C22×#634D311yescharacteristic, complemented
26C2×C3×#235—31no—
27C2×C3×#235—31no—
28C2×C3×#235—61no—
29C22×C3×Dic6—31no—
30#64135D311yescharacteristic, complemented
31C3×#926—31no—
32#63962—34no—
33#63961—34no—
34#63873—32no—
35#63866—32no—
36#63863—32no—
37#63860—32no—
38#63857—34no—
39#63856—34no—
40C23×H(3,𝔽3)C2311yescharacteristic
41C22×#272C2312yes—
42C2×C4×H(3,𝔽3)D414yes—
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45#6235—24no—
46C2×#1084C2312yes—
47C4⋊#272—22no—
48C4×#272—24no—
49#6229—22no—
50C2×(C3⋊Dic3)⋊C3—24no—
51C2×(C3⋊Dic3)⋊C3C2314yes—
52C2×(C3⋊Dic3)⋊C3C2314yes—
53#634⋊C3—24no—
54#634⋊C3—24no—
55#5825—91no—
56#5816—91no—
57#5526—91no—
58C22×C32×C4—21no—
59C22×C32×C4C2×D311yescharacteristic
60C22×C32×C4C2×D311yescharacteristic
61C22×C3⋊Dic3C2×D312yes—
62C2×#634—24no—
63C2×#634C2×D312yes—
64C2×#634C2×D312yes—
65C2×C3×#67—32no—
66C2×C3×#67—32no—
67C2×C3×#67—62no—
68C22×C3×Dic3—32no—
69C3×#235—68no—
70C3×#235—64no—
71C3×#235—64no—
72C2×C3×D12—31no—
73C2×C3×D12—31no—
74C2×C3×D12—61no—
75C2×C3×C4×D3—32no—
76C2×C3×C4×D3—32no—
77C2×C3×C4×D3—62no—
78C2×C3×Dic6—32no—
79C2×C3×Dic6—32no—
80C2×C3×Dic6—31no—
81C2×C3×Dic6—31no—
82C2×C3×Dic6—61no—
83C2×C3×Dic6—64no—
84#3757C2×D312yes—
85C4⋊(C3⋊Dic3)C2×D312yes—
86C4×C3⋊Dic3C2×D312yes—
87C3×#217—32no—
88C3×C4⋊Dic3—32no—
89C3×C4×Dic3—32no—
90#3724—38no—
91Dic3⋊Dic3—38no—
92(C2×D3)⋊Dic3—38no—
93(C2×D3)⋊Dic3—38no—
94C22×H(3,𝔽3)—22no—
95C22×H(3,𝔽3)C2×D412yes—
96C22×H(3,𝔽3)C2411yescharacteristic
97C2×#272—22no—
98C2×#272—24no—
99C4×H(3,𝔽3)C2×D412yes—
100C4×H(3,𝔽3)C2×D412yes—
101#1084—22no—
102(C3⋊Dic3)⋊C3—24no—
103(C3⋊Dic3)⋊C3—24no—
104C2×C3×#41—91no—
105C22×C3×Q8—91no—
106C2×#235—31no—
107C2×#235—31no—
108C2×#235—61no—
109C22×Dic6—31no—
110C22×Dic6—91no—
111C3×#117—91no—
112#947—92no—
113#946—94no—
114#944—92no—
115#944—92no—
116#936—92no—
117#935—94no—
118#929—94no—
119#926—31no—
120#926—91no—
121#924—94no—
122#897—94no—
123#886—94no—
124#875—92no—
125#869—92no—
126C23×C32—21no—
127C23×C32C22×D311yescharacteristic
128C23×C32C22×D311yescharacteristic
129C22×C3×D3—32no—
130C22×C3×D3—32no—
131C22×C3×D3—62no—
132C2×C32×C4—24no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4D1214yes—
136C2×C32×C4#6714yes—
137C2×C32×C4C22×D311yescharacteristic
138C2×C32×C4C22×D311yescharacteristic
139C2×C32×C4C22×D311yescharacteristic
140C2×C32×C4C22×D311yescharacteristic
141C2×C3⋊Dic3—24no—
142C2×C3⋊Dic3C22×D314yes—
143C2×C3⋊Dic3C22×D314yes—
144#634—24no—
145#634—24no—
146C3×#67—68no—
147C3×#67—64no—
148C3×#67—64no—
149C2×C3×Dic3—34no—
150C2×C3×Dic3—34no—
151C2×C3×Dic3—32no—
152C2×C3×Dic3—32no—
153C2×C3×Dic3—62no—
154C2×C3×Dic3—64no—
155C3×D12—64no—
156C3×D12—62no—
157C3×D12—62no—
158C3×C4×D3—68no—
159C3×C4×D3—64no—
160C3×C4×D3—64no—
161C3×Dic6—64no—
162C3×Dic6—64no—
163C3×Dic6—64no—
164C3×Dic6—62no—
165C3×Dic6—62no—
166#536—271noSylow
167C2×H(3,𝔽3)—22no—
168C2×H(3,𝔽3)C22×D411yescharacteristic
169C2×H(3,𝔽3)2−1+412yes—
170#272—42no—
171C3×#41—184no—
172C2×C3×Q8—92no—
173C2×C3×Q8—92no—
174C2×C3×Q8—91no—
175C2×C3×Q8—184no—
176C2×C3×D4—92no—
177C2×C3×D4—91no—
178C22×C3×C4—31no—
179C22×C3×C4—31no—
180C22×C3×C4—61no—
181C22×C3×C4—92no—
182C22×C3×C4—92no—
183C22×C3×C4D3211yescharacteristic
184C2×#67—32no—
185C2×#67—32no—
186C2×#67—62no—
187C22×Dic3—32no—
188C22×Dic3—92no—
189#235—64no—
190#235—64no—
191#235—68no—
192C2×D12—31no—
193C2×D12—31no—
194C2×D12—61no—
195C2×C4×D3—32no—
196C2×C4×D3—32no—
197C2×C4×D3—62no—
198C2×Dic6—31no—
199C2×Dic6—31no—
200C2×Dic6—32no—
201C2×Dic6—32no—
202C2×Dic6—61no—
203C2×Dic6—64no—
204C2×Dic6—92no—
205C2×Dic6—92no—
206C2×Dic6—184no—
207C3×C4⋊C4—92no—
208C3×C22⋊C4—92no—
209C3×C42—92no—
210#217—32no—
211#217—92no—
212#217—98no—
213(C2×D3)⋊C4—98no—
214(C2×D3)⋊C4—98no—
215C4⋊Dic3—32no—
216C4⋊Dic3—92no—
217C4⋊Dic3—98no—
218Dic3⋊C4—98no—
219Dic3⋊C4—98no—
220C4×Dic3—32no—
221C4×Dic3—92no—
222C22×C32—22no—
223C22×C32—22no—
224C22×C32—21no—
225C22×C32—22no—
226C22×C32—42no—
227C22×C32C2×D1212yes—
228C22×C32C2×#6712yes—
229C22×C32C23×D311yescharacteristic
230C22×C32C23×D311yescharacteristic
231C2×C3×D3—64no—
232C2×C3×D3—62no—
233C2×C3×D3—64no—
234C2×C3×D3—64no—
235C2×C3×D3—62no—
236C2×C3×D3—124no—
237C32×C4—22no—
238C32×C4—22no—
239C32×C4C2×D1212yes—
240C32×C4C2×D1212yes—
241C32×C4C2×#6712yes—
242C32×C4C2×#6712yes—
243C3⋊Dic3—24no—
244C3⋊Dic3—24no—
245C3×Dic3—64no—
246C3×Dic3—64no—
247C3×Dic3—62no—
248C3×Dic3—64no—
249C3×Dic3—62no—
250C2×#41—91no—
251C22×Q8—91no—
252C4⋊Q8—272no—
253#124—272no—
254#123—274no—
255C22⋊Q8—274no—
256#117—91no—
257H(3,𝔽3)#53611yescharacteristic, complemented, Sylow
258C23×C3—31no—
259C23×C3—31no—
260C23×C3—61no—
261C23×C3—92no—
262C23×C3C2×D3211yescharacteristic
263C22×D3—32no—
264C22×D3—32no—
265C22×D3—62no—
266C3×Q8—184no—
267C3×Q8—184no—
268C3×Q8—182no—
269C3×D4—184no—
270C3×D4—182no—
271C2×C3×C4—34no—
272C2×C3×C4—34no—
273C2×C3×C4—31no—
274C2×C3×C4—31no—
275C2×C3×C4—31no—
276C2×C3×C4—31no—
277C2×C3×C4—64no—
278C2×C3×C4—61no—
279C2×C3×C4—61no—
280C2×C3×C4—94no—
281C2×C3×C4—94no—
282C2×C3×C4—92no—
283C2×C3×C4—184no—
284C2×C3×C4—184no—
285C2×C3×C4#62614yes—
286C2×C3×C4C2×D3211yescharacteristic
287C2×C3×C4C2×D3211yescharacteristic
288#67—64no—
289#67—64no—
290#67—68no—
291C2×Dic3—32no—
292C2×Dic3—32no—
293C2×Dic3—34no—
294C2×Dic3—34no—
295C2×Dic3—62no—
296C2×Dic3—64no—
297C2×Dic3—94no—
298C2×Dic3—94no—
299C2×Dic3—184no—
300D12—62no—
301D12—62no—
302D12—64no—
303C4×D3—64no—
304C4×D3—64no—
305C4×D3—68no—
306Dic6—62no—
307Dic6—62no—
308Dic6—64no—
309Dic6—64no—
310Dic6—64no—
311Dic6—184no—
312Dic6—184no—
313C2×C32—22no—
314C2×C32—22no—
315C2×C32—22no—
316C2×C32—21no—
317C2×C32—42no—
318C2×C32C22×D1211yescharacteristic
319C2×C32#100712yes—
320C2×C32#101012yes—
321C2×C32C22×#6711yescharacteristic
322C3×D3—124no—
323C3×D3—122no—
324C3×D3—122no—
325#41—184no—
326C2×Q8—92no—
327C2×Q8—92no—
328C2×Q8—91no—
329C2×Q8—184no—
330C2×D4—92no—
331C2×D4—91no—
332C22×C4—92no—
333C22×C4—92no—
334C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
335C4⋊C4—92no—
336C4⋊C4—278no—
337C22⋊C4—92no—
338C22⋊C4—278no—
339C42—92no—
340C22×C3—22no—
341C22×C3—31no—
342C22×C3—31no—
343C22×C3—32no—
344C22×C3—32no—
345C22×C3—62no—
346C22×C3—62no—
347C22×C3—61no—
348C22×C3—62no—
349C22×C3—122no—
350C22×C3—184no—
351C22×C3—182no—
352C22×C3C2×#62612yes—
353C22×C3C22×D3211yescharacteristic
354C2×D3—62no—
355C2×D3—62no—
356C2×D3—64no—
357C2×D3—64no—
358C2×D3—64no—
359C2×D3—124no—
360C3×C4—32no—
361C3×C4—32no—
362C3×C4—32no—
363C3×C4—32no—
364C3×C4—62no—
365C3×C4—62no—
366C3×C4—184no—
367C3×C4—184no—
368C3×C4—182no—
369C3×C4C2×#62612yes—
370C3×C4C2×#62612yes—
371Dic3—62no—
372Dic3—62no—
373Dic3—64no—
374Dic3—64no—
375Dic3—64no—
376Dic3—184no—
377Dic3—184no—
378C32—21no—
379C32#552611yescharacteristic, complemented
380C32#581611yescharacteristic, complemented
381C23—92no—
382C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
383Q8—184no—
384Q8—184no—
385Q8—182no—
386D4—184no—
387D4—182no—
388C2×C4—94no—
389C2×C4—94no—
390C2×C4—92no—
391C2×C4—184no—
392C2×C4—184no—
393C2×C4#620014yes—
394C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
395C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
396C2×C3—22no—
397C2×C3—31no—
398C2×C3—31no—
399C2×C3—32no—
400C2×C3—32no—
401C2×C3—62no—
402C2×C3—62no—
403C2×C3—61no—
404C2×C3—62no—
405C2×C3—122no—
406C2×C3—362no—
407C2×C3#6429712yes—
408C2×C3C22×#62611yescharacteristic
409D3—122no—
410D3—122no—
411D3—124no—
412C22—22no—
413C22—182no—
414C22—184no—
415C22C2×#620012yes—
416C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
417C4—182no—
418C4—184no—
419C4—184no—
420C4C2×#620012yes—
421C4C2×#620012yes—
422C3—31no—
423C3—31no—
424C3—61no—
425C3#1059486611yescharacteristic
426C2—22no—
427C2—362no—
428C2#1172248212yes—
429C2C22×#620011yescharacteristic
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×#88740C2
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. #64135⋊C3C2semidirect
4. #11721301C2semidirect
5. #11721114C2semidirect
6. #11721094C2semidirect
7. #11721090C2semidirect
25. C22×#634D3semidirect
30. #64135D3semidirect
257. H(3,𝔽3)#536semidirect
379. C32#5526semidirect
380. C32#5816semidirect

Maximal subgroups

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

subgroupindex
C2×#887402
C22×#634⋊C32
#64135⋊C32
#117213012
#117211142
#117210942
#117210902
#105948563
#105948553