Finite Group Databasebeta

← #13150633

Subgroups — #13150633

Order 1728.

1101 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 1101 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
1C22×#6229C211yesmaximal, characteristic, complemented
2C2×#3828⋊C3C211yesmaximal, characteristic
3#64135⋊C3C211yesmaximal, characteristic, complemented
4#11721307C214yesmaximal, complemented
5#11721110C212yesmaximal, complemented
6#11721107C214yesmaximal, complemented
7#11721090C212yesmaximal, complemented
8#10594866—31nomaximal
9#10594855—31nomaximal
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C22×#1084C2212yes—
12C2×#6229—24no—
13C2×#6229C2212yes—
14C2×#6229C2212yes—
15C2×#638⋊C3C2212yes—
16#3828⋊C3—24no—
17C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
18C2×C4×(C3⋊D3)⋊C3C2212yes—
19C2×#634⋊C3C2211yescharacteristic
20#3757⋊C3C2212yes—
21C4⋊((C3⋊Dic3)⋊C3)C2212yes—
22C4×(C3⋊Dic3)⋊C3C2212yes—
23#88450C2218yes—
24#88448C2218yes—
25C2×#3828D311yescharacteristic
26C2×C3×#235—31no—
27C22×C3×Dic6—31no—
28C22×C3×Dic6—31no—
29C22×C3×Dic6—61no—
30#64135D311yescharacteristic, complemented
31C3×#926—31no—
32#63966—34no—
33#63961—34no—
34#63873—32no—
35#63871—34no—
36#63866—32no—
37#63864—32no—
38#63863—32no—
39#63856—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×#1084—24no—
46C2×#1084C2314yes—
47#6229—24no—
48#6229—24no—
49#638⋊C3—24no—
50C2×(C3⋊Dic3)⋊C3C2312yes—
51C2×(C3⋊Dic3)⋊C3C2314yes—
52C4⋊((C3⋊D3)⋊C3)—22no—
53C4×(C3⋊D3)⋊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—
62#3828—24no—
63C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
64C2×C4×C3⋊D3C2×D312yes—
65C2×#634C2×D311yescharacteristic
66C2×C3×#67—32no—
67C22×C3×Dic3—32no—
68C22×C3×Dic3—32no—
69C22×C3×Dic3—62no—
70C3×#235—64no—
71C2×C3×D12—31no—
72C2×C3×C4×D3—32no—
73C2×C3×Dic6—32no—
74C2×C3×Dic6—32no—
75C2×C3×Dic6—32no—
76C2×C3×Dic6—32no—
77C2×C3×Dic6—31no—
78C2×C3×Dic6—62no—
79C2×C3×Dic6—62no—
80C2×C3×Dic6—68no—
81C2×C3×Dic6—64no—
82C2×C3×Dic6—64no—
83#3757C2×D312yes—
84C4⋊(C3⋊Dic3)C2×D312yes—
85C4×C3⋊Dic3C2×D312yes—
86C3×#217—32no—
87C3×C4⋊Dic3—32no—
88C3×C4×Dic3—32no—
89#3724—38no—
90Dic3⋊Dic3—38no—
91#3722—38no—
92(C2×D3)⋊Dic3—38no—
93C22×H(3,𝔽3)—22no—
94C22×H(3,𝔽3)C2×D412yes—
95C22×H(3,𝔽3)C2411yescharacteristic
96C2×(C3⋊D3)⋊C3—22no—
97C2×(C3⋊D3)⋊C3—24no—
98C4×H(3,𝔽3)C2×D412yes—
99C4×H(3,𝔽3)C2×D412yes—
100#1084—24no—
101(C3⋊Dic3)⋊C3—22no—
102(C3⋊Dic3)⋊C3—24no—
103C2×C3×#41—91no—
104C22×C3×Q8—91no—
105C2×#235—31no—
106C2×#235—91no—
107C22×Dic6—31no—
108C22×Dic6—31no—
109C22×Dic6—61no—
110C3×#117—91no—
111#947—92no—
112#946—94no—
113#944—92no—
114#944—92no—
115#936—92no—
116#935—94no—
117#929—94no—
118#926—31no—
119#926—91no—
120#924—94no—
121#897—94no—
122#886—94no—
123#875—92no—
124#869—92no—
125C23×C32—21no—
126C23×C32C22×D311yescharacteristic
127C23×C32C22×D311yescharacteristic
128C22×C3⋊D3C22×D312yes—
129C22×C3×D3—32no—
130C2×C32×C4—24no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4D1214yes—
134C2×C32×C4#6714yes—
135C2×C32×C4C22×D311yescharacteristic
136C2×C32×C4C22×D311yescharacteristic
137C2×C32×C4C22×D311yescharacteristic
138C2×C32×C4C22×D311yescharacteristic
139#638—24no—
140C2×C3⋊Dic3C22×D314yes—
141C2×C3⋊Dic3C22×D312yes—
142C4⋊(C3⋊D3)—22no—
143C4×C3⋊D3—24no—
144#634—22no—
145C3×#67—64no—
146C2×C3×Dic3—34no—
147C2×C3×Dic3—32no—
148C2×C3×Dic3—34no—
149C2×C3×Dic3—34no—
150C2×C3×Dic3—64no—
151C2×C3×Dic3—68no—
152C2×C3×Dic3—64no—
153C2×C3×Dic3—64no—
154C3×D12—62no—
155C3×C4×D3—64no—
156C3×Dic6—68no—
157C3×Dic6—68no—
158C3×Dic6—62no—
159C3×Dic6—64no—
160C3×Dic6—64no—
161C3×Dic6—64no—
162C3×Dic6—64no—
163#536—271noSylow
164C2×H(3,𝔽3)—22no—
165C2×H(3,𝔽3)C22×D411yescharacteristic
166C2×H(3,𝔽3)2−1+412yes—
167(C3⋊D3)⋊C3—42no—
168C3×#41—184no—
169C2×C3×Q8—91no—
170C2×C3×Q8—92no—
171C2×C3×Q8—92no—
172C2×C3×Q8—184no—
173C2×C3×D4—91no—
174C2×C3×D4—92no—
175C22×C3×C4—31no—
176C22×C3×C4—31no—
177C22×C3×C4—61no—
178C22×C3×C4—92no—
179C22×C3×C4—92no—
180C22×C3×C4D3211yescharacteristic
181C2×#67—32no—
182C2×#67—92no—
183C22×Dic3—32no—
184C22×Dic3—32no—
185C22×Dic3—62no—
186#235—64no—
187#235—184no—
188C2×D12—31no—
189C2×D12—91no—
190C2×C4×D3—32no—
191C2×C4×D3—92no—
192C2×Dic6—32no—
193C2×Dic6—32no—
194C2×Dic6—32no—
195C2×Dic6—32no—
196C2×Dic6—31no—
197C2×Dic6—62no—
198C2×Dic6—62no—
199C2×Dic6—64no—
200C2×Dic6—64no—
201C2×Dic6—68no—
202C2×Dic6—91no—
203C3×C4⋊C4—92no—
204C3×C22⋊C4—92no—
205C3×C42—92no—
206#217—32no—
207#217—92no—
208#217—98no—
209(C2×D3)⋊C4—98no—
210(C2×D3)⋊C4—98no—
211C4⋊Dic3—32no—
212C4⋊Dic3—92no—
213C4⋊Dic3—98no—
214Dic3⋊C4—98no—
215Dic3⋊C4—98no—
216C4×Dic3—32no—
217C4×Dic3—92no—
218C22×C32—24no—
219C22×C32—22no—
220C22×C32—22no—
221C22×C32—21no—
222C22×C32—22no—
223C22×C32C2×D1212yes—
224C22×C32C2×#6712yes—
225C22×C32C23×D311yescharacteristic
226C22×C32C23×D311yescharacteristic
227C2×C3⋊D3—24no—
228C2×C3⋊D3—22no—
229C2×C3×D3—62no—
230C2×C3×D3—64no—
231C32×C4—22no—
232C32×C4—22no—
233C32×C4C2×D1212yes—
234C32×C4C2×D1212yes—
235C32×C4C2×#6712yes—
236C32×C4C2×#6712yes—
237C3⋊Dic3—24no—
238C3⋊Dic3—22no—
239C3×Dic3—68no—
240C3×Dic3—62no—
241C3×Dic3—64no—
242C3×Dic3—64no—
243C3×Dic3—64no—
244C2×#41—91no—
245C22×Q8—91no—
246C4⋊Q8—272no—
247#124—272no—
248#123—274no—
249C22⋊Q8—274no—
250#117—91no—
251H(3,𝔽3)#53611yescharacteristic, complemented, Sylow
252C23×C3—31no—
253C23×C3—31no—
254C23×C3—61no—
255C23×C3—92no—
256C23×C3C2×D3211yescharacteristic
257C22×D3—32no—
258C22×D3—92no—
259C3×Q8—182no—
260C3×Q8—184no—
261C3×Q8—184no—
262C3×D4—182no—
263C3×D4—184no—
264C2×C3×C4—34no—
265C2×C3×C4—34no—
266C2×C3×C4—31no—
267C2×C3×C4—31no—
268C2×C3×C4—31no—
269C2×C3×C4—31no—
270C2×C3×C4—64no—
271C2×C3×C4—61no—
272C2×C3×C4—61no—
273C2×C3×C4—92no—
274C2×C3×C4—94no—
275C2×C3×C4—94no—
276C2×C3×C4—184no—
277C2×C3×C4—184no—
278C2×C3×C4#62614yes—
279C2×C3×C4C2×D3211yescharacteristic
280C2×C3×C4C2×D3211yescharacteristic
281#67—64no—
282#67—184no—
283C2×Dic3—34no—
284C2×Dic3—34no—
285C2×Dic3—34no—
286C2×Dic3—32no—
287C2×Dic3—64no—
288C2×Dic3—64no—
289C2×Dic3—64no—
290C2×Dic3—68no—
291C2×Dic3—94no—
292C2×Dic3—92no—
293D12—62no—
294D12—182no—
295C4×D3—64no—
296C4×D3—184no—
297Dic6—64no—
298Dic6—64no—
299Dic6—64no—
300Dic6—64no—
301Dic6—68no—
302Dic6—68no—
303Dic6—62no—
304Dic6—182no—
305C2×C32—24no—
306C2×C32—22no—
307C2×C32—22no—
308C2×C32—22no—
309C2×C32—21no—
310C2×C32C22×D1211yescharacteristic
311C2×C32#101012yes—
312C2×C32#101012yes—
313C2×C32C22×#6711yescharacteristic
314C3⋊D3—42no—
315C3×D3—122no—
316#41—184no—
317C2×Q8—92no—
318C2×Q8—92no—
319C2×Q8—91no—
320C2×Q8—184no—
321C2×D4—92no—
322C2×D4—91no—
323C22×C4—92no—
324C22×C4—92no—
325C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
326C4⋊C4—92no—
327C4⋊C4—278no—
328C22⋊C4—92no—
329C22⋊C4—278no—
330C42—92no—
331C22×C3—22no—
332C22×C3—31no—
333C22×C3—31no—
334C22×C3—32no—
335C22×C3—32no—
336C22×C3—62no—
337C22×C3—62no—
338C22×C3—64no—
339C22×C3—61no—
340C22×C3—62no—
341C22×C3—182no—
342C22×C3—184no—
343C22×C3C2×#62612yes—
344C22×C3C22×D3211yescharacteristic
345C2×D3—64no—
346C2×D3—62no—
347C2×D3—182no—
348C2×D3—184no—
349C3×C4—32no—
350C3×C4—32no—
351C3×C4—32no—
352C3×C4—32no—
353C3×C4—62no—
354C3×C4—62no—
355C3×C4—182no—
356C3×C4—184no—
357C3×C4—184no—
358C3×C4C2×#62612yes—
359C3×C4C2×#62612yes—
360Dic3—64no—
361Dic3—64no—
362Dic3—68no—
363Dic3—64no—
364Dic3—62no—
365Dic3—182no—
366Dic3—184no—
367C32—21no—
368C32#552611yescharacteristic, complemented
369C32#582511yescharacteristic, complemented
370C23—92no—
371C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
372Q8—184no—
373Q8—184no—
374Q8—182no—
375D4—184no—
376D4—182no—
377C2×C4—94no—
378C2×C4—94no—
379C2×C4—92no—
380C2×C4—184no—
381C2×C4—184no—
382C2×C4#620014yes—
383C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
384C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
385C2×C3—22no—
386C2×C3—31no—
387C2×C3—31no—
388C2×C3—32no—
389C2×C3—32no—
390C2×C3—62no—
391C2×C3—62no—
392C2×C3—64no—
393C2×C3—61no—
394C2×C3—62no—
395C2×C3—362no—
396C2×C3#6429812yes—
397C2×C3C22×#62611yescharacteristic
398D3—122no—
399D3—362no—
400C22—22no—
401C22—182no—
402C22—184no—
403C22C2×#620012yes—
404C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
405C4—182no—
406C4—184no—
407C4—184no—
408C4C2×#620012yes—
409C4C2×#620012yes—
410C3—31no—
411C3—31no—
412C3—61no—
413C3#1059485611yescharacteristic
414C2—22no—
415C2—362no—
416C2#1172248312yes—
417C2C22×#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×#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
1. C22×#6229C2semidirect
3. #64135⋊C3C2semidirect
4. #11721307C2semidirect
5. #11721110C2semidirect
6. #11721107C2semidirect
7. #11721090C2semidirect
30. #64135D3semidirect
251. H(3,𝔽3)#536semidirect
368. C32#5526semidirect
369. C32#5825semidirect

Maximal subgroups

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

subgroupindex
C22×#62292
C2×#3828⋊C32
#64135⋊C32
#117213072
#117211102
#117211072
#117210902
#105948663
#105948553