Finite Group Databasebeta

← #13147869

Subgroups — #13147869

Order 1728.

892 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 892 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C4×C4⋊#272C211yesmaximal, characteristic
2#64078⋊C3C211yesmaximal, characteristic, complemented
3C4×#634⋊C3C211yesmaximal, characteristic, complemented
4#11721152C212yesmaximal, complemented
5#11721149C212yesmaximal
6#11721148C214yesmaximal, complemented
7#11721094C212yesmaximal, complemented
8#11721093C212yesmaximal, complemented
9#10593960—31nomaximal
10#10593956—31nomaximal
11C2×C4⋊#272C2211yescharacteristic
12C2×C4×#272C2212yes—
13C2×#634⋊C3C2211yescharacteristic
14C2×#634⋊C3C2212yes—
15C2×#6199C2214yes—
16C42×H(3,𝔽3)C2211yescharacteristic
17#88542C2212yes—
18C4⋊#1084C2211yescharacteristic
19C4⋊((C3⋊Dic3)⋊C3)C2212yes—
20C4⋊((C3⋊Dic3)⋊C3)C2212yes—
21C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
22#3750⋊C3C2212yes—
23C4×(C3⋊Dic3)⋊C3C2212yes—
24#88451C2214yes—
25#88449C2214yes—
26#88448C2214yes—
27#64078D311yescharacteristic, complemented
28C4×#634D311yescharacteristic, complemented
29C3×C4×D12—31no—
30C3×C4×D12—31no—
31C3×C4×D12—61no—
32C3×#869—31no—
33C3×C4×Dic6—31no—
34#63898—32no—
35#63896—34no—
36Dic3×D12—32no—
37D3×C4⋊Dic3—32no—
38D3×C4⋊Dic3—32no—
39#63887—34no—
40#63872—32no—
41#63869—32no—
42#63857—32no—
43#63856—32no—
44C22×#272C2312yes—
45C2×C4×H(3,𝔽3)C2311yescharacteristic
46C2×C4×H(3,𝔽3)C2311yescharacteristic
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48C2×#1084C2312yes—
49C4⋊#272Q814yescomplemented
50C4×#272—24no—
51C2×(C3⋊Dic3)⋊C3C2312yes—
52C2×(C3⋊Dic3)⋊C3C2314yes—
53C2×(C3⋊Dic3)⋊C3C2312yes—
54#634⋊C3—22no—
55#634⋊C3—42no—
56#6199—44no—
57#5730—91no—
58#5579—91no—
59#5543—91no—
60C2×#634C2×D311yescharacteristic
61C2×#634C2×D312yes—
62C2×C3×D12—31no—
63C2×C3×D12—31no—
64C2×C3×D12—61no—
65C2×C3×C4×D3—32no—
66C2×C3×C4×D3—32no—
67C2×C3×C4×D3—62no—
68C2×C3×Dic6—31no—
69C2×C3×Dic6—32no—
70C2×D3×Dic3—34no—
71C2×D3×Dic3—34no—
72C32×C42—21no—
73C32×C42C2×D311yescharacteristic
74C32×C42C2×D311yescharacteristic
75C4⋊(C3⋊Dic3)C2×D311yescharacteristic
76C4⋊(C3⋊Dic3)C2×D312yes—
77C4⋊(C3⋊Dic3)C2×D312yes—
78#3750C2×D312yes—
79C4×C3⋊Dic3C2×D312yes—
80C3×(C2×D3)⋊C4—32no—
81C3×(C2×D3)⋊C4—32no—
82C3×(C2×D3)⋊C4—62no—
83C3×C4⋊Dic3—32no—
84C3×C4⋊Dic3—31no—
85C3×C4⋊Dic3—31no—
86C3×C4⋊Dic3—32no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—61no—
89C3×Dic3⋊C4—32no—
90C3×C4×Dic3—32no—
91#3724—34no—
92Dic3⋊Dic3—34no—
93Dic3⋊Dic3—34no—
94Dic3⋊Dic3—34no—
95(C2×D3)⋊Dic3—34no—
96(C2×D3)⋊Dic3—34no—
97C22×H(3,𝔽3)C2411yescharacteristic
98C2×#272—22no—
99C2×#272—22no—
100C2×#272C2×Q814yes—
101C4×H(3,𝔽3)—21no—
102C4×H(3,𝔽3)C2×Q812yes—
103C4×H(3,𝔽3)#4112yes—
104#1084—22no—
105(C3⋊Dic3)⋊C3—22no—
106(C3⋊Dic3)⋊C3—22no—
107(C3⋊Dic3)⋊C3—24no—
108C3×C4⋊Q8—91no—
109C3×C4×Q8—91no—
110C3×C4×D4—91no—
111#946—94no—
112#946—92no—
113D4×Dic3—92no—
114C2×C4⋊Dic3—92no—
115#924—92no—
116#897—92no—
117#896—94no—
118#893—92no—
119D3×C4⋊C4—92no—
120D3×C4⋊C4—92no—
121#890—92no—
122#890—92no—
123#878—94no—
124C4×D12—31no—
125C4×D12—31no—
126C4×D12—61no—
127C4×D12—92no—
128#870—92no—
129#869—31no—
130#869—91no—
131C4×Dic6—31no—
132C4×Dic6—91no—
133C22×C3×D3—32no—
134C22×C3×D3—32no—
135C22×C3×D3—62no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4C22×D311yescharacteristic
140C2×C32×C4C22×D311yescharacteristic
141C2×C32×C4C22×D311yescharacteristic
142C2×C32×C4C22×D311yescharacteristic
143C2×C32×C4C22×D311yescharacteristic
144C2×C32×C4C22×D311yescharacteristic
145C2×C3⋊Dic3C22×D312yes—
146C2×C3⋊Dic3C22×D314yes—
147C2×C3⋊Dic3C22×D312yes—
148#634—22no—
149#634—42no—
150C2×C3×Dic3—32no—
151C2×C3×Dic3—34no—
152C2×C3×Dic3—32no—
153C2×C3×Dic3—32no—
154C2×C3×Dic3—32no—
155C2×C3×Dic3—62no—
156C3×D12—34no—
157C3×D12—34no—
158C3×D12—64no—
159C3×C4×D3—64no—
160C3×C4×D3—64no—
161C3×C4×D3—124no—
162C3×Dic6—62no—
163C3×Dic6—122no—
164D3×Dic3—124no—
165D3×Dic3—124no—
166#554—271noSylow
167C2×H(3,𝔽3)C22×Q811yescharacteristic
168C2×H(3,𝔽3)C2×#4111yescharacteristic
169C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
170#272—24no—
171C2×C3×Q8—91no—
172C2×C3×Q8—92no—
173C2×C3×D4—91no—
174C22×C3×C4—92no—
175C22×Dic3—94no—
176C2×D12—31no—
177C2×D12—31no—
178C2×D12—61no—
179C2×C4×D3—32no—
180C2×C4×D3—32no—
181C2×C4×D3—62no—
182C2×C4×D3—94no—
183C2×C4×D3—94no—
184C2×Dic6—31no—
185C2×Dic6—32no—
186C2×Dic6—91no—
187C2×Dic6—92no—
188C3×C4⋊C4—92no—
189C3×C4⋊C4—91no—
190C3×C4⋊C4—92no—
191C3×C4⋊C4—92no—
192C3×C4⋊C4—91no—
193C3×C22⋊C4—92no—
194C3×C42—31no—
195C3×C42—31no—
196C3×C42—61no—
197C3×C42—92no—
198C3×C42D3211yescharacteristic
199#217—94no—
200(C2×D3)⋊C4—32no—
201(C2×D3)⋊C4—32no—
202(C2×D3)⋊C4—62no—
203(C2×D3)⋊C4—94no—
204(C2×D3)⋊C4—94no—
205C4⋊Dic3—31no—
206C4⋊Dic3—31no—
207C4⋊Dic3—31no—
208C4⋊Dic3—32no—
209C4⋊Dic3—32no—
210C4⋊Dic3—61no—
211C4⋊Dic3—91no—
212C4⋊Dic3—92no—
213C4⋊Dic3—92no—
214C4⋊Dic3—94no—
215C4⋊Dic3—94no—
216Dic3⋊C4—32no—
217Dic3⋊C4—92no—
218Dic3⋊C4—94no—
219Dic3⋊C4—94no—
220Dic3⋊C4—94no—
221Dic3⋊C4—94no—
222C4×Dic3—32no—
223C4×Dic3—92no—
224C22×C32—21no—
225C22×C32C23×D311yescharacteristic
226C22×C32C23×D311yescharacteristic
227C2×C3×D3—34no—
228C2×C3×D3—34no—
229C2×C3×D3—64no—
230C2×C3×D3—62no—
231C2×C3×D3—62no—
232C2×C3×D3—62no—
233C2×C3×D3—62no—
234C2×C3×D3—122no—
235C2×C3×D3—122no—
236C32×C4—21no—
237C32×C4—21no—
238C32×C4—22no—
239C32×C4—22no—
240C32×C4—41no—
241C32×C4#23512yes—
242C32×C4D3×Q812yes—
243C32×C4D3×Q812yes—
244C32×C4#23912yes—
245C3⋊Dic3—22no—
246C3⋊Dic3—24no—
247C3⋊Dic3—22no—
248C3×Dic3—62no—
249C3×Dic3—62no—
250C3×Dic3—62no—
251C3×Dic3—62no—
252C3×Dic3—64no—
253C3×Dic3—122no—
254C4⋊Q8—91no—
255#125—272no—
256C22⋊Q8—274no—
257C22⋊Q8—272no—
258C4×Q8—91no—
259C4×D4—91no—
260C4×D4—272no—
261C2×C4⋊C4—272no—
262H(3,𝔽3)#55411yescharacteristic, complemented, Sylow
263C23×C3—92no—
264C22×D3—32no—
265C22×D3—32no—
266C22×D3—62no—
267C3×Q8—182no—
268C3×Q8—362no—
269C3×D4—94no—
270C2×C3×C4—31no—
271C2×C3×C4—31no—
272C2×C3×C4—31no—
273C2×C3×C4—31no—
274C2×C3×C4—31no—
275C2×C3×C4—31no—
276C2×C3×C4—61no—
277C2×C3×C4—61no—
278C2×C3×C4—61no—
279C2×C3×C4—92no—
280C2×C3×C4—92no—
281C2×C3×C4—94no—
282C2×C3×C4—92no—
283C2×C3×C4—184no—
284C2×C3×C4C2×D3211yescharacteristic
285C2×C3×C4C2×D3211yescharacteristic
286C2×C3×C4C2×D3211yescharacteristic
287C2×Dic3—32no—
288C2×Dic3—32no—
289C2×Dic3—32no—
290C2×Dic3—32no—
291C2×Dic3—34no—
292C2×Dic3—62no—
293C2×Dic3—92no—
294C2×Dic3—92no—
295C2×Dic3—94no—
296C2×Dic3—364no—
297D12—34no—
298D12—34no—
299D12—64no—
300C4×D3—64no—
301C4×D3—64no—
302C4×D3—124no—
303C4×D3—364no—
304C4×D3—364no—
305Dic6—62no—
306Dic6—122no—
307Dic6—182no—
308Dic6—362no—
309C2×C32—21no—
310C2×C32—21no—
311C2×C32—21no—
312C2×C32C2×#23511yescharacteristic
313C2×C32#100411yescharacteristic
314C2×C32C2×D3×Q811yescharacteristic
315C2×C32C2×D3×Q811yescharacteristic
316C2×C32C2×#23911yescharacteristic
317C2×C32#100911yescharacteristic
318C3×D3—64no—
319C3×D3—64no—
320C3×D3—124no—
321C2×Q8—91no—
322C2×Q8—92no—
323C2×D4—91no—
324C22×C4—92no—
325C22×C4—274no—
326C4⋊C4—91no—
327C4⋊C4—92no—
328C4⋊C4—91no—
329C4⋊C4—92no—
330C4⋊C4—92no—
331C4⋊C4—274no—
332C4⋊C4—274no—
333C22⋊C4—92no—
334C22⋊C4—274no—
335C42—92no—
336C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
337C22×C3—31no—
338C22×C3—31no—
339C22×C3—61no—
340C22×C3—94no—
341C22×C3—182no—
342C22×C3—182no—
343C22×C3C22×D3211yescharacteristic
344C2×D3—34no—
345C2×D3—34no—
346C2×D3—64no—
347C2×D3—62no—
348C2×D3—62no—
349C2×D3—62no—
350C2×D3—62no—
351C2×D3—122no—
352C2×D3—122no—
353C3×C4—21no—
354C3×C4—32no—
355C3×C4—32no—
356C3×C4—32no—
357C3×C4—32no—
358C3×C4—61no—
359C3×C4—61no—
360C3×C4—62no—
361C3×C4—62no—
362C3×C4—121no—
363C3×C4—182no—
364C3×C4—184no—
365C3×C4—182no—
366C3×C4—182no—
367C3×C4#379712yes—
368C3×C4#379912yes—
369Dic3—62no—
370Dic3—62no—
371Dic3—62no—
372Dic3—64no—
373Dic3—62no—
374Dic3—122no—
375Dic3—182no—
376Dic3—184no—
377Dic3—182no—
378C32—21no—
379C32#557911yescharacteristic, complemented
380C32#573011yescharacteristic, complemented
381C23—92no—
382Q8—182no—
383Q8—362no—
384D4—94no—
385C2×C4—92no—
386C2×C4—92no—
387C2×C4—92no—
388C2×C4—94no—
389C2×C4—184no—
390C2×C4—1084no—
391C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
392C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
393C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
394C2×C3—31no—
395C2×C3—31no—
396C2×C3—31no—
397C2×C3—31no—
398C2×C3—31no—
399C2×C3—31no—
400C2×C3—61no—
401C2×C3—61no—
402C2×C3—61no—
403C2×C3—184no—
404C2×C3C2×#379711yescharacteristic
405C2×C3C2×#379911yescharacteristic
406C2×C3#6430711yescharacteristic
407D3—64no—
408D3—64no—
409D3—124no—
410C22—94no—
411C22—182no—
412C22—182no—
413C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
414C4—21no—
415C4—182no—
416C4—182no—
417C4—182no—
418C4—184no—
419C4#8865112yes—
420C4#8865212yes—
421C3—31no—
422C3—31no—
423C3—61no—
424C3#1059395811yescharacteristic
425C2—184no—
426C2C2×#8865111yescharacteristic
427C2C2×#8865211yescharacteristic
428C2#1172249211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC2×C4⋊#272C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#554
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #64078⋊C3C2semidirect
3. C4×#634⋊C3C2semidirect
4. #11721152C2semidirect
6. #11721148C2semidirect
7. #11721094C2semidirect
8. #11721093C2semidirect
27. #64078D3semidirect
28. C4×#634D3semidirect
49. C4⋊#272Q8semidirect
262. H(3,𝔽3)#554semidirect
379. C32#5579semidirect
380. C32#5730semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊#2722
#64078⋊C32
C4×#634⋊C32
#117211522
#117211492
#117211482
#117210942
#117210932
#105939603
#105939563