Finite Group Databasebeta

← #13150892

Subgroups — #13150892

Order 1728.

1433 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 1433 conjugacy classes of subgroups — too many to draw. Only its 155 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×C4×#272C211yesmaximal, characteristic, complemented
2C2×#88677C211yesmaximal, characteristic
3#64138⋊C3C212yesmaximal, complemented
4#11721320C211yesmaximal, characteristic, complemented
5#11721307C212yesmaximal, complemented
6#11721161C212yesmaximal, complemented
7#11721147C212yesmaximal, complemented
8#11721104C214yesmaximal, complemented
9D3×#929—32nomaximal
10C23×#272C2211yescharacteristic
11C22×((C3⋊D3)⋊C3)⋊C2C2211yescharacteristic
12C22×C4×H(3,𝔽3)C2211yescharacteristic, complemented
13C22×#1084C2211yescharacteristic, complemented
14C2×C4×#272—24no—
15C2×C4×#272C2212yes—
16C2×C4×#272C2212yescomplemented
17C2×#638⋊C3C2212yes—
18#88677—24no—
19C2×#6200C2212yes—
20C2×#6199C2211yescharacteristic
21C2×#6199C2212yes—
22#3757⋊C3C2212yescomplemented
23#3752⋊C3C2214yescomplemented
24#3750⋊C3C2214yescomplemented
25#88451C2212yescomplemented
26#88450C2214yescomplemented
27#88448C2214yescomplemented
28C22×C3×C4×D3—32no—
29C22×C3×C4×D3—61no—
30C2×D3×#67—32no—
31#64138D312yescomplemented
32C3×#929—32no—
33D3×#217—32no—
34#63966—32no—
35#63961—32no—
36D3×(C2×D3)⋊C4—34no—
37#63905—34no—
38D3×Dic3⋊C4—34no—
39#63850—34no—
40C23×H(3,𝔽3)C2311yescharacteristic
41C22×#272—24no—
42C22×#272D414yes—
43C22×#272C2311yescharacteristic
44C22×#272C2311yescharacteristic
45C22×(C3⋊D3)⋊C3C2312yes—
46C2×((C3⋊D3)⋊C3)⋊C2—22no—
47C2×((C3⋊D3)⋊C3)⋊C2—24no—
48C2×C4×H(3,𝔽3)—22no—
49C2×C4×H(3,𝔽3)C2312yes—
50C2×#1084—22no—
51C2×#1084C2312yes—
52C4×#272—24no—
53C4×#272—24no—
54#638⋊C3—24no—
55C2×(C3⋊Dic3)⋊C3C2312yes—
56C2×(C3⋊Dic3)⋊C3C2314yes—
57#6200—24no—
58#6199—22no—
59#6199—24no—
60C2×#929—91no—
61D3×#123—92no—
62C23×C3×D3—32no—
63C23×C3×D3—61no—
64C22×D32—32no—
65C22×C32×C4—21no—
66C22×C32×C4C2×D312yescomplemented
67C2×#638C2×D312yes—
68C2×C3×#67—32no—
69C22×C3×Dic3—32no—
70C22×C3×Dic3—61no—
71C2×C3×C4×D3—34no—
72C2×C3×C4×D3—34no—
73C2×C3×C4×D3—68no—
74C2×C3×C4×D3—62no—
75C2×C3×C4×D3—62no—
76C2×C3×C4×D3—68no—
77D3×#67—68no—
78C2×#626—32no—
79C2×#625—32no—
80C2×D3×Dic3—32no—
81C2×D3×Dic3—34no—
82#3757C2×D312yescomplemented
83#3752C2×D314yescomplemented
84#3750C2×D314yescomplemented
85C3×#217—32no—
86C3×(C2×D3)⋊C4—34no—
87C3×Dic3⋊C4—34no—
88#3724—34no—
89Dic3⋊Dic3—34no—
90#3722—34no—
91(C2×D3)⋊Dic3—34no—
92(C2×D3)⋊Dic3—34no—
93C22×H(3,𝔽3)—22no—
94C22×H(3,𝔽3)C2×D412yes—
95C22×H(3,𝔽3)C2411yescharacteristic
96C2×#272—22no—
97C2×#272—28no—
98C2×#272C2×D412yes—
99C2×#272C2×D412yes—
100C2×#272#4114yes—
101C2×(C3⋊D3)⋊C3—22no—
102C2×(C3⋊D3)⋊C3—24no—
103((C3⋊D3)⋊C3)⋊C2—44no—
104C4×H(3,𝔽3)—22no—
105#1084—22no—
106(C3⋊Dic3)⋊C3—22no—
107(C3⋊Dic3)⋊C3—24no—
108C23×C3×C4—91no—
109C22×#67—91no—
110C2×D3×D4—92no—
111C22×C4×D3—32no—
112C22×C4×D3—61no—
113C3×#123—92no—
114C2×#217—91no—
115#935—92no—
116#929—32no—
117#929—92no—
118#929—94no—
119#929—92no—
120C2×(C2×D3)⋊C4—92no—
121C2×Dic3⋊C4—92no—
122#894—94no—
123D3×C4⋊C4—94no—
124#886—92no—
125#883—94no—
126D3×C22⋊C4—94no—
127D3×C22⋊C4—92no—
128C23×C32—21no—
129C23×C32C22×D312yes—
130C22×C3⋊D3C22×D312yes—
131C22×C3×D3—32no—
132C22×C3×D3—38no—
133C22×C3×D3—32no—
134C22×C3×D3—32no—
135C22×C3×D3—64no—
136C22×C3×D3—61no—
137C22×C3×D3—61no—
138C22×C3×D3—68no—
139C22×C3×D3—68no—
140C2×D32—68no—
141C2×D32—64no—
142C2×C32×C4—24no—
143C2×C32×C4—24no—
144C2×C32×C4—22no—
145C2×C32×C4C22×D314yes—
146#638—24no—
147C2×C3⋊Dic3C22×D314yes—
148C2×C3⋊Dic3C22×D312yes—
149C3×#67—64no—
150C2×C3×Dic3—34no—
151C2×C3×Dic3—32no—
152C2×C3×Dic3—34no—
153C2×C3×Dic3—62no—
154C2×C3×Dic3—64no—
155C2×C3×Dic3—64no—
156C3×C4×D3—68no—
157C3×C4×D3—68no—
158C3×C4×D3—68no—
159C3×C4×D3—68no—
160#626—64no—
161#625—64no—
162D3×Dic3—68no—
163D3×Dic3—64no—
164C2×#123—271noSylow
165C2×H(3,𝔽3)—22no—
166C2×H(3,𝔽3)C22×D411yescharacteristic
167C2×H(3,𝔽3)C2×#4112yes—
168#272—22no—
169#272#12314yescomplemented
170(C3⋊D3)⋊C3—42no—
171C24×C3—91no—
172C23×D3—32no—
173C23×D3—61no—
174C23×D3—91no—
175C23×D3—92no—
176C2×C3×D4—92no—
177C22×C3×C4—32no—
178C22×C3×C4—61no—
179C22×C3×C4—91no—
180C22×C3×C4—92no—
181C22×C3×C4—92no—
182C22×C3×C4—184no—
183C22×C3×C4D3211yescharacteristic
184C2×#67—32no—
185C2×#67—92no—
186C2×#67—92no—
187C2×#67—92no—
188C2×#67—184no—
189C22×Dic3—32no—
190C22×Dic3—61no—
191C22×Dic3—91no—
192C22×Dic3—92no—
193D3×D4—188no—
194C2×D12—92no—
195C2×C4×D3—34no—
196C2×C4×D3—34no—
197C2×C4×D3—62no—
198C2×C4×D3—62no—
199C2×C4×D3—68no—
200C2×C4×D3—68no—
201C2×C4×D3—94no—
202C2×C4×D3—92no—
203C3×C4⋊C4—94no—
204C3×C22⋊C4—94no—
205C3×C22⋊C4—92no—
206#217—32no—
207#217—92no—
208#217—92no—
209#217—94no—
210(C2×D3)⋊C4—34no—
211(C2×D3)⋊C4—94no—
212(C2×D3)⋊C4—94no—
213(C2×D3)⋊C4—94no—
214(C2×D3)⋊C4—94no—
215C4⋊Dic3—94no—
216Dic3⋊C4—34no—
217Dic3⋊C4—94no—
218Dic3⋊C4—94no—
219Dic3⋊C4—94no—
220C22×C32—24no—
221C22×C32—24no—
222C22×C32—21no—
223C22×C32—22no—
224C22×C32D3×D414yes—
225C22×C32C23×D312yes—
226C2×C3⋊D3—22no—
227C2×C3⋊D3—24no—
228C2×C3×D3—34no—
229C2×C3×D3—38no—
230C2×C3×D3—34no—
231C2×C3×D3—62no—
232C2×C3×D3—62no—
233C2×C3×D3—64no—
234C2×C3×D3—616no—
235C2×C3×D3—616no—
236C2×C3×D3—64no—
237C2×C3×D3—64no—
238C2×C3×D3—62no—
239C2×C3×D3—64no—
240D32—128no—
241C32×C4—24no—
242C32×C4—24no—
243C3⋊Dic3—22no—
244C3⋊Dic3—24no—
245C3×Dic3—64no—
246C3×Dic3—62no—
247C3×Dic3—64no—
248C3×Dic3—64no—
249C22×D4—271no—
250C23×C4—91no—
251#123—92no—
252#123—274no—
253#123—272no—
254C2×C4⋊C4—272no—
255C2×C22⋊C4—272no—
256C2×C22⋊C4—271no—
257H(3,𝔽3)C2×#12311yescharacteristic, complemented, Sylow
258C23×C3—32no—
259C23×C3—61no—
260C23×C3—92no—
261C23×C3—91no—
262C23×C3—91no—
263C23×C3—94no—
264C23×C3—184no—
265C23×C3C2×D3211yescharacteristic
266C22×D3—32no—
267C22×D3—32no—
268C22×D3—38no—
269C22×D3—32no—
270C22×D3—61no—
271C22×D3—61no—
272C22×D3—64no—
273C22×D3—68no—
274C22×D3—68no—
275C22×D3—92no—
276C22×D3—184no—
277C22×D3—188no—
278C22×D3—182no—
279C22×D3—184no—
280C3×D4—184no—
281C2×C3×C4—22no—
282C2×C3×C4—34no—
283C2×C3×C4—64no—
284C2×C3×C4—64no—
285C2×C3×C4—62no—
286C2×C3×C4—92no—
287C2×C3×C4—94no—
288C2×C3×C4—92no—
289C2×C3×C4—184no—
290C2×C3×C4—184no—
291C2×C3×C4—182no—
292C2×C3×C4C2×D3212yes—
293#67—64no—
294#67—184no—
295#67—184no—
296#67—184no—
297C2×Dic3—34no—
298C2×Dic3—32no—
299C2×Dic3—34no—
300C2×Dic3—62no—
301C2×Dic3—64no—
302C2×Dic3—64no—
303C2×Dic3—92no—
304C2×Dic3—94no—
305C2×Dic3—184no—
306C2×Dic3—182no—
307D12—184no—
308C4×D3—68no—
309C4×D3—68no—
310C4×D3—68no—
311C4×D3—68no—
312C4×D3—188no—
313C4×D3—184no—
314C2×C32—24no—
315C2×C32—24no—
316C2×C32—22no—
317C2×C32—21no—
318C2×C32C2×D3×D412yes—
319C2×C32D3×#4114yes—
320C3⋊D3—42no—
321C3×D3—38no—
322C3×D3—64no—
323C3×D3—64no—
324C3×D3—64no—
325C3×D3—122no—
326C24—91no—
327C24—271no—
328C2×D4—92no—
329C2×D4—272no—
330C2×D4—544no—
331C22×C4—91no—
332C22×C4—92no—
333C22×C4—92no—
334C22×C4—184no—
335C22×C4—272no—
336C22×C4—271no—
337C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
338C4⋊C4—94no—
339C4⋊C4—274no—
340C22⋊C4—92no—
341C22⋊C4—94no—
342C22⋊C4—272no—
343C22⋊C4—274no—
344C22×C3—22no—
345C22×C3—32no—
346C22×C3—34no—
347C22×C3—64no—
348C22×C3—64no—
349C22×C3—61no—
350C22×C3—62no—
351C22×C3—94no—
352C22×C3—92no—
353C22×C3—92no—
354C22×C3—188no—
355C22×C3—182no—
356C22×C3—184no—
357C22×C3—182no—
358C22×C3#381112yes—
359C22×C3C22×D3211yescharacteristic
360C2×D3—34no—
361C2×D3—38no—
362C2×D3—34no—
363C2×D3—62no—
364C2×D3—62no—
365C2×D3—64no—
366C2×D3—616no—
367C2×D3—64no—
368C2×D3—616no—
369C2×D3—64no—
370C2×D3—64no—
371C2×D3—62no—
372C2×D3—182no—
373C2×D3—184no—
374C2×D3—368no—
375C2×D3—364no—
376C3×C4—22no—
377C3×C4—64no—
378C3×C4—64no—
379C3×C4—182no—
380C3×C4—184no—
381C3×C4—182no—
382Dic3—64no—
383Dic3—64no—
384Dic3—64no—
385Dic3—62no—
386Dic3—182no—
387Dic3—184no—
388C32—21no—
389C32D3×#12312yescomplemented
390C23—91no—
391C23—91no—
392C23—94no—
393C23—92no—
394C23—184no—
395C23—542no—
396C23—544no—
397C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
398D4—184no—
399D4—544no—
400C2×C4—22no—
401C2×C4—94no—
402C2×C4—92no—
403C2×C4—92no—
404C2×C4—184no—
405C2×C4—184no—
406C2×C4—182no—
407C2×C4—542no—
408C2×C4—544no—
409C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
410C2×C3—22no—
411C2×C3—32no—
412C2×C3—34no—
413C2×C3—64no—
414C2×C3—64no—
415C2×C3—61no—
416C2×C3—62no—
417C2×C3—94no—
418C2×C3—182no—
419C2×C3—362no—
420C2×C3#6430612yes—
421C2×C3C2×#381111yescharacteristic
422D3—38no—
423D3—64no—
424D3—64no—
425D3—64no—
426D3—122no—
427D3—362no—
428C22—22no—
429C22—92no—
430C22—94no—
431C22—92no—
432C22—188no—
433C22—182no—
434C22—182no—
435C22—184no—
436C22—1084no—
437C22#8867712yes—
438C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
439C4—22no—
440C4—182no—
441C4—182no—
442C4—184no—
443C3—32no—
444C3—61no—
445C3#1059507911yescharacteristic
446C2—22no—
447C2—94no—
448C2—182no—
449C2—362no—
450C2#1172249112yes—
451C2C2×#8867711yescharacteristic
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×#88677C2
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#123
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×C4×#272C2semidirect
3. #64138⋊C3C2semidirect
4. #11721320C2semidirect
5. #11721307C2semidirect
6. #11721161C2semidirect
7. #11721147C2semidirect
8. #11721104C2semidirect
12. C22×C4×H(3,𝔽3)C22semidirect
13. C22×#1084C22semidirect
16. C2×C4×#272C22semidirect
22. #3757⋊C3C22semidirect
23. #3752⋊C3C22semidirect
24. #3750⋊C3C22semidirect
25. #88451C22semidirect
26. #88450C22semidirect
27. #88448C22semidirect
31. #64138D3semidirect
66. C22×C32×C4C2×D3semidirect
82. #3757C2×D3semidirect
83. #3752C2×D3semidirect
84. #3750C2×D3semidirect
169. #272#123semidirect
257. H(3,𝔽3)C2×#123semidirect
337. C22×C4((C3⋊D3)⋊C3)⋊C2semidirect
389. C32D3×#123semidirect

Maximal subgroups

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

subgroupindex
C22×C4×#2722
C2×#886772
#64138⋊C32
#117213202
#117213072
#117211612
#117211472
#117211042
D3×#9293