Finite Group Databasebeta

← #13140212

Subgroups — #13140212

Order 1728.

839 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 839 conjugacy classes of subgroups — too many to draw. Only its 125 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×#634⋊C3C211yesmaximal, characteristic, complemented
2C2×#88542C211yesmaximal, characteristic
3C2×#88451C212yesmaximal
4#63658⋊C3C211yesmaximal, characteristic, complemented
5#11720334C212yesmaximal, complemented
6#10591322—31nomaximal
7#10591304—31nomaximal
8C23×#272C2211yescharacteristic
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×#1084C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3C2212yes—
12C22×(C3⋊Dic3)⋊C3C2212yes—
13C2×#634⋊C3—22no—
14C2×#634⋊C3—24no—
15#88542—22no—
16#88451—24no—
17C22×#634D311yescharacteristic, complemented
18C22×C3×Dic6—31no—
19C2×C3×(C2×D3)⋊C4—31no—
20C2×C3×(C2×D3)⋊C4—31no—
21C2×C3×(C2×D3)⋊C4—61no—
22C2×(C2×D3)⋊Dic3—32no—
23C2×(C2×D3)⋊Dic3—32no—
24#63658D311yescharacteristic, complemented
25C3×C3⋊#95—31no—
26#63579—32no—
27#63579—32no—
28C23×H(3,𝔽3)C2311yescharacteristic
29C22×#272—21no—
30C22×#272—22no—
31C22×#272—21no—
32C22×#272—22no—
33C22×#272Q812yescomplemented
34C2×C4×H(3,𝔽3)—22no—
35C2×C4×H(3,𝔽3)D412yes—
36C2×#1084—21no—
37C2×#1084—22no—
38C2×(C3⋊Dic3)⋊C3—22no—
39C2×(C3⋊Dic3)⋊C3—22no—
40C2×(C3⋊Dic3)⋊C3—22no—
41C2×(C3⋊Dic3)⋊C3—22no—
42C2×(C3⋊Dic3)⋊C3—22no—
43C2×(C3⋊Dic3)⋊C3D414yes—
44#634⋊C3—28no—
45#5231—91no—
46#4947—91no—
47#4673—91no—
48C23×C3×D3—31no—
49C23×C3×D3—31no—
50C23×C3×D3—61no—
51C22×C32×C4—21no—
52C22×C32×C4C2×D311yescharacteristic
53C22×C32×C4C2×D311yescharacteristic
54C22×C3⋊Dic3C2×D312yes—
55C22×C3⋊Dic3C2×D312yes—
56C2×#634—22no—
57C2×#634—24no—
58C22×C3×Dic3—32no—
59C22×C3×Dic3—31no—
60C22×C3×Dic3—31no—
61C22×C3×Dic3—32no—
62C22×C3×Dic3—61no—
63C2×C3×Dic6—64no—
64C2×C3×Dic6—62no—
65C3×(C2×D3)⋊C4—64no—
66C3×(C2×D3)⋊C4—62no—
67C3×(C2×D3)⋊C4—62no—
68(C2×D3)⋊Dic3—64no—
69(C2×D3)⋊Dic3—64no—
70C22×H(3,𝔽3)C2×D411yescharacteristic
71C22×H(3,𝔽3)C2×D412yes—
72C22×H(3,𝔽3)C2×Q811yescharacteristic
73C22×H(3,𝔽3)#4112yes—
74C22×H(3,𝔽3)#4111yescharacteristic
75C2×#272—22no—
76C2×#272—24no—
77C2×#272—41no—
78C2×#272—41no—
79C2×#272—42no—
80C4×H(3,𝔽3)—22no—
81#1084—41no—
82(C3⋊Dic3)⋊C3—24no—
83(C3⋊Dic3)⋊C3—42no—
84C22×C3×Q8—91no—
85C22×Dic6—31no—
86C22×Dic6—91no—
87C2×C3×C22⋊C4—91no—
88C2×#217—92no—
89C2×(C2×D3)⋊C4—31no—
90C2×(C2×D3)⋊C4—31no—
91C2×(C2×D3)⋊C4—61no—
92C2×(C2×D3)⋊C4—92no—
93C2×(C2×D3)⋊C4—92no—
94C3×#95—91no—
95C3⋊#95—31no—
96C3⋊#95—91no—
97C3⋊#95—92no—
98C3⋊#95—92no—
99C3⋊#95—92no—
100C23×C32—21no—
101C23×C32C22×D311yescharacteristic
102C23×C32C22×D311yescharacteristic
103C22×C3×D3—32no—
104C22×C3×D3—32no—
105C22×C3×D3—62no—
106C22×C3×D3—62no—
107C22×C3×D3—62no—
108C22×C3×D3—61no—
109C22×C3×D3—62no—
110C22×C3×D3—62no—
111C22×C3×D3—61no—
112C22×C3×D3—61no—
113C22×C3×D3—62no—
114C22×C3×D3—62no—
115C22×C3×D3—61no—
116C22×C3×D3—122no—
117C22×C3×D3—122no—
118C2×C32×C4—22no—
119C2×C32×C4—22no—
120C2×C32×C4—22no—
121C2×C32×C4—42no—
122C2×C32×C4D1212yes—
123C2×C32×C4#6712yes—
124C2×C3⋊Dic3—22no—
125C2×C3⋊Dic3—22no—
126C2×C3⋊Dic3—22no—
127C2×C3⋊Dic3—22no—
128C2×C3⋊Dic3—22no—
129C2×C3⋊Dic3#6714yes—
130#634—28no—
131C2×C3×Dic3—34no—
132C2×C3×Dic3—62no—
133C2×C3×Dic3—62no—
134C2×C3×Dic3—62no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—62no—
137C2×C3×Dic3—62no—
138C2×C3×Dic3—62no—
139C2×C3×Dic3—62no—
140C2×C3×Dic3—62no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—122no—
143C3×Dic6—68no—
144#393—271noSylow
145C2×H(3,𝔽3)#12011yescharacteristic
146C2×H(3,𝔽3)C22⋊Q812yes—
147C2×H(3,𝔽3)C22⋊Q811yescharacteristic
148C2×H(3,𝔽3)#12411yescharacteristic
149C2×H(3,𝔽3)#12412yes—
150#272—42no—
151C24×C3—91no—
152C23×D3—31no—
153C23×D3—31no—
154C23×D3—61no—
155C2×C3×Q8—182no—
156C2×C3×Q8—184no—
157C22×C3×C4—31no—
158C22×C3×C4—31no—
159C22×C3×C4—61no—
160C22×C3×C4—92no—
161C22×C3×C4—92no—
162C22×C3×C4—91no—
163C22×C3×C4D3211yescharacteristic
164C22×Dic3—32no—
165C22×Dic3—31no—
166C22×Dic3—31no—
167C22×Dic3—32no—
168C22×Dic3—61no—
169C22×Dic3—92no—
170C22×Dic3—92no—
171C2×Dic6—64no—
172C2×Dic6—62no—
173C2×Dic6—184no—
174C2×Dic6—182no—
175C3×C22⋊C4—182no—
176#217—184no—
177(C2×D3)⋊C4—62no—
178(C2×D3)⋊C4—62no—
179(C2×D3)⋊C4—64no—
180(C2×D3)⋊C4—184no—
181(C2×D3)⋊C4—184no—
182C22×C32—21no—
183C22×C32—21no—
184C22×C32—22no—
185C22×C32—21no—
186C22×C32—22no—
187C22×C32C2×D1211yescharacteristic
188C22×C32#23512yes—
189C22×C32D3×D412yes—
190C22×C32#23711yescharacteristic
191C22×C32D3×Q811yescharacteristic
192C22×C32D3×Q811yescharacteristic
193C22×C32#23911yescharacteristic
194C22×C32#23912yes—
195C22×C32C2×#6711yescharacteristic
196C22×C32C2×#6712yes—
197C2×C3×D3—64no—
198C2×C3×D3—64no—
199C2×C3×D3—62no—
200C2×C3×D3—62no—
201C2×C3×D3—64no—
202C2×C3×D3—124no—
203C2×C3×D3—122no—
204C2×C3×D3—124no—
205C2×C3×D3—122no—
206C2×C3×D3—121no—
207C2×C3×D3—121no—
208C2×C3×D3—122no—
209C2×C3×D3—121no—
210C2×C3×D3—122no—
211C2×C3×D3—121no—
212C32×C4—22no—
213C32×C4—22no—
214C32×C4—42no—
215C3⋊Dic3—24no—
216C3⋊Dic3—42no—
217C3×Dic3—64no—
218C3×Dic3—122no—
219C3×Dic3—121no—
220C3×Dic3—122no—
221C3×Dic3—121no—
222C22×Q8—91no—
223C2×C22⋊C4—91no—
224C2×C22⋊C4—272no—
225#95—91no—
226#95—272no—
227H(3,𝔽3)#39311yescharacteristic, complemented, Sylow
228C23×C3—31no—
229C23×C3—31no—
230C23×C3—61no—
231C23×C3—92no—
232C23×C3—181no—
233C23×C3—182no—
234C23×C3—181no—
235C23×C3—182no—
236C23×C3C2×D3211yescharacteristic
237C22×D3—32no—
238C22×D3—32no—
239C22×D3—62no—
240C22×D3—61no—
241C22×D3—61no—
242C22×D3—62no—
243C22×D3—62no—
244C22×D3—61no—
245C22×D3—61no—
246C22×D3—62no—
247C22×D3—62no—
248C22×D3—62no—
249C22×D3—62no—
250C22×D3—122no—
251C22×D3—122no—
252C3×Q8—188no—
253C2×C3×C4—22no—
254C2×C3×C4—32no—
255C2×C3×C4—32no—
256C2×C3×C4—62no—
257C2×C3×C4—62no—
258C2×C3×C4—62no—
259C2×C3×C4—94no—
260C2×C3×C4—122no—
261C2×C3×C4—182no—
262C2×C3×C4—182no—
263C2×C3×C4—182no—
264C2×C3×C4—182no—
265C2×C3×C4—182no—
266C2×C3×C4—182no—
267C2×C3×C4—181no—
268C2×C3×C4#62612yes—
269C2×Dic3—34no—
270C2×Dic3—62no—
271C2×Dic3—62no—
272C2×Dic3—61no—
273C2×Dic3—61no—
274C2×Dic3—62no—
275C2×Dic3—62no—
276C2×Dic3—62no—
277C2×Dic3—62no—
278C2×Dic3—62no—
279C2×Dic3—62no—
280C2×Dic3—94no—
281C2×Dic3—122no—
282C2×Dic3—182no—
283C2×Dic3—182no—
284C2×Dic3—182no—
285C2×Dic3—182no—
286C2×Dic3—182no—
287Dic6—68no—
288Dic6—188no—
289C2×C32—21no—
290C2×C32—21no—
291C2×C32—22no—
292C2×C32—22no—
293C2×C32—21no—
294C2×C32#87511yescharacteristic
295C2×C32#88211yescharacteristic
296C2×C32#88512yes—
297C2×C32#89612yes—
298C2×C32#89711yescharacteristic
299C2×C32#94611yescharacteristic
300C2×C32#94612yes—
301C2×C32#94711yescharacteristic
302C2×C32#94712yes—
303C2×C32#95311yescharacteristic
304C3×D3—124no—
305C3×D3—122no—
306C3×D3—122no—
307C24—91no—
308C2×Q8—184no—
309C2×Q8—182no—
310C22×C4—92no—
311C22×C4—92no—
312C22×C4—91no—
313C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
314C22⋊C4—182no—
315C22⋊C4—544no—
316C22×C3—31no—
317C22×C3—31no—
318C22×C3—31no—
319C22×C3—31no—
320C22×C3—32no—
321C22×C3—32no—
322C22×C3—31no—
323C22×C3—31no—
324C22×C3—32no—
325C22×C3—32no—
326C22×C3—61no—
327C22×C3—61no—
328C22×C3—62no—
329C22×C3—61no—
330C22×C3—62no—
331C22×C3—182no—
332C22×C3—184no—
333C22×C3—361no—
334C22×C3—362no—
335C22×C3—361no—
336C22×C3#379611yescharacteristic
337C22×C3#379711yescharacteristic
338C22×C3#379912yes—
339C22×C3C2×#62611yescharacteristic
340C22×C3D3×#6712yes—
341C2×D3—62no—
342C2×D3—62no—
343C2×D3—64no—
344C2×D3—64no—
345C2×D3—64no—
346C2×D3—124no—
347C2×D3—121no—
348C2×D3—121no—
349C2×D3—121no—
350C2×D3—121no—
351C2×D3—122no—
352C2×D3—122no—
353C2×D3—122no—
354C2×D3—122no—
355C2×D3—124no—
356C3×C4—22no—
357C3×C4—62no—
358C3×C4—62no—
359C3×C4—122no—
360C3×C4—184no—
361C3×C4—362no—
362C3×C4—361no—
363Dic3—64no—
364Dic3—121no—
365Dic3—121no—
366Dic3—122no—
367Dic3—122no—
368Dic3—184no—
369Dic3—362no—
370C32—21no—
371C32#467311yescharacteristic, complemented
372C32#523111yescharacteristic, complemented
373C23—92no—
374C23—181no—
375C23—182no—
376C23—181no—
377C23—182no—
378C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
379Q8—188no—
380C2×C4—22no—
381C2×C4—94no—
382C2×C4—182no—
383C2×C4—182no—
384C2×C4—182no—
385C2×C4—182no—
386C2×C4—182no—
387C2×C4—181no—
388C2×C4—182no—
389C2×C4#620012yes—
390C2×C3—31no—
391C2×C3—31no—
392C2×C3—31no—
393C2×C3—31no—
394C2×C3—31no—
395C2×C3—31no—
396C2×C3—32no—
397C2×C3—32no—
398C2×C3—32no—
399C2×C3—32no—
400C2×C3—61no—
401C2×C3—61no—
402C2×C3—61no—
403C2×C3—62no—
404C2×C3—62no—
405C2×C3—362no—
406C2×C3#6386411yescharacteristic
407C2×C3#6387111yescharacteristic
408C2×C3#6388812yes—
409C2×C3#6390712yes—
410C2×C3#6397711yescharacteristic
411D3—122no—
412D3—122no—
413D3—124no—
414C22—182no—
415C22—184no—
416C22—361no—
417C22—361no—
418C22—362no—
419C22#8864911yescharacteristic
420C22#8865111yescharacteristic
421C22#8865212yes—
422C22C2×#620011yescharacteristic
423C22#8867612yes—
424C4—22no—
425C4—184no—
426C4—361no—
427C4—362no—
428C3—31no—
429C3—31no—
430C3—61no—
431C3#1059131011yescharacteristic
432C2—362no—
433C2#1172109411yescharacteristic
434C2#1172111411yescharacteristic
435C2#1172114812yes—
436C2#1172115512yes—
437C2#1172132111yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC23×H(3,𝔽3)C23
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC23×#272C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#393
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×#634⋊C3C2semidirect
4. #63658⋊C3C2semidirect
5. #11720334C2semidirect
17. C22×#634D3semidirect
24. #63658D3semidirect
33. C22×#272Q8semidirect
227. H(3,𝔽3)#393semidirect
371. C32#4673semidirect
372. C32#5231semidirect

Maximal subgroups

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

subgroupindex
C22×#634⋊C32
C2×#885422
C2×#884512
#63658⋊C32
#117203342
#105913223
#105913043