Finite Group Databasebeta

← #13140151

Subgroups — #13140151

Order 1728.

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

normal subgroups only

This group has 925 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×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
2C2×#88540C211yesmaximal, characteristic, complemented
3C2×#88450C212yesmaximal, complemented
4C2×#88450C212yesmaximal, complemented
5#63658⋊C3C211yesmaximal, characteristic, complemented
6#10591321—31nomaximal
7#10591300—31nomaximal
8C23×(C3⋊D3)⋊C3C2212yes—
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×#1084C2211yescharacteristic
11C22×#1084C2211yescharacteristic
12C22×(C3⋊Dic3)⋊C3C2212yes—
13C2×C4⋊((C3⋊D3)⋊C3)—24no—
14C2×C4⋊((C3⋊D3)⋊C3)—22no—
15#88540—22no—
16#88450—24no—
17#88450—24no—
18C22×C4⋊(C3⋊D3)D311yescharacteristic
19C22×C3×D12—31no—
20C2×C3×Dic3⋊C4—31no—
21C2×C3×Dic3⋊C4—31no—
22C2×C3×Dic3⋊C4—61no—
23C2×#3722—32no—
24C2×#3722—32no—
25C2×(C2×D3)⋊Dic3—32no—
26C2×(C2×D3)⋊Dic3—32no—
27#63658D311yescharacteristic, complemented
28C3×C3⋊#95—31no—
29C23×H(3,𝔽3)C2311yescharacteristic
30C22×(C3⋊D3)⋊C3—22no—
31C22×(C3⋊D3)⋊C3—22no—
32C22×(C3⋊D3)⋊C3—22no—
33C22×(C3⋊D3)⋊C3—22no—
34C22×(C3⋊D3)⋊C3—22no—
35C22×(C3⋊D3)⋊C3—22no—
36C22×(C3⋊D3)⋊C3D414yes—
37C2×C4×H(3,𝔽3)—22no—
38C2×C4×H(3,𝔽3)D412yes—
39C2×#1084—21no—
40C2×#1084—21no—
41C2×#1084—21no—
42C2×#1084—22no—
43C2×#1084D412yescomplemented
44C2×(C3⋊Dic3)⋊C3—22no—
45C2×(C3⋊Dic3)⋊C3—22no—
46C2×(C3⋊Dic3)⋊C3—22no—
47C4⋊((C3⋊D3)⋊C3)—28no—
48#5222—91no—
49#4991—91no—
50#4672—91no—
51C23×C3⋊D3C2×D312yes—
52C23×C3×D3—32no—
53C22×C32×C4—21no—
54C22×C32×C4C2×D311yescharacteristic
55C22×C32×C4C2×D311yescharacteristic
56C22×C3⋊Dic3C2×D312yes—
57C2×C4⋊(C3⋊D3)—24no—
58C2×C4⋊(C3⋊D3)—22no—
59C22×C3×Dic3—31no—
60C22×C3×Dic3—31no—
61C22×C3×Dic3—31no—
62C22×C3×Dic3—31no—
63C22×C3×Dic3—32no—
64C22×C3×Dic3—61no—
65C22×C3×Dic3—61no—
66C2×C3×D12—64no—
67C2×C3×D12—62no—
68C3×Dic3⋊C4—64no—
69C3×Dic3⋊C4—62no—
70C3×Dic3⋊C4—62no—
71#3722—64no—
72#3722—64no—
73(C2×D3)⋊Dic3—64no—
74(C2×D3)⋊Dic3—64no—
75C22×H(3,𝔽3)C2×D412yes—
76C22×H(3,𝔽3)C2×D411yescharacteristic
77C22×H(3,𝔽3)C2×D411yescharacteristic
78C22×H(3,𝔽3)#4111yescharacteristic
79C22×H(3,𝔽3)#4112yes—
80C2×(C3⋊D3)⋊C3—24no—
81C2×(C3⋊D3)⋊C3—24no—
82C2×(C3⋊D3)⋊C3—24no—
83C2×(C3⋊D3)⋊C3—42no—
84C2×(C3⋊D3)⋊C3—42no—
85C2×(C3⋊D3)⋊C3—42no—
86C2×(C3⋊D3)⋊C3—42no—
87C4×H(3,𝔽3)—22no—
88#1084—22no—
89#1084—41no—
90(C3⋊Dic3)⋊C3—42no—
91C22×C3×D4—91no—
92C22×D12—31no—
93C22×D12—91no—
94C2×C3×C4⋊C4—91no—
95C2×#217—92no—
96C2×#217—92no—
97C2×(C2×D3)⋊C4—92no—
98C2×(C2×D3)⋊C4—92no—
99C2×(C2×D3)⋊C4—92no—
100C2×(C2×D3)⋊C4—92no—
101C2×Dic3⋊C4—31no—
102C2×Dic3⋊C4—31no—
103C2×Dic3⋊C4—61no—
104C3×#95—91no—
105C3⋊#95—31no—
106C3⋊#95—91no—
107C23×C32—21no—
108C23×C32C22×D311yescharacteristic
109C23×C32C22×D311yescharacteristic
110C22×C3⋊D3—22no—
111C22×C3⋊D3—22no—
112C22×C3⋊D3—22no—
113C22×C3⋊D3—22no—
114C22×C3⋊D3—22no—
115C22×C3⋊D3—22no—
116C22×C3⋊D3#6714yes—
117C22×C3×D3—34no—
118C22×C3×D3—62no—
119C22×C3×D3—62no—
120C22×C3×D3—62no—
121C22×C3×D3—62no—
122C22×C3×D3—62no—
123C22×C3×D3—62no—
124C2×C32×C4—22no—
125C2×C32×C4—22no—
126C2×C32×C4—22no—
127C2×C32×C4—42no—
128C2×C32×C4D1212yes—
129C2×C32×C4#6712yes—
130C2×C3⋊Dic3—22no—
131C2×C3⋊Dic3—22no—
132C2×C3⋊Dic3—22no—
133C4⋊(C3⋊D3)—28no—
134C2×C3×Dic3—32no—
135C2×C3×Dic3—32no—
136C2×C3×Dic3—62no—
137C2×C3×Dic3—62no—
138C2×C3×Dic3—62no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—62no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—62no—
146C2×C3×Dic3—62no—
147C2×C3×Dic3—62no—
148C2×C3×Dic3—62no—
149C2×C3×Dic3—62no—
150C2×C3×Dic3—61no—
151C2×C3×Dic3—122no—
152C3×D12—68no—
153#394—271noSylow
154C2×H(3,𝔽3)#12011yescharacteristic
155C2×H(3,𝔽3)#12112yes—
156C2×H(3,𝔽3)#12111yescharacteristic
157C2×H(3,𝔽3)#12312yes—
158C2×H(3,𝔽3)#12411yescharacteristic
159(C3⋊D3)⋊C3—44no—
160C24×C3—92no—
161C23×D3—32no—
162C23×D3—92no—
163C2×C3×D4—182no—
164C2×C3×D4—184no—
165C22×C3×C4—31no—
166C22×C3×C4—31no—
167C22×C3×C4—61no—
168C22×C3×C4—91no—
169C22×C3×C4—92no—
170C22×C3×C4—91no—
171C22×C3×C4D3211yescharacteristic
172C22×Dic3—31no—
173C22×Dic3—31no—
174C22×Dic3—31no—
175C22×Dic3—31no—
176C22×Dic3—32no—
177C22×Dic3—61no—
178C22×Dic3—61no—
179C22×Dic3—92no—
180C2×D12—64no—
181C2×D12—62no—
182C2×D12—182no—
183C2×D12—184no—
184C3×C4⋊C4—182no—
185#217—184no—
186#217—184no—
187(C2×D3)⋊C4—184no—
188(C2×D3)⋊C4—184no—
189(C2×D3)⋊C4—184no—
190(C2×D3)⋊C4—184no—
191Dic3⋊C4—62no—
192Dic3⋊C4—62no—
193Dic3⋊C4—64no—
194C22×C32—22no—
195C22×C32—21no—
196C22×C32—21no—
197C22×C32—21no—
198C22×C32—22no—
199C22×C32C2×D1211yescharacteristic
200C22×C32#23512yes—
201C22×C32D3×D411yescharacteristic
202C22×C32D3×D411yescharacteristic
203C22×C32D3×D412yes—
204C22×C32#23712yes—
205C22×C32#23711yescharacteristic
206C22×C32#23911yescharacteristic
207C22×C32C2×#6711yescharacteristic
208C22×C32C2×#6712yes—
209C2×C3⋊D3—24no—
210C2×C3⋊D3—24no—
211C2×C3⋊D3—24no—
212C2×C3⋊D3—42no—
213C2×C3⋊D3—42no—
214C2×C3⋊D3—42no—
215C2×C3⋊D3—42no—
216C2×C3×D3—64no—
217C2×C3×D3—64no—
218C2×C3×D3—64no—
219C2×C3×D3—122no—
220C2×C3×D3—122no—
221C2×C3×D3—122no—
222C2×C3×D3—122no—
223C32×C4—22no—
224C32×C4—22no—
225C32×C4—42no—
226C3⋊Dic3—42no—
227C3×Dic3—64no—
228C3×Dic3—62no—
229C3×Dic3—62no—
230C3×Dic3—122no—
231C3×Dic3—121no—
232C3×Dic3—122no—
233C3×Dic3—121no—
234C22×D4—91no—
235C2×C4⋊C4—91no—
236C2×C22⋊C4—272no—
237C2×C22⋊C4—272no—
238#95—91no—
239H(3,𝔽3)#39411yescharacteristic, complemented, Sylow
240C23×C3—31no—
241C23×C3—31no—
242C23×C3—61no—
243C23×C3—94no—
244C23×C3—182no—
245C23×C3—182no—
246C23×C3—182no—
247C23×C3—182no—
248C23×C3—182no—
249C23×C3—182no—
250C23×C3C2×D3211yescharacteristic
251C22×D3—34no—
252C22×D3—62no—
253C22×D3—62no—
254C22×D3—62no—
255C22×D3—62no—
256C22×D3—62no—
257C22×D3—62no—
258C22×D3—94no—
259C22×D3—182no—
260C22×D3—182no—
261C22×D3—182no—
262C22×D3—182no—
263C22×D3—182no—
264C22×D3—182no—
265C3×D4—188no—
266C2×C3×C4—22no—
267C2×C3×C4—32no—
268C2×C3×C4—32no—
269C2×C3×C4—62no—
270C2×C3×C4—62no—
271C2×C3×C4—62no—
272C2×C3×C4—92no—
273C2×C3×C4—122no—
274C2×C3×C4—181no—
275C2×C3×C4—181no—
276C2×C3×C4—182no—
277C2×C3×C4—182no—
278C2×C3×C4—182no—
279C2×C3×C4—181no—
280C2×C3×C4—182no—
281C2×C3×C4#62612yes—
282C2×Dic3—32no—
283C2×Dic3—32no—
284C2×Dic3—62no—
285C2×Dic3—61no—
286C2×Dic3—61no—
287C2×Dic3—61no—
288C2×Dic3—61no—
289C2×Dic3—62no—
290C2×Dic3—62no—
291C2×Dic3—61no—
292C2×Dic3—61no—
293C2×Dic3—62no—
294C2×Dic3—62no—
295C2×Dic3—62no—
296C2×Dic3—62no—
297C2×Dic3—62no—
298C2×Dic3—62no—
299C2×Dic3—122no—
300C2×Dic3—182no—
301C2×Dic3—182no—
302C2×Dic3—182no—
303D12—68no—
304D12—188no—
305C2×C32—21no—
306C2×C32—22no—
307C2×C32—22no—
308C2×C32—21no—
309C2×C32—21no—
310C2×C32#87511yescharacteristic
311C2×C32#88211yescharacteristic
312C2×C32#88412yes—
313C2×C32#89412yes—
314C2×C32#89511yescharacteristic
315C2×C32#93512yes—
316C2×C32#93611yescharacteristic
317C2×C32#93811yescharacteristic
318C2×C32#93912yes—
319C2×C32#95311yescharacteristic
320C3⋊D3—44no—
321C3×D3—124no—
322C24—92no—
323C2×D4—184no—
324C2×D4—182no—
325C22×C4—91no—
326C22×C4—92no—
327C22×C4—91no—
328C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
329C4⋊C4—182no—
330C22⋊C4—544no—
331C22⋊C4—544no—
332C22×C3—32no—
333C22×C3—32no—
334C22×C3—31no—
335C22×C3—31no—
336C22×C3—31no—
337C22×C3—31no—
338C22×C3—31no—
339C22×C3—31no—
340C22×C3—32no—
341C22×C3—32no—
342C22×C3—62no—
343C22×C3—61no—
344C22×C3—61no—
345C22×C3—61no—
346C22×C3—62no—
347C22×C3—184no—
348C22×C3—184no—
349C22×C3—184no—
350C22×C3—362no—
351C22×C3—362no—
352C22×C3—362no—
353C22×C3—362no—
354C22×C3#379512yes—
355C22×C3#379611yescharacteristic
356C22×C3#380211yescharacteristic
357C22×C3C2×#62611yescharacteristic
358C22×C3D3×#6712yes—
359C2×D3—64no—
360C2×D3—64no—
361C2×D3—64no—
362C2×D3—122no—
363C2×D3—122no—
364C2×D3—122no—
365C2×D3—122no—
366C2×D3—184no—
367C2×D3—184no—
368C2×D3—184no—
369C2×D3—362no—
370C2×D3—362no—
371C2×D3—362no—
372C2×D3—362no—
373C3×C4—22no—
374C3×C4—62no—
375C3×C4—62no—
376C3×C4—122no—
377C3×C4—182no—
378C3×C4—362no—
379C3×C4—361no—
380Dic3—62no—
381Dic3—62no—
382Dic3—64no—
383Dic3—121no—
384Dic3—121no—
385Dic3—122no—
386Dic3—122no—
387Dic3—362no—
388C32—21no—
389C32#467211yescharacteristic, complemented
390C32#522211yescharacteristic, complemented
391C23—94no—
392C23—182no—
393C23—182no—
394C23—182no—
395C23—182no—
396C23—182no—
397C23—182no—
398C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
399D4—188no—
400C2×C4—22no—
401C2×C4—92no—
402C2×C4—181no—
403C2×C4—181no—
404C2×C4—182no—
405C2×C4—182no—
406C2×C4—181no—
407C2×C4—182no—
408C2×C4—182no—
409C2×C4#620012yes—
410C2×C3—31no—
411C2×C3—31no—
412C2×C3—31no—
413C2×C3—31no—
414C2×C3—32no—
415C2×C3—32no—
416C2×C3—32no—
417C2×C3—32no—
418C2×C3—31no—
419C2×C3—31no—
420C2×C3—61no—
421C2×C3—61no—
422C2×C3—62no—
423C2×C3—62no—
424C2×C3—61no—
425C2×C3—364no—
426C2×C3#6386011yescharacteristic
427C2×C3#6388412yes—
428C2×C3#6388512yes—
429C2×C3#6391611yescharacteristic
430C2×C3#6397711yescharacteristic
431D3—124no—
432D3—364no—
433C22—184no—
434C22—184no—
435C22—184no—
436C22—362no—
437C22—362no—
438C22—362no—
439C22—362no—
440C22#8864812yes—
441C22#8864911yescharacteristic
442C22#8865511yescharacteristic
443C22C2×#620011yescharacteristic
444C22#8867612yes—
445C4—22no—
446C4—182no—
447C4—361no—
448C4—362no—
449C3—31no—
450C3—31no—
451C3—61no—
452C3#1059130211yescharacteristic
453C2—364no—
454C2#1172111011yescharacteristic
455C2#1172112512yes—
456C2#1172112612yes—
457C2#1172115911yescharacteristic
458C2#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 radicalC22×C4⋊((C3⋊D3)⋊C3)C2
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#394
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
2. C2×#88540C2semidirect
3. C2×#88450C2semidirect
4. C2×#88450C2semidirect
5. #63658⋊C3C2semidirect
27. #63658D3semidirect
43. C2×#1084D4semidirect
239. H(3,𝔽3)#394semidirect
389. C32#4672semidirect
390. C32#5222semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊((C3⋊D3)⋊C3)2
C2×#885402
C2×#884502
#63658⋊C32
#105913213
#105913003