Finite Group Databasebeta

← #13140173

Subgroups — #13140173

Order 1728.

909 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 909 conjugacy classes of subgroups — too many to draw. Only its 127 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×#3750⋊C3C211yesmaximal, characteristic, complemented
3C2×#88450C212yesmaximal, complemented
4C2×#88450C212yesmaximal, complemented
5#11720574C211yesmaximal, characteristic, complemented
6#10591321—31nomaximal
7#10591302—31nomaximal
8C23×(C3⋊D3)⋊C3C2212yes—
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×#1084C2212yes—
11C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
12C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
13C2×C4⋊((C3⋊D3)⋊C3)—24no—
14C2×C4⋊((C3⋊D3)⋊C3)—22no—
15#3750⋊C3—22no—
16#88450—24no—
17#88450—24no—
18C22×C4⋊(C3⋊D3)D311yescharacteristic
19C22×C3×D12—31no—
20C2×#3750D311yescharacteristic, complemented
21C2×C3×Dic3⋊C4—31no—
22C2×#3722—32no—
23C2×#3722—32no—
24C2×(C2×D3)⋊Dic3—32no—
25C2×(C2×D3)⋊Dic3—32no—
26C3×C3⋊#95—31no—
27C3×C3⋊#95—31no—
28C3×C3⋊#95—61no—
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—22no—
40C2×#1084—22no—
41C2×#1084—22no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3—22no—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3D412yescomplemented
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×D311yescharacteristic
57C22×C3⋊Dic3C2×D311yescharacteristic
58C2×C4⋊(C3⋊D3)—24no—
59C2×C4⋊(C3⋊D3)—22no—
60C22×C3×Dic3—31no—
61C22×C3×Dic3—31no—
62C22×C3×Dic3—32no—
63C22×C3×Dic3—32no—
64C22×C3×Dic3—62no—
65C2×C3×D12—64no—
66C2×C3×D12—62no—
67#3750—22no—
68C3×Dic3⋊C4—62no—
69#3722—64no—
70#3722—64no—
71(C2×D3)⋊Dic3—64no—
72(C2×D3)⋊Dic3—64no—
73C22×H(3,𝔽3)C2×D412yes—
74C22×H(3,𝔽3)C2×D411yescharacteristic
75C22×H(3,𝔽3)C2×D411yescharacteristic
76C22×H(3,𝔽3)#4111yescharacteristic
77C22×H(3,𝔽3)#4112yes—
78C2×(C3⋊D3)⋊C3—24no—
79C2×(C3⋊D3)⋊C3—24no—
80C2×(C3⋊D3)⋊C3—24no—
81C2×(C3⋊D3)⋊C3—42no—
82C2×(C3⋊D3)⋊C3—42no—
83C2×(C3⋊D3)⋊C3—42no—
84C2×(C3⋊D3)⋊C3—42no—
85C4×H(3,𝔽3)—22no—
86#1084—42no—
87(C3⋊Dic3)⋊C3—22no—
88(C3⋊Dic3)⋊C3—41no—
89C22×C3×D4—91no—
90C22×D12—31no—
91C22×D12—91no—
92C2×C3×C4⋊C4—91no—
93C2×#217—92no—
94C2×#217—92no—
95C2×(C2×D3)⋊C4—92no—
96C2×(C2×D3)⋊C4—92no—
97C2×(C2×D3)⋊C4—92no—
98C2×(C2×D3)⋊C4—92no—
99C2×Dic3⋊C4—31no—
100C2×Dic3⋊C4—91no—
101C3×#95—91no—
102C3⋊#95—31no—
103C3⋊#95—31no—
104C3⋊#95—61no—
105C23×C32—21no—
106C23×C32C22×D311yescharacteristic
107C23×C32C22×D311yescharacteristic
108C22×C3⋊D3—22no—
109C22×C3⋊D3—22no—
110C22×C3⋊D3—22no—
111C22×C3⋊D3—22no—
112C22×C3⋊D3—22no—
113C22×C3⋊D3—22no—
114C22×C3⋊D3#6714yes—
115C22×C3×D3—34no—
116C22×C3×D3—62no—
117C22×C3×D3—62no—
118C22×C3×D3—62no—
119C22×C3×D3—62no—
120C22×C3×D3—62no—
121C22×C3×D3—62no—
122C2×C32×C4—22no—
123C2×C32×C4—22no—
124C2×C32×C4—22no—
125C2×C32×C4—42no—
126C2×C32×C4#6712yes—
127C2×C32×C4#6712yes—
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3—21no—
130C2×C3⋊Dic3—22no—
131C2×C3⋊Dic3—21no—
132C2×C3⋊Dic3D1212yescomplemented
133C4⋊(C3⋊D3)—28no—
134C2×C3×Dic3—32no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—61no—
137C2×C3×Dic3—64no—
138C2×C3×Dic3—62no—
139C2×C3×Dic3—62no—
140C2×C3×Dic3—62no—
141C2×C3×Dic3—62no—
142C2×C3×Dic3—62no—
143C2×C3×Dic3—62no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—62no—
146C2×C3×Dic3—122no—
147C2×C3×Dic3—122no—
148C3×D12—68no—
149#394—271noSylow
150C2×H(3,𝔽3)#12011yescharacteristic
151C2×H(3,𝔽3)#12112yes—
152C2×H(3,𝔽3)#12111yescharacteristic
153C2×H(3,𝔽3)#12312yes—
154C2×H(3,𝔽3)#12411yescharacteristic
155(C3⋊D3)⋊C3—44no—
156C24×C3—92no—
157C23×D3—32no—
158C23×D3—92no—
159C2×C3×D4—182no—
160C2×C3×D4—184no—
161C22×C3×C4—31no—
162C22×C3×C4—31no—
163C22×C3×C4—61no—
164C22×C3×C4—91no—
165C22×C3×C4—91no—
166C22×C3×C4—92no—
167C22×C3×C4D3211yescharacteristic
168C22×Dic3—31no—
169C22×Dic3—32no—
170C22×Dic3—32no—
171C22×Dic3—31no—
172C22×Dic3—62no—
173C22×Dic3—91no—
174C22×Dic3—91no—
175C2×D12—64no—
176C2×D12—62no—
177C2×D12—182no—
178C2×D12—184no—
179C3×C4⋊C4—182no—
180#217—184no—
181#217—184no—
182(C2×D3)⋊C4—184no—
183(C2×D3)⋊C4—184no—
184(C2×D3)⋊C4—184no—
185(C2×D3)⋊C4—184no—
186Dic3⋊C4—62no—
187Dic3⋊C4—182no—
188C22×C32—21no—
189C22×C32—22no—
190C22×C32—21no—
191C22×C32—21no—
192C22×C32—22no—
193C22×C32C2×D1211yescharacteristic
194C22×C32#23511yescharacteristic
195C22×C32D3×D411yescharacteristic
196C22×C32D3×D412yes—
197C22×C32#23711yescharacteristic
198C22×C32#23712yes—
199C22×C32#23912yes—
200C22×C32C2×#6711yescharacteristic
201C22×C32C2×#6711yescharacteristic
202C22×C32C2×#6712yes—
203C2×C3⋊D3—24no—
204C2×C3⋊D3—24no—
205C2×C3⋊D3—24no—
206C2×C3⋊D3—42no—
207C2×C3⋊D3—42no—
208C2×C3⋊D3—42no—
209C2×C3⋊D3—42no—
210C2×C3×D3—64no—
211C2×C3×D3—64no—
212C2×C3×D3—64no—
213C2×C3×D3—122no—
214C2×C3×D3—122no—
215C2×C3×D3—122no—
216C2×C3×D3—122no—
217C32×C4—22no—
218C32×C4—22no—
219C32×C4—42no—
220C3⋊Dic3—22no—
221C3⋊Dic3—41no—
222C3×Dic3—62no—
223C3×Dic3—124no—
224C3×Dic3—122no—
225C3×Dic3—122no—
226C3×Dic3—121no—
227C22×D4—91no—
228C2×C4⋊C4—91no—
229C2×C22⋊C4—272no—
230C2×C22⋊C4—272no—
231#95—91no—
232H(3,𝔽3)#39411yescharacteristic, complemented, Sylow
233C23×C3—31no—
234C23×C3—31no—
235C23×C3—61no—
236C23×C3—94no—
237C23×C3—182no—
238C23×C3—182no—
239C23×C3—182no—
240C23×C3—182no—
241C23×C3—182no—
242C23×C3—182no—
243C23×C3C2×D3211yescharacteristic
244C22×D3—34no—
245C22×D3—62no—
246C22×D3—62no—
247C22×D3—62no—
248C22×D3—62no—
249C22×D3—62no—
250C22×D3—62no—
251C22×D3—94no—
252C22×D3—182no—
253C22×D3—182no—
254C22×D3—182no—
255C22×D3—182no—
256C22×D3—182no—
257C22×D3—182no—
258C3×D4—188no—
259C2×C3×C4—22no—
260C2×C3×C4—32no—
261C2×C3×C4—32no—
262C2×C3×C4—62no—
263C2×C3×C4—62no—
264C2×C3×C4—62no—
265C2×C3×C4—92no—
266C2×C3×C4—122no—
267C2×C3×C4—181no—
268C2×C3×C4—181no—
269C2×C3×C4—182no—
270C2×C3×C4—181no—
271C2×C3×C4—182no—
272C2×C3×C4—182no—
273C2×C3×C4—182no—
274C2×C3×C4#62512yes—
275C2×Dic3—32no—
276C2×Dic3—61no—
277C2×Dic3—61no—
278C2×Dic3—62no—
279C2×Dic3—62no—
280C2×Dic3—62no—
281C2×Dic3—62no—
282C2×Dic3—62no—
283C2×Dic3—62no—
284C2×Dic3—64no—
285C2×Dic3—61no—
286C2×Dic3—62no—
287C2×Dic3—92no—
288C2×Dic3—122no—
289C2×Dic3—122no—
290C2×Dic3—181no—
291C2×Dic3—181no—
292C2×Dic3—181no—
293C2×Dic3—182no—
294D12—68no—
295D12—188no—
296C2×C32—21no—
297C2×C32—22no—
298C2×C32—22no—
299C2×C32—21no—
300C2×C32—21no—
301C2×C32#89412yes—
302C2×C32#89512yes—
303C2×C32#93011yescharacteristic
304C2×C32#93512yes—
305C2×C32#93611yescharacteristic
306C2×C32#93711yescharacteristic
307C2×C32#93811yescharacteristic
308C2×C32#93912yes—
309C2×C32#94711yescharacteristic
310C2×C32#95311yescharacteristic
311C3⋊D3—44no—
312C3×D3—124no—
313C24—92no—
314C2×D4—184no—
315C2×D4—182no—
316C22×C4—91no—
317C22×C4—92no—
318C22×C4—91no—
319C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
320C4⋊C4—182no—
321C22⋊C4—544no—
322C22⋊C4—544no—
323C22×C3—31no—
324C22×C3—31no—
325C22×C3—32no—
326C22×C3—32no—
327C22×C3—31no—
328C22×C3—31no—
329C22×C3—31no—
330C22×C3—31no—
331C22×C3—32no—
332C22×C3—32no—
333C22×C3—61no—
334C22×C3—62no—
335C22×C3—61no—
336C22×C3—61no—
337C22×C3—62no—
338C22×C3—184no—
339C22×C3—184no—
340C22×C3—184no—
341C22×C3—362no—
342C22×C3—362no—
343C22×C3—362no—
344C22×C3—362no—
345C22×C3#379511yescharacteristic
346C22×C3#379612yes—
347C22×C3D3×D1211yescharacteristic
348C22×C3C2×#62511yescharacteristic
349C22×C3D3×#6712yes—
350C2×D3—64no—
351C2×D3—64no—
352C2×D3—64no—
353C2×D3—122no—
354C2×D3—122no—
355C2×D3—122no—
356C2×D3—122no—
357C2×D3—184no—
358C2×D3—184no—
359C2×D3—184no—
360C2×D3—362no—
361C2×D3—362no—
362C2×D3—362no—
363C2×D3—362no—
364C3×C4—22no—
365C3×C4—62no—
366C3×C4—62no—
367C3×C4—122no—
368C3×C4—182no—
369C3×C4—361no—
370C3×C4—362no—
371Dic3—62no—
372Dic3—122no—
373Dic3—122no—
374Dic3—124no—
375Dic3—121no—
376Dic3—182no—
377Dic3—361no—
378C32—21no—
379C32#499111yescharacteristic, complemented
380C32#522211yescharacteristic, complemented
381C23—94no—
382C23—182no—
383C23—182no—
384C23—182no—
385C23—182no—
386C23—182no—
387C23—182no—
388C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
389D4—188no—
390C2×C4—22no—
391C2×C4—92no—
392C2×C4—181no—
393C2×C4—181no—
394C2×C4—182no—
395C2×C4—182no—
396C2×C4—181no—
397C2×C4—182no—
398C2×C4—182no—
399C2×C4#619812yes—
400C2×C3—31no—
401C2×C3—31no—
402C2×C3—31no—
403C2×C3—31no—
404C2×C3—32no—
405C2×C3—32no—
406C2×C3—32no—
407C2×C3—32no—
408C2×C3—31no—
409C2×C3—31no—
410C2×C3—61no—
411C2×C3—61no—
412C2×C3—62no—
413C2×C3—62no—
414C2×C3—61no—
415C2×C3—364no—
416C2×C3#6386311yescharacteristic
417C2×C3#6388412yes—
418C2×C3#6388612yes—
419C2×C3#6390811yescharacteristic
420C2×C3#6397611yescharacteristic
421D3—124no—
422D3—364no—
423C22—184no—
424C22—184no—
425C22—184no—
426C22—362no—
427C22—362no—
428C22—362no—
429C22—362no—
430C22#8864811yescharacteristic
431C22#8864912yes—
432C22#8865411yescharacteristic
433C22C2×#619811yescharacteristic
434C22#8867612yes—
435C4—22no—
436C4—182no—
437C4—361no—
438C4—362no—
439C3—31no—
440C3—31no—
441C3—61no—
442C3#1059130011yescharacteristic
443C2—364no—
444C2#1172111311yescharacteristic
445C2#1172112512yes—
446C2#1172112712yes—
447C2#1172114111yescharacteristic
448C2#1172131611yescharacteristic
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×#3750⋊C3C2semidirect
3. C2×#88450C2semidirect
4. C2×#88450C2semidirect
5. #11720574C2semidirect
20. C2×#3750D3semidirect
46. C2×(C3⋊Dic3)⋊C3D4semidirect
132. C2×C3⋊Dic3D12semidirect
232. H(3,𝔽3)#394semidirect
379. C32#4991semidirect
380. 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×#3750⋊C32
C2×#884502
#117205742
#105913213
#105913023