Finite Group Databasebeta

← #13140241

Subgroups — #13140241

Order 1728.

1011 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 1011 conjugacy classes of subgroups — too many to draw. Only its 139 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×#6198C211yesmaximal, characteristic
2C2×C4×#1084C211yesmaximal, characteristic, complemented
3C2×#3752⋊C3C212yesmaximal, complemented
4C2×#88450C212yesmaximal, complemented
5C2×#88449C211yesmaximal, characteristic, complemented
6#10591383—32nomaximal
7C23×(C3⋊D3)⋊C3C2212yes—
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×#1084C2211yescharacteristic
10C22×#1084C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3C2212yes—
12C2×#6198—22no—
13C2×#6198—22no—
14C2×#6198C414yescomplemented
15C4×#1084—22no—
16#3752⋊C3—24no—
17#88450—24no—
18#88449—22no—
19C22×#626—32no—
20C2×#3752D312yescomplemented
21C2×C3×(C2×D3)⋊C4—32no—
22C2×C3×C4×Dic3—32no—
23C2×C3×C4×Dic3—61no—
24C2×Dic3⋊Dic3—32no—
25C2×#3722—32no—
26C2×(C2×D3)⋊Dic3—32no—
27C23×H(3,𝔽3)C2311yescharacteristic
28C22×(C3⋊D3)⋊C3—22no—
29C22×(C3⋊D3)⋊C3—22no—
30C22×(C3⋊D3)⋊C3—22no—
31C22×(C3⋊D3)⋊C3—22no—
32C22×(C3⋊D3)⋊C3—22no—
33C22×(C3⋊D3)⋊C3—22no—
34C22×(C3⋊D3)⋊C3C2×C414yes—
35C2×C4×H(3,𝔽3)—21no—
36C2×C4×H(3,𝔽3)—21no—
37C2×C4×H(3,𝔽3)D412yescomplemented
38C2×#1084—21no—
39C2×#1084—21no—
40C2×#1084C2×C412yes—
41C2×#1084D412yes—
42C2×#1084D412yes—
43C2×#1084D412yescomplemented
44C2×(C3⋊Dic3)⋊C3—22no—
45C2×(C3⋊Dic3)⋊C3—22no—
46C2×(C3⋊Dic3)⋊C3—22no—
47#6198—28no—
48#4957—92no—
49#4939—91no—
50C23×C3⋊D3C2×D312yes—
51C23×C3×D3—32no—
52C22×C32×C4—21no—
53C22×C32×C4C2×D312yes—
54C22×C3⋊Dic3C2×D312yes—
55C22×C3×Dic3—32no—
56C22×C3×Dic3—32no—
57C22×C3×Dic3—32no—
58C22×C3×Dic3—61no—
59C22×C3×Dic3—61no—
60C2×#626—38no—
61C2×#626—64no—
62C2×#626—64no—
63#3752—24no—
64C3×(C2×D3)⋊C4—64no—
65C3×C4×Dic3—64no—
66C3×C4×Dic3—64no—
67Dic3⋊Dic3—64no—
68#3722—64no—
69(C2×D3)⋊Dic3—64no—
70C22×H(3,𝔽3)C22×C411yescharacteristic
71C22×H(3,𝔽3)C2×D411yescharacteristic
72C22×H(3,𝔽3)C2×D411yescharacteristic
73C22×H(3,𝔽3)C2×D411yescharacteristic
74C22×H(3,𝔽3)C2×D411yescharacteristic
75C22×H(3,𝔽3)#4111yescharacteristic
76C22×H(3,𝔽3)#4111yescharacteristic
77C2×(C3⋊D3)⋊C3—24no—
78C2×(C3⋊D3)⋊C3—24no—
79C2×(C3⋊D3)⋊C3—24no—
80C2×(C3⋊D3)⋊C3—42no—
81C2×(C3⋊D3)⋊C3—42no—
82C2×(C3⋊D3)⋊C3—42no—
83C2×(C3⋊D3)⋊C3—42no—
84C4×H(3,𝔽3)—22no—
85#1084—22no—
86#1084C22⋊C414yescomplemented
87(C3⋊Dic3)⋊C3—42no—
88C22×#67—92no—
89C22×D12—91no—
90C2×C3×C22⋊C4—92no—
91C2×C3×C42—91no—
92C2×#217—92no—
93C2×(C2×D3)⋊C4—32no—
94C2×(C2×D3)⋊C4—92no—
95C2×(C2×D3)⋊C4—92no—
96C2×(C2×D3)⋊C4—92no—
97C2×C4⋊Dic3—91no—
98C2×Dic3⋊C4—92no—
99C2×C4×Dic3—32no—
100C2×C4×Dic3—61no—
101C23×C32—21no—
102C23×C32C22×D312yes—
103C22×C3⋊D3—22no—
104C22×C3⋊D3—22no—
105C22×C3⋊D3—22no—
106C22×C3⋊D3—22no—
107C22×C3⋊D3—22no—
108C22×C3⋊D3—22no—
109C22×C3⋊D3C4×D314yes—
110C22×C3×D3—34no—
111C22×C3×D3—62no—
112C22×C3×D3—62no—
113C22×C3×D3—62no—
114C22×C3×D3—62no—
115C22×C3×D3—62no—
116C22×C3×D3—62no—
117C2×C32×C4—22no—
118C2×C32×C4—22no—
119C2×C32×C4—22no—
120C2×C32×C4—22no—
121C2×C32×C4—22no—
122C2×C32×C4#6714yescomplemented
123C2×C3⋊Dic3—22no—
124C2×C3⋊Dic3—22no—
125C2×C3⋊Dic3—22no—
126C2×C3×Dic3—34no—
127C2×C3×Dic3—34no—
128C2×C3×Dic3—34no—
129C2×C3×Dic3—34no—
130C2×C3×Dic3—62no—
131C2×C3×Dic3—62no—
132C2×C3×Dic3—62no—
133C2×C3×Dic3—62no—
134C2×C3×Dic3—62no—
135C2×C3×Dic3—62no—
136C2×C3×Dic3—62no—
137C2×C3×Dic3—62no—
138C2×C3×Dic3—62no—
139C2×C3×Dic3—62no—
140C2×C3×Dic3—62no—
141#626—616no—
142#390—271noSylow
143C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
144C2×H(3,𝔽3)C4×D411yescharacteristic
145C2×H(3,𝔽3)C4×D411yescharacteristic
146C2×H(3,𝔽3)#12111yescharacteristic
147C2×H(3,𝔽3)#12111yescharacteristic
148C2×H(3,𝔽3)#12411yescharacteristic
149C2×H(3,𝔽3)#12711yescharacteristic
150(C3⋊D3)⋊C3—44no—
151C24×C3—92no—
152C23×D3—32no—
153C23×D3—92no—
154C22×C3×C4—32no—
155C22×C3×C4—61no—
156C22×C3×C4—91no—
157C22×C3×C4—91no—
158C22×C3×C4—92no—
159C22×C3×C4D3211yescharacteristic
160C2×#67—98no—
161C2×#67—184no—
162C2×#67—184no—
163C22×Dic3—32no—
164C22×Dic3—32no—
165C22×Dic3—32no—
166C22×Dic3—61no—
167C22×Dic3—61no—
168C22×Dic3—92no—
169C2×D12—94no—
170C2×D12—182no—
171C2×D12—182no—
172C3×C22⋊C4—184no—
173C3×C42—182no—
174#217—184no—
175(C2×D3)⋊C4—64no—
176(C2×D3)⋊C4—184no—
177(C2×D3)⋊C4—184no—
178(C2×D3)⋊C4—184no—
179C4⋊Dic3—182no—
180Dic3⋊C4—184no—
181C4×Dic3—64no—
182C4×Dic3—64no—
183C22×C32—21no—
184C22×C32—21no—
185C22×C32—21no—
186C22×C32—21no—
187C22×C32—21no—
188C22×C32—21no—
189C22×C32—21no—
190C22×C32C2×C4×D312yes—
191C22×C32#23512yes—
192C22×C32D3×D412yes—
193C22×C32D3×D412yes—
194C22×C32D3×D412yes—
195C22×C32#23712yes—
196C22×C32C2×#6712yes—
197C2×C3⋊D3—24no—
198C2×C3⋊D3—24no—
199C2×C3⋊D3—24no—
200C2×C3⋊D3—42no—
201C2×C3⋊D3—42no—
202C2×C3⋊D3—42no—
203C2×C3⋊D3—42no—
204C2×C3×D3—64no—
205C2×C3×D3—64no—
206C2×C3×D3—64no—
207C2×C3×D3—122no—
208C2×C3×D3—122no—
209C2×C3×D3—122no—
210C2×C3×D3—122no—
211C32×C4—24no—
212C32×C4—24no—
213C3⋊Dic3—42no—
214C3×Dic3—38no—
215C3×Dic3—64no—
216C3×Dic3—64no—
217C3×Dic3—64no—
218C3×Dic3—122no—
219C22×D4—271no—
220C2×C4⋊C4—271no—
221C2×C22⋊C4—92no—
222C2×C22⋊C4—272no—
223C2×C42—91no—
224H(3,𝔽3)#39011yescharacteristic, complemented, Sylow
225C23×C3—32no—
226C23×C3—61no—
227C23×C3—94no—
228C23×C3—182no—
229C23×C3—182no—
230C23×C3—182no—
231C23×C3—182no—
232C23×C3—182no—
233C23×C3—182no—
234C23×C3C2×D3211yescharacteristic
235C22×D3—34no—
236C22×D3—62no—
237C22×D3—62no—
238C22×D3—62no—
239C22×D3—62no—
240C22×D3—62no—
241C22×D3—62no—
242C22×D3—94no—
243C22×D3—182no—
244C22×D3—182no—
245C22×D3—182no—
246C22×D3—182no—
247C22×D3—182no—
248C22×D3—182no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—34no—
252C2×C3×C4—62no—
253C2×C3×C4—62no—
254C2×C3×C4—62no—
255C2×C3×C4—62no—
256C2×C3×C4—62no—
257C2×C3×C4—92no—
258C2×C3×C4—92no—
259C2×C3×C4—92no—
260C2×C3×C4—92no—
261C2×C3×C4—181no—
262C2×C3×C4—181no—
263C2×C3×C4—182no—
264C2×C3×C4—182no—
265C2×C3×C4—182no—
266C2×C3×C4#62512yes—
267#67—1816no—
268C2×Dic3—34no—
269C2×Dic3—34no—
270C2×Dic3—34no—
271C2×Dic3—34no—
272C2×Dic3—62no—
273C2×Dic3—62no—
274C2×Dic3—62no—
275C2×Dic3—62no—
276C2×Dic3—62no—
277C2×Dic3—62no—
278C2×Dic3—62no—
279C2×Dic3—62no—
280C2×Dic3—62no—
281C2×Dic3—62no—
282C2×Dic3—62no—
283C2×Dic3—182no—
284C2×Dic3—182no—
285C2×Dic3—182no—
286D12—188no—
287C2×C32—21no—
288C2×C32—21no—
289C2×C32—21no—
290C2×C32—21no—
291C2×C32—21no—
292C2×C32—21no—
293C2×C32—21no—
294C2×C32D3×C22⋊C412yes—
295C2×C32#67⋊C412yes—
296C2×C32#88412yes—
297C2×C32#88512yes—
298C2×C32C4×#6712yes—
299C2×C32#93912yes—
300C2×C32#94012yes—
301C3⋊D3—44no—
302C3×D3—124no—
303C24—92no—
304C2×D4—274no—
305C2×D4—542no—
306C2×D4—542no—
307C22×C4—91no—
308C22×C4—91no—
309C22×C4—92no—
310C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
311C4⋊C4—542no—
312C22⋊C4—184no—
313C22⋊C4—544no—
314C42—182no—
315C22×C3—32no—
316C22×C3—32no—
317C22×C3—32no—
318C22×C3—32no—
319C22×C3—32no—
320C22×C3—32no—
321C22×C3—32no—
322C22×C3—61no—
323C22×C3—61no—
324C22×C3—61no—
325C22×C3—61no—
326C22×C3—61no—
327C22×C3—61no—
328C22×C3—61no—
329C22×C3—184no—
330C22×C3—184no—
331C22×C3—184no—
332C22×C3—362no—
333C22×C3—362no—
334C22×C3—362no—
335C22×C3—362no—
336C22×C3#379811yescharacteristic
337C22×C3C4×D3211yescharacteristic
338C22×C3#380211yescharacteristic
339C22×C3#380211yescharacteristic
340C22×C3#380511yescharacteristic
341C22×C3C2×#62511yescharacteristic
342C22×C3#381111yescharacteristic
343C2×D3—64no—
344C2×D3—64no—
345C2×D3—64no—
346C2×D3—122no—
347C2×D3—122no—
348C2×D3—122no—
349C2×D3—122no—
350C2×D3—184no—
351C2×D3—184no—
352C2×D3—184no—
353C2×D3—362no—
354C2×D3—362no—
355C2×D3—362no—
356C2×D3—362no—
357C3×C4—22no—
358C3×C4—64no—
359C3×C4—64no—
360C3×C4—94no—
361C3×C4—182no—
362C3×C4—362no—
363Dic3—38no—
364Dic3—64no—
365Dic3—64no—
366Dic3—64no—
367Dic3—122no—
368Dic3—362no—
369C32—21no—
370C32#495712yescomplemented
371C23—94no—
372C23—182no—
373C23—182no—
374C23—182no—
375C23—182no—
376C23—182no—
377C23—182no—
378C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
379D4—548no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—92no—
383C2×C4—92no—
384C2×C4—92no—
385C2×C4—92no—
386C2×C4—181no—
387C2×C4—181no—
388C2×C4—182no—
389C2×C4—182no—
390C2×C4—182no—
391C2×C4#619812yescomplemented
392C2×C3—32no—
393C2×C3—32no—
394C2×C3—32no—
395C2×C3—32no—
396C2×C3—32no—
397C2×C3—32no—
398C2×C3—32no—
399C2×C3—61no—
400C2×C3—61no—
401C2×C3—61no—
402C2×C3—61no—
403C2×C3—61no—
404C2×C3—61no—
405C2×C3—61no—
406C2×C3—364no—
407C2×C3C4×#62511yescharacteristic
408C2×C3#6390211yescharacteristic
409C2×C3#6391211yescharacteristic
410C2×C3#6391411yescharacteristic
411C2×C3#6391511yescharacteristic
412C2×C3#6392111yescharacteristic
413C2×C3#6398011yescharacteristic
414D3—124no—
415D3—364no—
416C22—184no—
417C22—184no—
418C22—184no—
419C22—362no—
420C22—362no—
421C22—362no—
422C22—362no—
423C22#8865011yescharacteristic
424C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
425C22#8865511yescharacteristic
426C22#8865511yescharacteristic
427C22#8867211yescharacteristic
428C22C2×#619811yescharacteristic
429C22#8867711yescharacteristic
430C4—22no—
431C4—94no—
432C4—182no—
433C4—362no—
434C3—32no—
435C3—61no—
436C3#1059138511yescharacteristic
437C2—364no—
438C2C4×#619811yescharacteristic
439C2#1172113811yescharacteristic
440C2#1172114211yescharacteristic
441C2#1172114411yescharacteristic
442C2#1172114511yescharacteristic
443C2#1172116111yescharacteristic
444C2#1172131811yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC22×#6198C2
cosocleC23×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#390
LCSG ▷ C22×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×C4×#1084C2semidirect
3. C2×#3752⋊C3C2semidirect
4. C2×#88450C2semidirect
5. C2×#88449C2semidirect
14. C2×#6198C4semidirect
20. C2×#3752D3semidirect
37. C2×C4×H(3,𝔽3)D4semidirect
43. C2×#1084D4semidirect
86. #1084C22⋊C4semidirect
122. C2×C32×C4#67semidirect
224. H(3,𝔽3)#390semidirect
370. C32#4957semidirect
391. C2×C4#6198semidirect

Maximal subgroups

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

subgroupindex
C22×#61982
C2×C4×#10842
C2×#3752⋊C32
C2×#884502
C2×#884492
#105913833