Finite Group Databasebeta

← #13140245

Subgroups — #13140245

Order 1728.

995 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 995 conjugacy classes of subgroups — too many to draw. Only its 135 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×#88540C211yesmaximal, characteristic, complemented
3C2×#3752⋊C3C212yesmaximal, complemented
4C2×#88450C212yesmaximal, complemented
5C2×#88447C211yesmaximal, characteristic, complemented
6#10591384—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
15#88540—22no—
16#3752⋊C3—24no—
17#88450—24no—
18#88447—22no—
19C22×#626—32no—
20C2×#3752D312yescomplemented
21C2×C3×(C2×D3)⋊C4—32no—
22C2×C3×Dic3⋊C4—32no—
23C2×C3×Dic3⋊C4—61no—
24C2×#3722—32no—
25C2×(C2×D3)⋊Dic3—32no—
26C2×Dic32—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)—22no—
37C2×#1084—21no—
38C2×#1084—22no—
39C2×#1084C2×C412yes—
40C2×#1084D412yes—
41C2×#1084D412yes—
42C2×(C3⋊Dic3)⋊C3—22no—
43C2×(C3⋊Dic3)⋊C3—22no—
44C2×(C3⋊Dic3)⋊C3D414yescomplemented
45#6198—28no—
46#4988—91no—
47#4957—92no—
48C23×C3⋊D3C2×D312yes—
49C23×C3×D3—32no—
50C22×C32×C4—21no—
51C22×C32×C4C2×D312yes—
52C22×C3⋊Dic3C2×D312yes—
53C22×C3×Dic3—32no—
54C22×C3×Dic3—32no—
55C22×C3×Dic3—32no—
56C22×C3×Dic3—61no—
57C22×C3×Dic3—61no—
58C2×#626—38no—
59C2×#626—64no—
60C2×#626—64no—
61#3752—24no—
62C3×(C2×D3)⋊C4—64no—
63C3×Dic3⋊C4—64no—
64C3×Dic3⋊C4—64no—
65#3722—64no—
66(C2×D3)⋊Dic3—64no—
67Dic32—64no—
68C22×H(3,𝔽3)C22×C411yescharacteristic
69C22×H(3,𝔽3)C2×D411yescharacteristic
70C22×H(3,𝔽3)C2×D411yescharacteristic
71C22×H(3,𝔽3)C2×D412yes—
72C22×H(3,𝔽3)#4112yes—
73C2×(C3⋊D3)⋊C3—24no—
74C2×(C3⋊D3)⋊C3—24no—
75C2×(C3⋊D3)⋊C3—24no—
76C2×(C3⋊D3)⋊C3—42no—
77C2×(C3⋊D3)⋊C3—42no—
78C2×(C3⋊D3)⋊C3—42no—
79C2×(C3⋊D3)⋊C3—42no—
80C4×H(3,𝔽3)—41no—
81#1084—41no—
82#1084C22⋊C414yescomplemented
83(C3⋊Dic3)⋊C3—24no—
84C22×#67—92no—
85C22×D12—91no—
86C2×C3×C4⋊C4—91no—
87C2×C3×C22⋊C4—92no—
88C2×#217—92no—
89C2×(C2×D3)⋊C4—32no—
90C2×(C2×D3)⋊C4—92no—
91C2×(C2×D3)⋊C4—92no—
92C2×(C2×D3)⋊C4—92no—
93C2×Dic3⋊C4—32no—
94C2×Dic3⋊C4—61no—
95C2×C4×Dic3—91no—
96C2×C4×Dic3—92no—
97C23×C32—21no—
98C23×C32C22×D312yes—
99C22×C3⋊D3—22no—
100C22×C3⋊D3—22no—
101C22×C3⋊D3—22no—
102C22×C3⋊D3—22no—
103C22×C3⋊D3—22no—
104C22×C3⋊D3—22no—
105C22×C3⋊D3C4×D314yes—
106C22×C3×D3—34no—
107C22×C3×D3—62no—
108C22×C3×D3—62no—
109C22×C3×D3—62no—
110C22×C3×D3—62no—
111C22×C3×D3—62no—
112C22×C3×D3—62no—
113C2×C32×C4—22no—
114C2×C32×C4—22no—
115C2×C32×C4—22no—
116C2×C32×C4—22no—
117C2×C32×C4—42no—
118C2×C3⋊Dic3—22no—
119C2×C3⋊Dic3—22no—
120C2×C3⋊Dic3#6714yescomplemented
121C2×C3×Dic3—34no—
122C2×C3×Dic3—34no—
123C2×C3×Dic3—34no—
124C2×C3×Dic3—34no—
125C2×C3×Dic3—62no—
126C2×C3×Dic3—62no—
127C2×C3×Dic3—62no—
128C2×C3×Dic3—62no—
129C2×C3×Dic3—62no—
130C2×C3×Dic3—62no—
131C2×C3×Dic3—62no—
132C2×C3×Dic3—62no—
133C2×C3×Dic3—62no—
134C2×C3×Dic3—122no—
135#626—616no—
136#390—271noSylow
137C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
138C2×H(3,𝔽3)C4×D412yes—
139C2×H(3,𝔽3)#12112yes—
140C2×H(3,𝔽3)#12411yescharacteristic
141C2×H(3,𝔽3)#12711yescharacteristic
142(C3⋊D3)⋊C3—44no—
143C24×C3—92no—
144C23×D3—32no—
145C23×D3—92no—
146C22×C3×C4—32no—
147C22×C3×C4—61no—
148C22×C3×C4—91no—
149C22×C3×C4—92no—
150C22×C3×C4—91no—
151C22×C3×C4D3211yescharacteristic
152C2×#67—98no—
153C2×#67—184no—
154C2×#67—184no—
155C22×Dic3—32no—
156C22×Dic3—32no—
157C22×Dic3—32no—
158C22×Dic3—61no—
159C22×Dic3—61no—
160C22×Dic3—92no—
161C2×D12—94no—
162C2×D12—182no—
163C2×D12—182no—
164C3×C4⋊C4—182no—
165C3×C22⋊C4—184no—
166#217—184no—
167(C2×D3)⋊C4—64no—
168(C2×D3)⋊C4—184no—
169(C2×D3)⋊C4—184no—
170(C2×D3)⋊C4—184no—
171Dic3⋊C4—64no—
172Dic3⋊C4—64no—
173C4×Dic3—184no—
174C4×Dic3—182no—
175C22×C32—22no—
176C22×C32—21no—
177C22×C32—21no—
178C22×C32—22no—
179C22×C32—21no—
180C22×C32C2×C4×D312yes—
181C22×C32#23512yes—
182C22×C32D3×D412yes—
183C22×C32D3×D412yes—
184C22×C32D3×D412yes—
185C22×C32#23712yes—
186C22×C32C2×#6712yes—
187C2×C3⋊D3—24no—
188C2×C3⋊D3—24no—
189C2×C3⋊D3—24no—
190C2×C3⋊D3—42no—
191C2×C3⋊D3—42no—
192C2×C3⋊D3—42no—
193C2×C3⋊D3—42no—
194C2×C3×D3—64no—
195C2×C3×D3—64no—
196C2×C3×D3—64no—
197C2×C3×D3—122no—
198C2×C3×D3—122no—
199C2×C3×D3—122no—
200C2×C3×D3—122no—
201C32×C4—42no—
202C32×C4—42no—
203C3⋊Dic3—24no—
204C3×Dic3—38no—
205C3×Dic3—64no—
206C3×Dic3—64no—
207C3×Dic3—122no—
208C3×Dic3—122no—
209C22×D4—271no—
210C2×C4⋊C4—91no—
211C2×C22⋊C4—92no—
212C2×C22⋊C4—272no—
213C2×C42—271no—
214H(3,𝔽3)#39011yescharacteristic, complemented, Sylow
215C23×C3—32no—
216C23×C3—61no—
217C23×C3—94no—
218C23×C3—182no—
219C23×C3—182no—
220C23×C3—182no—
221C23×C3—182no—
222C23×C3—182no—
223C23×C3—182no—
224C23×C3C2×D3211yescharacteristic
225C22×D3—34no—
226C22×D3—62no—
227C22×D3—62no—
228C22×D3—62no—
229C22×D3—62no—
230C22×D3—62no—
231C22×D3—62no—
232C22×D3—94no—
233C22×D3—182no—
234C22×D3—182no—
235C22×D3—182no—
236C22×D3—182no—
237C22×D3—182no—
238C22×D3—182no—
239C2×C3×C4—22no—
240C2×C3×C4—21no—
241C2×C3×C4—62no—
242C2×C3×C4—62no—
243C2×C3×C4—62no—
244C2×C3×C4—62no—
245C2×C3×C4—92no—
246C2×C3×C4—92no—
247C2×C3×C4—92no—
248C2×C3×C4—94no—
249C2×C3×C4—122no—
250C2×C3×C4—182no—
251C2×C3×C4—182no—
252C2×C3×C4—182no—
253C2×C3×C4—181no—
254#67—1816no—
255C2×Dic3—34no—
256C2×Dic3—34no—
257C2×Dic3—34no—
258C2×Dic3—34no—
259C2×Dic3—62no—
260C2×Dic3—62no—
261C2×Dic3—62no—
262C2×Dic3—62no—
263C2×Dic3—62no—
264C2×Dic3—62no—
265C2×Dic3—62no—
266C2×Dic3—62no—
267C2×Dic3—62no—
268C2×Dic3—94no—
269C2×Dic3—122no—
270C2×Dic3—182no—
271C2×Dic3—182no—
272D12—188no—
273C2×C32—22no—
274C2×C32—22no—
275C2×C32—21no—
276C2×C32—21no—
277C2×C32—21no—
278C2×C32D3×C22⋊C412yes—
279C2×C32#67⋊C412yes—
280C2×C32#88412yes—
281C2×C32#88512yes—
282C2×C32C4×#6712yes—
283C2×C32#93912yes—
284C2×C32#94012yes—
285C3⋊D3—44no—
286C3×D3—124no—
287C24—92no—
288C2×D4—274no—
289C2×D4—542no—
290C2×D4—542no—
291C22×C4—91no—
292C22×C4—92no—
293C22×C4—91no—
294C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
295C4⋊C4—182no—
296C22⋊C4—184no—
297C22⋊C4—544no—
298C42—542no—
299C22×C3—32no—
300C22×C3—32no—
301C22×C3—32no—
302C22×C3—32no—
303C22×C3—32no—
304C22×C3—32no—
305C22×C3—32no—
306C22×C3—62no—
307C22×C3—61no—
308C22×C3—61no—
309C22×C3—62no—
310C22×C3—61no—
311C22×C3—184no—
312C22×C3—184no—
313C22×C3—184no—
314C22×C3—362no—
315C22×C3—362no—
316C22×C3—362no—
317C22×C3—362no—
318C22×C3#379512yes—
319C22×C3C4×D3211yescharacteristic
320C22×C3#380211yescharacteristic
321C22×C3D3×#6712yes—
322C22×C3#381111yescharacteristic
323C2×D3—64no—
324C2×D3—64no—
325C2×D3—64no—
326C2×D3—122no—
327C2×D3—122no—
328C2×D3—122no—
329C2×D3—122no—
330C2×D3—184no—
331C2×D3—184no—
332C2×D3—184no—
333C2×D3—362no—
334C2×D3—362no—
335C2×D3—362no—
336C2×D3—362no—
337C3×C4—41no—
338C3×C4—94no—
339C3×C4—122no—
340C3×C4—122no—
341C3×C4—184no—
342C3×C4—361no—
343Dic3—38no—
344Dic3—64no—
345Dic3—64no—
346Dic3—122no—
347Dic3—122no—
348Dic3—184no—
349C32—21no—
350C32#495712yescomplemented
351C23—94no—
352C23—182no—
353C23—182no—
354C23—182no—
355C23—182no—
356C23—182no—
357C23—182no—
358C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
359D4—548no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—92no—
363C2×C4—92no—
364C2×C4—92no—
365C2×C4—94no—
366C2×C4—182no—
367C2×C4—182no—
368C2×C4—182no—
369C2×C4—181no—
370C2×C3—32no—
371C2×C3—32no—
372C2×C3—32no—
373C2×C3—32no—
374C2×C3—32no—
375C2×C3—32no—
376C2×C3—32no—
377C2×C3—61no—
378C2×C3—62no—
379C2×C3—62no—
380C2×C3—61no—
381C2×C3—61no—
382C2×C3—364no—
383C2×C3#6387912yes—
384C2×C3#6388512yes—
385C2×C3#6391311yescharacteristic
386C2×C3#6392111yescharacteristic
387C2×C3#6397911yescharacteristic
388D3—124no—
389D3—364no—
390C22—184no—
391C22—184no—
392C22—184no—
393C22—362no—
394C22—362no—
395C22—362no—
396C22—362no—
397C22#8864812yes—
398C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
399C22#8865511yescharacteristic
400C22#8867612yes—
401C22#8867711yescharacteristic
402C4—41no—
403C4—94no—
404C4—184no—
405C4—361no—
406C3—32no—
407C3—61no—
408C3#1059138611yescharacteristic
409C2—364no—
410C2#1172112212yes—
411C2#1172112612yes—
412C2#1172114311yescharacteristic
413C2#1172116111yescharacteristic
414C2#1172131711yescharacteristic
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×#88540C2semidirect
3. C2×#3752⋊C3C2semidirect
4. C2×#88450C2semidirect
5. C2×#88447C2semidirect
14. C2×#6198C4semidirect
20. C2×#3752D3semidirect
44. C2×(C3⋊Dic3)⋊C3D4semidirect
82. #1084C22⋊C4semidirect
120. C2×C3⋊Dic3#67semidirect
214. H(3,𝔽3)#390semidirect
350. C32#4957semidirect

Maximal subgroups

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

subgroupindex
C22×#61982
C2×#885402
C2×#3752⋊C32
C2×#884502
C2×#884472
#105913843