Finite Group Databasebeta

← #13140191

Subgroups — #13140191

Order 1728.

732 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 93 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
2#11721824C211yesmaximal, characteristic, complemented
3C2×#88438C212yesmaximal, complemented
4#63657⋊C3C211yesmaximal, characteristic, complemented
5#11720308C212yesmaximal, complemented
6#10591183—31nomaximal
7#10591169—31nomaximal
8C23×(C3⋊D3)⋊C3—21no—
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C2×#6229C2211yescharacteristic, complemented
11C2×C4⋊((C3⋊D3)⋊C3)—21no—
12C2×C4⋊((C3⋊D3)⋊C3)—24no—
13C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
14#88552—21no—
15C4⋊#1084C2211yescharacteristic, complemented
16#88540—21no—
17C2×(C3⋊(C3⋊C8))⋊C3C2212yescomplemented
18#88438—24no—
19C22×C4⋊(C3⋊D3)D311yescharacteristic
20C22×C3×D12—31no—
21C3×#924—31no—
22C3×#924—31no—
23C3×#924—61no—
24C2×#3717—32no—
25C2×#3715—32no—
26#63657D311yescharacteristic, complemented
27C3×#830—31no—
28#63564—32no—
29#63562—32no—
30C23×H(3,𝔽3)D411yescharacteristic
31C22×(C3⋊D3)⋊C3—24no—
32C22×(C3⋊D3)⋊C3—22no—
33C22×(C3⋊D3)⋊C3—42no—
34C22×(C3⋊D3)⋊C3—42no—
35C2×C4×H(3,𝔽3)—22no—
36C2×C4×H(3,𝔽3)D411yescharacteristic
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084—21no—
39C2×#1084—21no—
40#6229—22no—
41C4⋊((C3⋊D3)⋊C3)—24no—
42C4⋊((C3⋊D3)⋊C3)—22no—
43C4⋊((C3⋊D3)⋊C3)D414yes—
44(C3⋊(C3⋊C8))⋊C3—22no—
45#5218—91no—
46#5046—91no—
47#4731—91no—
48C23×C3⋊D3—21no—
49C23×C3×D3—61no—
50C22×C32×C4—21no—
51C22×C32×C4C2×D311yescharacteristic
52C22×C32×C4C2×D311yescharacteristic
53C2×C4⋊(C3⋊D3)—21no—
54C2×C4⋊(C3⋊D3)—24no—
55C2×C4⋊(C3⋊D3)C2×D312yes—
56C2×C3×D12—32no—
57C2×C3×D12—64no—
58C2×C3×D12—61no—
59C2×C3×Dic6—31no—
60C2×C3×Dic6—31no—
61C2×C3×Dic6—61no—
62C2×C3⋊(C3⋊C8)C2×D312yescomplemented
63C3×#217—62no—
64C3×#217—61no—
65C3×#217—61no—
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—61no—
69C3×Dic3⋊C4—62no—
70C3×Dic3⋊C4—61no—
71C3×Dic3⋊C4—61no—
72C2×C3×C3⋊C8—32no—
73#3717—64no—
74#3715—64no—
75C22×H(3,𝔽3)—21no—
76C22×H(3,𝔽3)—21no—
77C22×H(3,𝔽3)SD1612yescomplemented
78C22×H(3,𝔽3)C2×D411yescharacteristic
79C2×(C3⋊D3)⋊C3—24no—
80C2×(C3⋊D3)⋊C3—48no—
81C2×(C3⋊D3)⋊C3—42no—
82C2×(C3⋊D3)⋊C3—42no—
83C4×H(3,𝔽3)—21no—
84C4×H(3,𝔽3)C2×D412yes—
85#1084—41no—
86#1084—41no—
87C22×C3×D4—91no—
88C22×D12—31no—
89C22×D12—91no—
90C3×C22⋊Q8—91no—
91C2×#215—92no—
92C2×#214—92no—
93#924—31no—
94#924—31no—
95#924—61no—
96C2×#204—92no—
97C3×C22⋊C8—91no—
98#832—92no—
99#830—31no—
100#830—91no—
101#821—92no—
102#809—92no—
103C23×C32—21no—
104C23×C32D1211yescharacteristic
105C23×C32#6711yescharacteristic
106C22×C3⋊D3—24no—
107C22×C3⋊D3—22no—
108C22×C3⋊D3—42no—
109C22×C3⋊D3—42no—
110C22×C3×D3—64no—
111C22×C3×D3—62no—
112C22×C3×D3—122no—
113C22×C3×D3—122no—
114C2×C32×C4—22no—
115C2×C32×C4—22no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—42no—
119C2×C32×C4D1211yescharacteristic
120C2×C32×C4#6711yescharacteristic
121C2×C32×C4C22×D311yescharacteristic
122C2×C32×C4C22×D311yescharacteristic
123C4⋊(C3⋊D3)—24no—
124C4⋊(C3⋊D3)—22no—
125C4⋊(C3⋊D3)#6714yes—
126C2×C3×Dic3—62no—
127C2×C3×Dic3—62no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C3×D12—34no—
133C3×D12—64no—
134C3×D12—62no—
135C3×Dic6—62no—
136C3×Dic6—62no—
137C3×Dic6—122no—
138C3⋊(C3⋊C8)—22no—
139C3×C3⋊C8—62no—
140#450—271noSylow
141C2×H(3,𝔽3)—22no—
142C2×H(3,𝔽3)#12011yescharacteristic
143C2×H(3,𝔽3)C2×SD1611yescharacteristic
144C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
145(C3⋊D3)⋊C3—44no—
146C24×C3—181no—
147C23×D3—61no—
148C23×D3—181no—
149C2×C3×Q8—91no—
150C2×C3×D4—92no—
151C2×C3×D4—181no—
152C2×C3×D4—184no—
153C22×C3×C4—31no—
154C22×C3×C4—31no—
155C22×C3×C4—61no—
156C22×C3×C4D3211yescharacteristic
157C2×D12—32no—
158C2×D12—64no—
159C2×D12—61no—
160C2×D12—92no—
161C2×D12—184no—
162C2×D12—181no—
163C2×Dic6—31no—
164C2×Dic6—31no—
165C2×Dic6—61no—
166C2×C3×C8—92no—
167C3×C4⋊C4—91no—
168C3×C4⋊C4—181no—
169C3×C22⋊C4—181no—
170#217—61no—
171#217—61no—
172#217—62no—
173#215—184no—
174#214—184no—
175C4⋊Dic3—31no—
176C4⋊Dic3—31no—
177C4⋊Dic3—61no—
178Dic3⋊C4—61no—
179Dic3⋊C4—61no—
180Dic3⋊C4—62no—
181C2×C3⋊C8—32no—
182C2×C3⋊C8—92no—
183#204—184no—
184C22×C32—21no—
185C22×C32—21no—
186C22×C32—21no—
187C22×C32—21no—
188C22×C32—22no—
189C22×C32—21no—
190C22×C32—41no—
191C22×C32—41no—
192C22×C32#20412yescomplemented
193C22×C32#21412yescomplemented
194C22×C32C2×D1211yescharacteristic
195C22×C32C2×#6711yescharacteristic
196C2×C3⋊D3—24no—
197C2×C3⋊D3—48no—
198C2×C3⋊D3—42no—
199C2×C3⋊D3—42no—
200C2×C3×D3—64no—
201C2×C3×D3—128no—
202C2×C3×D3—122no—
203C2×C3×D3—122no—
204C32×C4—21no—
205C32×C4—21no—
206C32×C4—22no—
207C32×C4—41no—
208C32×C4D3×D412yes—
209C32×C4C2×#6712yes—
210C3×Dic3—122no—
211C3×Dic3—122no—
212C3×Dic3—121no—
213C3×Dic3—121no—
214C3×Dic3—121no—
215C3×Dic3—121no—
216C22×D4—91no—
217C2×SD16—272no—
218C22⋊Q8—91no—
219#102—272no—
220C22⋊C8—91no—
221H(3,𝔽3)#45011yescharacteristic, complemented, Sylow
222C23×C3—31no—
223C23×C3—31no—
224C23×C3—61no—
225C23×C3—182no—
226C23×C3—184no—
227C23×C3—362no—
228C23×C3—362no—
229C23×C3#62611yescharacteristic
230C22×D3—64no—
231C22×D3—62no—
232C22×D3—122no—
233C22×D3—122no—
234C22×D3—182no—
235C22×D3—184no—
236C22×D3—362no—
237C22×D3—362no—
238C3×Q8—182no—
239C3×D4—94no—
240C3×D4—182no—
241C3×D4—184no—
242C2×C3×C4—22no—
243C2×C3×C4—31no—
244C2×C3×C4—31no—
245C2×C3×C4—31no—
246C2×C3×C4—31no—
247C2×C3×C4—62no—
248C2×C3×C4—62no—
249C2×C3×C4—61no—
250C2×C3×C4—61no—
251C2×C3×C4—122no—
252C2×C3×C4—181no—
253C2×C3×C4—181no—
254C2×C3×C4#62611yescharacteristic
255C2×C3×C4C2×D3211yescharacteristic
256C2×Dic3—61no—
257C2×Dic3—61no—
258C2×Dic3—61no—
259C2×Dic3—61no—
260C2×Dic3—62no—
261C2×Dic3—62no—
262D12—34no—
263D12—64no—
264D12—62no—
265D12—94no—
266D12—182no—
267D12—184no—
268Dic6—62no—
269Dic6—62no—
270Dic6—122no—
271C3×C8—182no—
272C3⋊C8—62no—
273C3⋊C8—182no—
274C2×C32—22no—
275C2×C32—22no—
276C2×C32—21no—
277C2×C32—21no—
278C2×C32—21no—
279C2×C32—42no—
280C2×C32#88211yescharacteristic
281C2×C32C2×#20411yescharacteristic
282C2×C32#90811yescharacteristic
283C2×C32#93211yescharacteristic
284C2×C32C2×#21411yescharacteristic
285C2×C32#95311yescharacteristic
286C3⋊D3—44no—
287C3×D3—124no—
288C24—181no—
289C2×Q8—91no—
290C2×D4—92no—
291C2×D4—184no—
292C2×D4—181no—
293C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
294SD16—544no—
295C2×C8—92no—
296C4⋊C4—91no—
297C4⋊C4—181no—
298C22⋊C4—181no—
299C22×C3—21no—
300C22×C3—21no—
301C22×C3—31no—
302C22×C3—31no—
303C22×C3—32no—
304C22×C3—32no—
305C22×C3—61no—
306C22×C3—61no—
307C22×C3—61no—
308C22×C3—61no—
309C22×C3—61no—
310C22×C3—62no—
311C22×C3—121no—
312C22×C3—121no—
313C22×C3—184no—
314C22×C3—362no—
315C22×C3—362no—
316C22×C3—368no—
317C22×C3#371612yes—
318C22×C3C2×#62611yescharacteristic
319C2×D3—64no—
320C2×D3—128no—
321C2×D3—122no—
322C2×D3—122no—
323C2×D3—184no—
324C2×D3—362no—
325C2×D3—362no—
326C2×D3—368no—
327C3×C4—21no—
328C3×C4—32no—
329C3×C4—32no—
330C3×C4—61no—
331C3×C4—61no—
332C3×C4—62no—
333C3×C4—121no—
334C3×C4—361no—
335C3×C4—361no—
336C3×C4D3×#6712yes—
337Dic3—121no—
338Dic3—121no—
339Dic3—121no—
340Dic3—121no—
341Dic3—122no—
342Dic3—122no—
343C32—21no—
344C32#473111yescharacteristic, complemented
345C32#521811yescharacteristic, complemented
346C23—184no—
347C23—182no—
348C23—362no—
349C23—362no—
350C23#620011yescharacteristic
351Q8—182no—
352D4—94no—
353D4—184no—
354D4—182no—
355C2×C4—22no—
356C2×C4—181no—
357C2×C4—181no—
358C2×C4#620011yescharacteristic
359C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
360C8—182no—
361C2×C3—22no—
362C2×C3—31no—
363C2×C3—31no—
364C2×C3—31no—
365C2×C3—31no—
366C2×C3—31no—
367C2×C3—31no—
368C2×C3—62no—
369C2×C3—62no—
370C2×C3—61no—
371C2×C3—61no—
372C2×C3—61no—
373C2×C3—122no—
374C2×C3—364no—
375C2×C3C2×#371611yescharacteristic
376C2×C3#6383011yescharacteristic
377C2×C3#6397711yescharacteristic
378D3—124no—
379D3—364no—
380C22—21no—
381C22—21no—
382C22—184no—
383C22—368no—
384C22—362no—
385C22—362no—
386C22#8843812yescomplemented
387C22C2×#620011yescharacteristic
388C4—21no—
389C4—361no—
390C4—361no—
391C4#8867612yes—
392C3—31no—
393C3—31no—
394C3—61no—
395C3#1059117211yescharacteristic
396C2—22no—
397C2—364no—
398C2#1172107111yescharacteristic
399C2C2×#8843811yescharacteristic
400C2#1172132111yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radicalC22×C4⋊((C3⋊D3)⋊C3)C2
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#450
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. #11721824C2semidirect
3. C2×#88438C2semidirect
4. #63657⋊C3C2semidirect
5. #11720308C2semidirect
10. C2×#6229C22semidirect
15. C4⋊#1084C22semidirect
17. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
26. #63657D3semidirect
62. C2×C3⋊(C3⋊C8)C2×D3semidirect
77. C22×H(3,𝔽3)SD16semidirect
192. C22×C32#204semidirect
193. C22×C32#214semidirect
221. H(3,𝔽3)#450semidirect
344. C32#4731semidirect
345. C32#5218semidirect
386. C22#88438semidirect

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
#117218242
C2×#884382
#63657⋊C32
#117203082
#105911833
#105911693