Finite Group Databasebeta

← C2×#11721189 #13151308

Subgroups — C2×#11721189 #13151308

Order 1728.

1067 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1067 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C2×#88744C211yesmaximal, characteristic
2C2×#88651C211yesmaximal, characteristic, complemented
3C2×#3754⋊C3C212yesmaximal, complemented
4#11721189C218yesmaximal, direct product factor
5C2×#88439C212yesmaximal, complemented
6C2×#88435C211yesmaximal, characteristic, complemented
7C2×#63929—32nomaximal
8C2×D4×H(3,𝔽3)C2211yescharacteristic
9C2×#6235—21no—
10C22×#1084—21no—
11#88744—22no—
12#88744C2214yes—
13C2×C4×#272C2211yescharacteristic
14C2×#6229C2211yescharacteristic
15C2×#634⋊C3C2212yes—
16C2×#6199—21no—
17C2×#6196—21no—
18#88651—22no—
19#88651C2214yescomplemented
20#3754⋊C3C2218yescomplemented
21C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
22#88439C2218yescomplemented
23#88435C2214yescomplemented
24C2×C3×#237—32no—
25C2×C3×#237—61no—
26C2×D3×Dic6—32no—
27C2×#3754D312yescomplemented
28C2×C3×#214—32no—
29#63929—316no—
30C2×#3719—32no—
31C2×#3718—32no—
32C2×#3711—32no—
33C23×H(3,𝔽3)—21no—
34C22×#272D411yescharacteristic
35D4×H(3,𝔽3)—21no—
36D4×H(3,𝔽3)C2312yes—
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38#6235—24no—
39C2×#1084—21no—
40C2×#1084—24no—
41C2×#1084D411yescharacteristic
42C4×#272D412yes—
43C4×#272C2312yes—
44#6229—21no—
45#6229C2312yes—
46C2×(C3⋊Dic3)⋊C3—22no—
47#634⋊C3—22no—
48#634⋊C3C2314yes—
49#6199—24no—
50#6196—24no—
51(C3⋊(C3⋊C8))⋊C3C2314yes—
52C2×#951—91no—
53C2×#915—92no—
54C2×C32×D4—21no—
55C2×C32×D4C2×D312yes—
56C2×#634C2×D312yes—
57C2×C3×#67—62no—
58C2×C3×#67—62no—
59C22×C3×Dic3—62no—
60C22×C3×Dic3—62no—
61C3×#237—38no—
62C3×#237—64no—
63C3×#237—64no—
64C3×#237—64no—
65C2×C3×C4×D3—32no—
66C2×C3×C4×D3—61no—
67C2×C3×Dic6—32no—
68C2×C3×Dic6—32no—
69C2×C3×Dic6—61no—
70C2×#627—62no—
71C2×D3×Dic3—62no—
72D3×Dic6—38no—
73D3×Dic6—64no—
74#3754C2×D318yescomplemented
75C2×C3⋊(C3⋊C8)C2×D312yes—
76C3×#214—38no—
77C2×C3×C3⋊C8—32no—
78#3719—38no—
79#3718—38no—
80#3711—38no—
81C22×H(3,𝔽3)—22no—
82C22×H(3,𝔽3)—42no—
83C22×H(3,𝔽3)C2×D411yescharacteristic
84C2×#272—22no—
85C2×#272C2×D412yes—
86C4×H(3,𝔽3)C2×D411yescharacteristic
87C4×H(3,𝔽3)C2411yescharacteristic
88#1084—22no—
89#1084C2×D412yes—
90(C3⋊Dic3)⋊C3—24no—
91C2×C3×#41—91no—
92C2×D3×Q8—92no—
93C2×#237—32no—
94C2×#237—61no—
95C22×Dic6—91no—
96C2×C3×SD16—92no—
97#951—98no—
98C2×#216—92no—
99C2×#216—92no—
100C2×#214—32no—
101C2×#214—92no—
102C2×#208—91no—
103#915—916no—
104C2×Dic12—92no—
105C2×ΓL(1,𝔽25)—92no—
106C23×C32—22no—
107C23×C32—22no—
108C22×C3×D3—32no—
109C22×C3×D3—61no—
110C32×D4—22no—
111C32×D4—22no—
112C32×D4—22no—
113C32×D4C22×D314yes—
114C2×C32×C4—21no—
115C2×C32×C4C22×D312yes—
116C2×C3⋊Dic3—22no—
117#634—22no—
118#634C22×D314yes—
119C3×#67—68no—
120C3×#67—68no—
121C2×C3×Dic3—32no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—68no—
124C2×C3×Dic3—62no—
125C2×C3×Dic3—68no—
126C2×C3×Dic3—62no—
127C2×C3×Dic3—62no—
128C3×C4×D3—34no—
129C3×C4×D3—34no—
130C3×C4×D3—62no—
131C3×C4×D3—62no—
132C3×Dic6—34no—
133C3×Dic6—34no—
134C3×Dic6—62no—
135C3×Dic6—62no—
136C3×Dic6—62no—
137C3×Dic6—62no—
138#627—68no—
139D3×Dic3—68no—
140C3⋊(C3⋊C8)C22×D314yes—
141C3×C3⋊C8—34no—
142C2×#137—271noSylow
143C2×H(3,𝔽3)—42no—
144C2×H(3,𝔽3)#13712yescomplemented
145C2×H(3,𝔽3)C22×D411yescharacteristic
146#272—22no—
147C3×#41—94no—
148C3×#41—182no—
149C2×C3×Q8—92no—
150C2×C3×Q8—91no—
151C2×C3×D4—32no—
152C2×C3×D4—61no—
153C2×C3×D4—181no—
154C2×C3×D4D3211yescharacteristic
155C22×C3×C4—91no—
156C22×C3×C4—181no—
157C2×#67—62no—
158C2×#67—62no—
159C22×Dic3—62no—
160C22×Dic3—62no—
161C22×Dic3—181no—
162D3×Q8—98no—
163D3×Q8—184no—
164#237—38no—
165#237—64no—
166#237—64no—
167#237—64no—
168C2×C4×D3—32no—
169C2×C4×D3—61no—
170C2×C4×D3—182no—
171C2×Dic6—32no—
172C2×Dic6—32no—
173C2×Dic6—61no—
174C2×Dic6—92no—
175C2×Dic6—94no—
176C2×Dic6—182no—
177C2×Dic6—182no—
178C2×Dic6—181no—
179C3×SD16—98no—
180C2×C3×C8—92no—
181#216—98no—
182#216—98no—
183#214—38no—
184#214—98no—
185#208—94no—
186C2×C3⋊C8—32no—
187C2×C3⋊C8—92no—
188Dic12—98no—
189ΓL(1,𝔽25)—98no—
190C22×C32—24no—
191C22×C32—24no—
192C22×C32—21no—
193C22×C32—44no—
194C22×C32—44no—
195C22×C32D3×D412yes—
196C2×C3×D3—34no—
197C2×C3×D3—62no—
198C2×C3×D3—64no—
199C2×C3×D3—122no—
200C32×C4—21no—
201C32×C4—21no—
202C32×C4D3×D412yes—
203C32×C4C23×D312yes—
204C3⋊Dic3—24no—
205C3×Dic3—34no—
206C3×Dic3—62no—
207C3×Dic3—64no—
208C3×Dic3—64no—
209C3×Dic3—64no—
210C2×#41—91no—
211C22×Q8—271no—
212#137—278no—
213C2×Q16—272no—
214C2×SD16—92no—
215C2×M16—271no—
216H(3,𝔽3)C2×#13711yescharacteristic, complemented, Sylow
217C23×C3—21no—
218C23×C3—62no—
219C23×C3—62no—
220C23×C3—91no—
221C22×D3—32no—
222C22×D3—61no—
223C3×Q8—94no—
224C3×Q8—92no—
225C3×Q8—182no—
226C3×Q8—181no—
227C3×D4—21no—
228C3×D4—34no—
229C3×D4—62no—
230C3×D4—62no—
231C3×D4—62no—
232C3×D4—184no—
233C3×D4C2×D3212yes—
234C2×C3×C4—32no—
235C2×C3×C4—61no—
236C2×C3×C4—91no—
237C2×C3×C4—92no—
238C2×C3×C4—92no—
239C2×C3×C4—184no—
240C2×C3×C4—182no—
241C2×C3×C4—181no—
242C2×C3×C4C2×D3211yescharacteristic
243#67—68no—
244#67—68no—
245C2×Dic3—32no—
246C2×Dic3—61no—
247C2×Dic3—68no—
248C2×Dic3—62no—
249C2×Dic3—62no—
250C2×Dic3—68no—
251C2×Dic3—62no—
252C2×Dic3—182no—
253C2×Dic3—184no—
254C4×D3—34no—
255C4×D3—34no—
256C4×D3—62no—
257C4×D3—62no—
258C4×D3—188no—
259Dic6—34no—
260Dic6—34no—
261Dic6—62no—
262Dic6—62no—
263Dic6—62no—
264Dic6—62no—
265Dic6—94no—
266Dic6—182no—
267Dic6—188no—
268Dic6—184no—
269C3×C8—94no—
270C3⋊C8—34no—
271C3⋊C8—94no—
272C2×C32—21no—
273C2×C32—22no—
274C2×C32—44no—
275C2×C32—44no—
276C2×C32#91514yescomplemented
277C2×C32C2×D3×D412yes—
278C3×D3—64no—
279C3×D3—122no—
280#41—94no—
281#41—182no—
282C2×Q8—92no—
283C2×Q8—91no—
284C2×Q8—274no—
285C2×Q8—541no—
286C2×Q8—542no—
287C2×D4—181no—
288C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
289C22×C4—91no—
290C22×C4—181no—
291C22×C4—541no—
292Q16—278no—
293SD16—98no—
294M16—274no—
295C2×C8—92no—
296C22×C3—22no—
297C22×C3—32no—
298C22×C3—42no—
299C22×C3—64no—
300C22×C3—64no—
301C22×C3—61no—
302C22×C3—92no—
303C22×C3—124no—
304C22×C3—124no—
305C22×C3—182no—
306C22×C3#381111yescharacteristic
307C2×D3—34no—
308C2×D3—62no—
309C2×D3—64no—
310C2×D3—122no—
311C3×C4—32no—
312C3×C4—32no—
313C3×C4—61no—
314C3×C4—61no—
315C3×C4—92no—
316C3×C4—184no—
317C3×C4—182no—
318C3×C4#381111yescharacteristic
319C3×C4C22×D3211yescharacteristic
320Dic3—34no—
321Dic3—62no—
322Dic3—64no—
323Dic3—64no—
324Dic3—64no—
325Dic3—184no—
326C32—21no—
327C32C2×#91512yescomplemented
328C23—21no—
329C23—91no—
330Q8—94no—
331Q8—92no—
332Q8—182no—
333Q8—181no—
334Q8—544no—
335D4—21no—
336D4—184no—
337D4C2×((C3⋊D3)⋊C3)⋊C212yes—
338C2×C4—91no—
339C2×C4—92no—
340C2×C4—92no—
341C2×C4—184no—
342C2×C4—181no—
343C2×C4—182no—
344C2×C4—544no—
345C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
346C8—94no—
347C2×C3—32no—
348C2×C3—34no—
349C2×C3—42no—
350C2×C3—61no—
351C2×C3—62no—
352C2×C3—124no—
353C2×C3—124no—
354C2×C3—182no—
355C2×C3#6393112yes—
356C2×C3C2×#381111yescharacteristic
357D3—64no—
358D3—122no—
359C22—22no—
360C22—42no—
361C22—92no—
362C22—182no—
363C22#8867711yescharacteristic
364C4—92no—
365C4—182no—
366C4—184no—
367C4#8867711yescharacteristic
368C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
369C3—32no—
370C3—61no—
371C3C2×#6393111yescharacteristic
372C2—42no—
373C2—182no—
374C2#1172118912yesdirect product factor
375C2C2×#8867711yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC4×H(3,𝔽3)C24
FrattiniC3×C4C22×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC2×#88744C2
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#137
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×#88651C2semidirect
3. C2×#3754⋊C3C2semidirect
4. #11721189C2direct (N × H)
5. C2×#88439C2semidirect
6. C2×#88435C2semidirect
19. #88651C22semidirect
20. #3754⋊C3C22semidirect
22. #88439C22semidirect
23. #88435C22semidirect
27. C2×#3754D3semidirect
74. #3754C2×D3semidirect
144. C2×H(3,𝔽3)#137semidirect
216. H(3,𝔽3)C2×#137semidirect
276. C2×C32#915semidirect
327. C32C2×#915semidirect
374. C2#11721189direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#887442
C2×#886512
C2×#3754⋊C32
#117211892
C2×#884392
C2×#884352
C2×#639293