Finite Group Databasebeta

← C2×#11721058 #13150592

Subgroups — C2×#11721058 #13150592

Order 1728.

971 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3828⋊C3C211yesmaximal, characteristic
2C22×#634⋊C3C211yesmaximal, characteristic, complemented
3C2×#88538C211yesmaximal, characteristic, complemented
4C2×#88433C212yesmaximal, complemented
5#11721058C218yesmaximal, direct product factor
6C2×#88431C212yesmaximal, complemented
7C2×#63833—31nomaximal
8C2×#63831—31nomaximal
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C2×#638⋊C3—21no—
11C22×(C3⋊Dic3)⋊C3—21no—
12#3828⋊C3—22no—
13#3828⋊C3C2214yes—
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15C2×C4×(C3⋊D3)⋊C3—21no—
16C2×#634⋊C3—22no—
17C2×#634⋊C3—21no—
18C2×#634⋊C3C2212yes—
19C2×#634⋊C3C2211yescharacteristic
20C2×#634⋊C3C2214yescomplemented
21#88538C2214yescomplemented
22C2×#6180C2212yes—
23#88433C2218yescomplemented
24#88431C2218yescomplemented
25C2×#3828D311yescharacteristic
26C22×#634D311yescharacteristic, complemented
27C2×C3×#235—31no—
28C22×C3×Dic6—31no—
29C2×C3×#208—31no—
30C2×C3×#208—31no—
31C2×C3×#208—61no—
32C2×#3719—32no—
33C2×#3719—32no—
34#63833—38no—
35C2×#3717—32no—
36#63831—38no—
37C2×#3716—32no—
38C23×H(3,𝔽3)D411yescharacteristic
39C22×(C3⋊D3)⋊C3—21no—
40C2×C4×H(3,𝔽3)D411yescharacteristic
41C2×C4×H(3,𝔽3)D412yes—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×C4×H(3,𝔽3)C2312yes—
44#638⋊C3—24no—
45C2×(C3⋊Dic3)⋊C3—22no—
46C2×(C3⋊Dic3)⋊C3—24no—
47C2×(C3⋊Dic3)⋊C3—21no—
48C4⋊((C3⋊D3)⋊C3)—21no—
49C4⋊((C3⋊D3)⋊C3)C2312yes—
50C4×(C3⋊D3)⋊C3—24no—
51#634⋊C3—22no—
52#634⋊C3—24no—
53#634⋊C3—21no—
54#634⋊C3C2314yes—
55#634⋊C3C2312yes—
56#6180C2314yes—
57C2×#951—91no—
58C2×#942—91no—
59C2×#909—91no—
60C22×C32×C4—21no—
61C22×C32×C4C2×D311yescharacteristic
62C22×C32×C4C2×D311yescharacteristic
63C2×#638—21no—
64C22×C3⋊Dic3—21no—
65#3828—22no—
66#3828C2×D314yes—
67C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
68C2×C4×C3⋊D3—21no—
69C2×#634—22no—
70C2×#634—21no—
71C2×#634C2×D312yes—
72C2×#634C2×D311yescharacteristic
73C2×#634C2×D314yescomplemented
74C2×C3×#67—61no—
75C22×C3×Dic3—61no—
76C3×#235—34no—
77C3×#235—62no—
78C2×C3×D12—31no—
79C2×C3×C4×D3—61no—
80C2×C3×Dic6—32no—
81C2×C3×Dic6—34no—
82C2×C3×Dic6—31no—
83C2×C3×Dic6—61no—
84C2×C3×Dic6—62no—
85C3×#208—34no—
86C3×#208—34no—
87C3×#208—64no—
88C2×C3×C3⋊C8—32no—
89C2×C3×C3⋊C8—32no—
90C2×C3×C3⋊C8—62no—
91#3719—38no—
92#3719—38no—
93#3717—38no—
94#3716—38no—
95C22×H(3,𝔽3)—22no—
96C22×H(3,𝔽3)C2×D412yes—
97C22×H(3,𝔽3)C2×D411yescharacteristic
98C2×(C3⋊D3)⋊C3—22no—
99C2×(C3⋊D3)⋊C3—42no—
100C4×H(3,𝔽3)C2×D411yescharacteristic
101C4×H(3,𝔽3)C2×D412yes—
102C4×H(3,𝔽3)C2411yescharacteristic
103(C3⋊Dic3)⋊C3—24no—
104(C3⋊Dic3)⋊C3—22no—
105C2×C3×#41—91no—
106C22×C3×Q8—91no—
107C2×#235—31no—
108C2×#235—91no—
109C22×Dic6—31no—
110C22×Dic6—91no—
111C2×C3×M16—91no—
112#951—98no—
113C2×#216—92no—
114C2×#216—92no—
115#942—98no—
116C2×#215—92no—
117C2×#214—92no—
118C2×#208—31no—
119C2×#208—31no—
120C2×#208—61no—
121#909—98no—
122C2×Dic12—92no—
123C2×#204—92no—
124C23×C32—21no—
125C23×C32#6711yescharacteristic
126C23×C32#6711yescharacteristic
127C22×C3⋊D3—21no—
128C22×C3×D3—61no—
129C2×C32×C4—21no—
130C2×C32×C4—22no—
131C2×C32×C4—21no—
132C2×C32×C4—22no—
133C2×C32×C4#6711yescharacteristic
134C2×C32×C4#6712yes—
135C2×C32×C4#6711yescharacteristic
136C2×C32×C4#6712yes—
137C2×C32×C4C22×D311yescharacteristic
138C2×C32×C4C22×D311yescharacteristic
139C2×C32×C4C22×D312yes—
140C2×C32×C4C22×D312yes—
141#638—24no—
142C2×C3⋊Dic3—22no—
143C2×C3⋊Dic3—24no—
144C2×C3⋊Dic3—21no—
145C4⋊(C3⋊D3)—21no—
146C4⋊(C3⋊D3)C22×D312yes—
147C4×C3⋊D3—24no—
148#634—22no—
149#634—24no—
150#634—21no—
151#634C22×D314yes—
152#634C22×D312yes—
153C3×#67—64no—
154C2×C3×Dic3—64no—
155C2×C3×Dic3—62no—
156C2×C3×Dic3—61no—
157C3×D12—32no—
158C3×D12—61no—
159C3×C4×D3—64no—
160C3×Dic6—34no—
161C3×Dic6—32no—
162C3×Dic6—62no—
163C3×Dic6—64no—
164C3×Dic6—61no—
165C3×C3⋊C8—34no—
166C3×C3⋊C8—34no—
167C3×C3⋊C8—64no—
168C2×#137—271noSylow
169C2×H(3,𝔽3)—22no—
170C2×H(3,𝔽3)#13712yescomplemented
171C2×H(3,𝔽3)C22×D411yescharacteristic
172(C3⋊D3)⋊C3—42no—
173C3×#41—94no—
174C3×#41—182no—
175C2×C3×Q8—92no—
176C2×C3×Q8—94no—
177C2×C3×Q8—91no—
178C2×C3×Q8—181no—
179C2×C3×Q8—182no—
180C2×C3×D4—91no—
181C2×C3×D4—181no—
182C22×C3×C4—31no—
183C22×C3×C4—31no—
184C22×C3×C4—61no—
185C22×C3×C4—181no—
186C22×C3×C4—181no—
187C22×C3×C4D3211yescharacteristic
188C2×#67—61no—
189C2×#67—181no—
190C22×Dic3—61no—
191C22×Dic3—181no—
192#235—34no—
193#235—62no—
194#235—94no—
195#235—182no—
196C2×D12—31no—
197C2×D12—91no—
198C2×C4×D3—61no—
199C2×C4×D3—181no—
200C2×Dic6—32no—
201C2×Dic6—31no—
202C2×Dic6—34no—
203C2×Dic6—61no—
204C2×Dic6—62no—
205C2×Dic6—92no—
206C2×Dic6—91no—
207C2×Dic6—94no—
208C2×Dic6—181no—
209C2×Dic6—182no—
210C3×M16—94no—
211C2×C3×C8—92no—
212#216—98no—
213#216—98no—
214#215—98no—
215#214—98no—
216#208—34no—
217#208—34no—
218#208—64no—
219C2×C3⋊C8—32no—
220C2×C3⋊C8—32no—
221C2×C3⋊C8—62no—
222Dic12—98no—
223#204—98no—
224C22×C32—22no—
225C22×C32—22no—
226C22×C32—21no—
227C22×C32—22no—
228C22×C32—42no—
229C22×C32C2×#6712yes—
230C22×C32C2×#6712yes—
231C22×C32C2×#6711yescharacteristic
232C22×C32C2×#6711yescharacteristic
233C2×C3⋊D3—22no—
234C2×C3⋊D3—42no—
235C2×C3×D3—62no—
236C2×C3×D3—122no—
237C32×C4—21no—
238C32×C4—21no—
239C32×C4—22no—
240C32×C4C2×#6711yescharacteristic
241C32×C4C2×#6712yes—
242C32×C4C2×#6711yescharacteristic
243C32×C4C2×#6712yes—
244C32×C4C23×D311yescharacteristic
245C32×C4C23×D311yescharacteristic
246C3⋊Dic3—24no—
247C3⋊Dic3—22no—
248C3×Dic3—64no—
249C3×Dic3—62no—
250C2×#41—91no—
251C22×Q8—91no—
252#137—278no—
253C2×Q16—272no—
254C2×SD16—272no—
255C2×M16—91no—
256H(3,𝔽3)C2×#13711yescharacteristic, complemented, Sylow
257C23×C3—31no—
258C23×C3—31no—
259C23×C3—61no—
260C23×C3—181no—
261C23×C3#62511yescharacteristic
262C22×D3—61no—
263C22×D3—181no—
264C3×Q8—94no—
265C3×Q8—92no—
266C3×Q8—182no—
267C3×Q8—184no—
268C3×Q8—181no—
269C3×D4—92no—
270C3×D4—181no—
271C3×D4—184no—
272C2×C3×C4—31no—
273C2×C3×C4—32no—
274C2×C3×C4—31no—
275C2×C3×C4—32no—
276C2×C3×C4—31no—
277C2×C3×C4—31no—
278C2×C3×C4—32no—
279C2×C3×C4—32no—
280C2×C3×C4—62no—
281C2×C3×C4—61no—
282C2×C3×C4—61no—
283C2×C3×C4—62no—
284C2×C3×C4—184no—
285C2×C3×C4—181no—
286C2×C3×C4—184no—
287C2×C3×C4—182no—
288C2×C3×C4#62511yescharacteristic
289C2×C3×C4#62512yes—
290C2×C3×C4C2×D3211yescharacteristic
291C2×C3×C4C2×D3212yes—
292#67—64no—
293#67—184no—
294C2×Dic3—64no—
295C2×Dic3—62no—
296C2×Dic3—61no—
297C2×Dic3—184no—
298C2×Dic3—182no—
299C2×Dic3—181no—
300D12—32no—
301D12—61no—
302D12—92no—
303D12—181no—
304C4×D3—64no—
305C4×D3—184no—
306Dic6—34no—
307Dic6—32no—
308Dic6—62no—
309Dic6—64no—
310Dic6—61no—
311Dic6—94no—
312Dic6—92no—
313Dic6—184no—
314Dic6—182no—
315Dic6—181no—
316C3×C8—94no—
317C3⋊C8—34no—
318C3⋊C8—34no—
319C3⋊C8—64no—
320C2×C32—22no—
321C2×C32—22no—
322C2×C32—21no—
323C2×C32—22no—
324C2×C32—42no—
325C2×C32#94212yescomplemented
326C2×C32#95112yescomplemented
327C2×C32C22×#6711yescharacteristic
328C2×C32C22×#6711yescharacteristic
329C3⋊D3—42no—
330C3×D3—122no—
331#41—94no—
332#41—182no—
333C2×Q8—92no—
334C2×Q8—91no—
335C2×Q8—94no—
336C2×Q8—182no—
337C2×Q8—181no—
338C2×D4—91no—
339C2×D4—181no—
340C22×C4—181no—
341C22×C4—181no—
342C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
343Q16—278no—
344SD16—278no—
345M16—94no—
346C2×C8—92no—
347C22×C3—22no—
348C22×C3—32no—
349C22×C3—32no—
350C22×C3—31no—
351C22×C3—31no—
352C22×C3—62no—
353C22×C3—62no—
354C22×C3—62no—
355C22×C3—61no—
356C22×C3—122no—
357C22×C3—182no—
358C22×C3—362no—
359C22×C3C2×#62512yes—
360C22×C3C2×#62511yescharacteristic
361C2×D3—62no—
362C2×D3—122no—
363C2×D3—182no—
364C2×D3—362no—
365C3×C4—31no—
366C3×C4—31no—
367C3×C4—32no—
368C3×C4—31no—
369C3×C4—31no—
370C3×C4—32no—
371C3×C4—61no—
372C3×C4—62no—
373C3×C4—61no—
374C3×C4—182no—
375C3×C4—184no—
376C3×C4C2×#62511yescharacteristic
377C3×C4C2×#62512yes—
378C3×C4C22×D3211yescharacteristic
379Dic3—64no—
380Dic3—62no—
381Dic3—184no—
382Dic3—182no—
383C32—21no—
384C32C2×#94211yescharacteristic, complemented
385C32C2×#95111yescharacteristic, complemented
386C23—181no—
387C23#619811yescharacteristic
388Q8—94no—
389Q8—92no—
390Q8—184no—
391Q8—182no—
392Q8—181no—
393D4—92no—
394D4—184no—
395D4—181no—
396C2×C4—184no—
397C2×C4—181no—
398C2×C4—182no—
399C2×C4—184no—
400C2×C4#619812yes—
401C2×C4#619811yescharacteristic
402C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
403C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
404C8—94no—
405C2×C3—22no—
406C2×C3—31no—
407C2×C3—31no—
408C2×C3—32no—
409C2×C3—32no—
410C2×C3—62no—
411C2×C3—62no—
412C2×C3—61no—
413C2×C3—62no—
414C2×C3—122no—
415C2×C3—362no—
416C2×C3#6382712yes—
417C2×C3C22×#62511yescharacteristic
418D3—122no—
419D3—362no—
420C22—22no—
421C22—182no—
422C22—362no—
423C22C2×#619812yes—
424C22C2×#619811yescharacteristic
425C4—182no—
426C4—184no—
427C4C2×#619812yes—
428C4C2×#619811yescharacteristic
429C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
430C3—31no—
431C3—31no—
432C3—61no—
433C3C2×#6382711yescharacteristic
434C2—22no—
435C2—362no—
436C2#1172105812yesdirect product factor
437C2C22×#619811yescharacteristic
subgroupG/H
centerC22C2×#6198
commutator, cosocleC4×H(3,𝔽3)C24
FrattiniC3×C4C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#625
prime radicalC2×#3828⋊C3C2
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#137
LCSG ▷ 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. C22×#634⋊C3C2semidirect
3. C2×#88538C2semidirect
4. C2×#88433C2semidirect
5. #11721058C2direct (N × H)
6. C2×#88431C2semidirect
20. C2×#634⋊C3C22semidirect
21. #88538C22semidirect
23. #88433C22semidirect
24. #88431C22semidirect
26. C22×#634D3semidirect
73. C2×#634C2×D3semidirect
170. C2×H(3,𝔽3)#137semidirect
256. H(3,𝔽3)C2×#137semidirect
325. C2×C32#942semidirect
326. C2×C32#951semidirect
384. C32C2×#942semidirect
385. C32C2×#951semidirect
436. C2#11721058direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#3828⋊C32
C22×#634⋊C32
C2×#885382
C2×#884332
#117210582
C2×#884312
C2×#638333
C2×#638313