Finite Group Databasebeta

← C2×#11723554 #13177665

Subgroups — C2×#11723554 #13177665

Order 1728.

2204 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89078C211yesmaximal, characteristic
2C2×#89019C211yesmaximal, characteristic, complemented
3C2×C3×#3754C211yesmaximal, characteristic, complemented
4C2×C32×#214C211yesmaximal, characteristic, complemented
5#11723554C218yesmaximal, direct product factor
6C2×#88800C211yesmaximal, characteristic, complemented
7C2×#88799C211yesmaximal, characteristic, complemented
8C2×#88788C211yesmaximal, characteristic, complemented
9C2×#64124—31nomaximal
10C2×#63931—34nomaximal
11C2×C33×D4C2211yescharacteristic
12C2×#6312—21no—
13C22×C3⋊(C3⋊Dic3)—21no—
14#89078—22no—
15#89078C2214yes—
16C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
17C2×#6308C2211yescharacteristic
18C2×C3×#634C2211yescharacteristic
19C2×C32×Dic6C2211yescharacteristic
20C2×#6293—21no—
21C2×#6289—21no—
22#89019—22no—
23#89019C2214yescomplemented
24C3×#3754C2214yescomplemented
25C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
26C32×#214C2214yescomplemented
27C2×C32×C3⋊C8C2211yescharacteristic
28#88800C2214yescomplemented
29#88799C2214yescomplemented
30#88788C2214yescomplemented
31C2×Q8×C3⋊D3—31no—
32C2×#3830—31no—
33C2×#3830—34no—
34C2×#3830—64no—
35C2×#3797—34no—
36C2×C32×SD16—31no—
37C2×#3756—31no—
38C2×#3754D311yescharacteristic, complemented
39#64124—38no—
40C2×#3746—31no—
41C2×#3743—31no—
42C2×C3×#214—34no—
43C2×C3×#214D314yescomplemented
44#63931—332no—
45C2×#3719—34no—
46C2×#3719—34no—
47C2×#3712—34no—
48C23×C33—21no—
49C22×C3⋊(C3⋊D3)D411yescharacteristic
50C33×D4—21no—
51C33×D4C2312yes—
52C2×C33×C4C2311yescharacteristic
53#6312—24no—
54C2×C3⋊(C3⋊Dic3)—21no—
55C2×C3⋊(C3⋊Dic3)—24no—
56C2×C3⋊(C3⋊Dic3)D411yescharacteristic
57C4×C3⋊(C3⋊D3)D412yes—
58C4×C3⋊(C3⋊D3)C2312yes—
59#6308—21no—
60#6308C2312yes—
61C2×C3×C3⋊Dic3—21no—
62C3×#634—21no—
63C3×#634C2312yes—
64C2×C32×Dic3—21no—
65C32×Dic6—21no—
66C32×Dic6C2312yes—
67#6293—24no—
68#6289—24no—
69C3×C3⋊(C3⋊C8)C2312yes—
70C32×C3⋊C8C2312yes—
71C2×#915—94no—
72C2×#915—91no—
73C2×C32×Q8—31no—
74C2×C32×D4—24no—
75C2×C32×D4C2×D314yes—
76C2×C32×D4C2×D311yescharacteristic
77C2×#638—61no—
78C2×#638—64no—
79C2×#638—68no—
80C22×C3⋊Dic3—61no—
81C22×C3⋊Dic3—64no—
82C22×C3⋊Dic3—68no—
83Q8×C3⋊D3—34no—
84Q8×C3⋊D3—62no—
85#3830—34no—
86#3830—316no—
87#3830—62no—
88#3830—68no—
89#3830—616no—
90#3830—616no—
91C2×C4×C3⋊D3—31no—
92C2×C4×C3⋊D3—34no—
93C2×C4×C3⋊D3—61no—
94C2×C4×C3⋊D3—64no—
95C2×#634—31no—
96C2×#634—34no—
97C2×#634—61no—
98C2×#634—64no—
99C2×#634C2×D311yescharacteristic
100C2×C3×Dic6—34no—
101C2×C3×Dic6C2×D314yes—
102C2×#627—64no—
103C2×#624—64no—
104#3797—316no—
105#3797—68no—
106C32×SD16—34no—
107C2×C32×C8—31no—
108#3756—34no—
109#3754C2×D314yescomplemented
110C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
111#3746—34no—
112#3743—34no—
113C3×#214—316no—
114C3×#214C2×D3116yescomplemented
115C2×C3×C3⋊C8—34no—
116C2×C3×C3⋊C8C2×D314yes—
117#3719—316no—
118#3719—316no—
119#3712—316no—
120C22×C33—22no—
121C22×C33—42no—
122C22×C33C2×D411yescharacteristic
123C2×C3⋊(C3⋊D3)—22no—
124C2×C3⋊(C3⋊D3)C2×D412yes—
125C33×C4C2×D411yescharacteristic
126C33×C4C2411yescharacteristic
127C3⋊(C3⋊Dic3)—22no—
128C3⋊(C3⋊Dic3)C2×D412yes—
129C3×C3⋊Dic3—22no—
130C32×Dic3—22no—
131C2×D3×Q8—94no—
132C2×D3×Q8—91no—
133C2×#237—94no—
134C2×#237—91no—
135C2×#237—184no—
136C2×C3×SD16—34no—
137C2×C3×SD16—91no—
138C2×#216—94no—
139C2×#216—91no—
140C2×#214—34no—
141C2×#214C3⋊D311yescharacteristic, complemented
142#915—932no—
143#915—98no—
144C2×Dic12—94no—
145C2×Dic12—91no—
146C2×ΓL(1,𝔽25)—94no—
147C2×ΓL(1,𝔽25)—91no—
148C23×C32—28no—
149C23×C32—24no—
150C23×C32—21no—
151C22×C3⋊D3—31no—
152C22×C3⋊D3—34no—
153C22×C3⋊D3—64no—
154C32×Q8—32no—
155C32×Q8—61no—
156C32×D4—28no—
157C32×D4—24no—
158C32×D4—21no—
159C32×D4—28no—
160C32×D4C22×D318yes—
161C32×D4C22×D312yes—
162C2×C32×C4—24no—
163C2×C32×C4—61no—
164C2×C32×C4C22×D314yes—
165C2×C32×C4C22×D311yescharacteristic
166#638—64no—
167#638—616no—
168#638—632no—
169C2×C3⋊Dic3—21no—
170C2×C3⋊Dic3—31no—
171C2×C3⋊Dic3—34no—
172C2×C3⋊Dic3—64no—
173C2×C3⋊Dic3—64no—
174C2×C3⋊Dic3—616no—
175C2×C3⋊Dic3—68no—
176C2×C3⋊Dic3—632no—
177C2×C3⋊Dic3—64no—
178C2×C3⋊Dic3—61no—
179C4×C3⋊D3—32no—
180C4×C3⋊D3—32no—
181C4×C3⋊D3—38no—
182C4×C3⋊D3—38no—
183C4×C3⋊D3—68no—
184C4×C3⋊D3—68no—
185C4×C3⋊D3—64no—
186#634—21no—
187#634—32no—
188#634—38no—
189#634—68no—
190#634—64no—
191#634—68no—
192#634—64no—
193#634—61no—
194#634C22×D312yes—
195C2×C3×Dic3—24no—
196C2×C3×Dic3—64no—
197C3×Dic6—24no—
198C3×Dic6—38no—
199C3×Dic6—64no—
200C3×Dic6C22×D318yes—
201#627—616no—
202#624—616no—
203C32×C8—32no—
204C3⋊(C3⋊C8)C22×D312yes—
205C3×C3⋊C8—38no—
206C3×C3⋊C8C22×D318yes—
207C2×#137—271noSylow
208C2×C33—42no—
209C2×C33#13712yescomplemented
210C2×C33C22×D411yescharacteristic
211C3⋊(C3⋊D3)—22no—
212C2×C3×Q8—34no—
213C2×C3×Q8—91no—
214C2×C3×D4—24no—
215C2×C3×D4D3214yes—
216C2×C3×D4C2×C3⋊D311yescharacteristic
217C2×#67—184no—
218C2×#67—181no—
219C2×#67—188no—
220C22×Dic3—184no—
221C22×Dic3—181no—
222C22×Dic3—188no—
223D3×Q8—916no—
224D3×Q8—94no—
225D3×Q8—182no—
226D3×Q8—188no—
227#237—916no—
228#237—94no—
229#237—1816no—
230#237—188no—
231#237—182no—
232#237—1816no—
233C2×C4×D3—94no—
234C2×C4×D3—91no—
235C2×C4×D3—184no—
236C2×C4×D3—181no—
237C2×C4×D3—184no—
238C2×Dic6—34no—
239C2×Dic6—94no—
240C2×Dic6—91no—
241C2×Dic6—184no—
242C2×Dic6—181no—
243C2×Dic6—184no—
244C2×Dic6C2×C3⋊D311yescharacteristic
245C3×SD16—316no—
246C3×SD16—94no—
247C2×C3×C8—34no—
248C2×C3×C8—91no—
249#216—916no—
250#216—94no—
251#214—316no—
252#214C2×C3⋊D314yescomplemented
253C2×C3⋊C8—34no—
254C2×C3⋊C8C2×C3⋊D311yescharacteristic
255Dic12—916no—
256Dic12—94no—
257ΓL(1,𝔽25)—916no—
258ΓL(1,𝔽25)—94no—
259C22×C32—216no—
260C22×C32—28no—
261C22×C32—22no—
262C22×C32—24no—
263C22×C32—416no—
264C22×C32—48no—
265C22×C32—42no—
266C22×C32D3×D414yes—
267C22×C32D3×D411yescharacteristic
268C2×C3⋊D3—32no—
269C2×C3⋊D3—38no—
270C2×C3⋊D3—68no—
271C2×C3⋊D3—62no—
272C2×C3⋊D3—68no—
273C2×C3⋊D3—128no—
274C32×C4—24no—
275C32×C4—24no—
276C32×C4—62no—
277C32×C4D3×D414yes—
278C32×C4D3×D411yescharacteristic
279C32×C4C23×D314yes—
280C32×C4C23×D311yescharacteristic
281C3⋊Dic3—22no—
282C3⋊Dic3—32no—
283C3⋊Dic3—38no—
284C3⋊Dic3—68no—
285C3⋊Dic3—616no—
286C3⋊Dic3—68no—
287C3⋊Dic3—62no—
288C3×Dic3—28no—
289C3×Dic3—68no—
290C2×#41—271no—
291C22×Q8—271no—
292#137—278no—
293C2×Q16—271no—
294C2×Q16—271no—
295C2×SD16—31no—
296C2×SD16—91no—
297C2×M16—271no—
298C33C2×#13711yescharacteristic, complemented, Sylow
299C23×C3—28no—
300C23×C3—24no—
301C23×C3—21no—
302C22×D3—94no—
303C22×D3—91no—
304C22×D3—184no—
305C3×Q8—38no—
306C3×Q8—64no—
307C3×Q8—92no—
308C3×Q8—181no—
309C3×D4—28no—
310C3×D4—24no—
311C3×D4—21no—
312C3×D4—28no—
313C3×D4C2×D3218yes—
314C3×D4C22×C3⋊D312yes—
315C2×C3×C4—24no—
316C2×C3×C4—64no—
317C2×C3×C4—181no—
318C2×C3×C4C2×D3214yes—
319C2×C3×C4C22×C3⋊D311yescharacteristic
320#67—1816no—
321#67—184no—
322#67—1832no—
323C2×Dic3—21no—
324C2×Dic3—64no—
325C2×Dic3—94no—
326C2×Dic3—91no—
327C2×Dic3—184no—
328C2×Dic3—1816no—
329C2×Dic3—184no—
330C2×Dic3—188no—
331C2×Dic3—1832no—
332C2×Dic3—184no—
333C2×Dic3—181no—
334C4×D3—98no—
335C4×D3—92no—
336C4×D3—98no—
337C4×D3—92no—
338C4×D3—188no—
339C4×D3—188no—
340C4×D3—184no—
341C4×D3—1816no—
342Dic6—21no—
343Dic6—38no—
344Dic6—64no—
345Dic6—98no—
346Dic6—92no—
347Dic6—188no—
348Dic6—184no—
349Dic6—1816no—
350Dic6—188no—
351Dic6—184no—
352Dic6—181no—
353Dic6C22×C3⋊D312yes—
354C3×C8—38no—
355C3×C8—92no—
356C3⋊C8—38no—
357C3⋊C8C22×C3⋊D312yes—
358C2×C32—24no—
359C2×C32—28no—
360C2×C32—416no—
361C2×C32—48no—
362C2×C32—42no—
363C2×C32#91518yescomplemented
364C2×C32#91512yescomplemented
365C2×C32C2×D3×D414yes—
366C2×C32C2×D3×D411yescharacteristic
367C3⋊D3—62no—
368C3⋊D3—68no—
369C3⋊D3—128no—
370#41—274no—
371#41—542no—
372C2×Q8—31no—
373C2×Q8—91no—
374C2×Q8—274no—
375C2×Q8—271no—
376C2×Q8—541no—
377C2×Q8—542no—
378C2×D4—541no—
379C2×D4D3×C3⋊D311yescharacteristic
380C22×C4—271no—
381C22×C4—541no—
382C22×C4—541no—
383Q16—274no—
384Q16—274no—
385SD16—34no—
386SD16—94no—
387M16—274no—
388C2×C8—31no—
389C2×C8—91no—
390C22×C3—216no—
391C22×C3—28no—
392C22×C3—22no—
393C22×C3—24no—
394C22×C3—416no—
395C22×C3—42no—
396C22×C3—48no—
397C22×C3#381114yes—
398C22×C3D4×C3⋊D311yescharacteristic
399C2×D3—98no—
400C2×D3—92no—
401C2×D3—188no—
402C2×D3—182no—
403C2×D3—188no—
404C2×D3—368no—
405C3×C4—24no—
406C3×C4—24no—
407C3×C4—68no—
408C3×C4—182no—
409C3×C4#381114yes—
410C3×C4D4×C3⋊D311yescharacteristic
411C3×C4C22×D3214yes—
412C3×C4C23×C3⋊D311yescharacteristic
413Dic3—22no—
414Dic3—68no—
415Dic3—98no—
416Dic3—92no—
417Dic3—188no—
418Dic3—1816no—
419Dic3—188no—
420Dic3—182no—
421C32—24no—
422C32C2×#91514yescomplemented
423C32C2×#91511yescharacteristic, complemented
424C23—21no—
425C23—271no—
426Q8—32no—
427Q8—61no—
428Q8—92no—
429Q8—181no—
430Q8—272no—
431Q8—544no—
432Q8—541no—
433D4—21no—
434D4—544no—
435D4C2×D3×C3⋊D312yes—
436C2×C4—61no—
437C2×C4—181no—
438C2×C4—271no—
439C2×C4—272no—
440C2×C4—272no—
441C2×C4—544no—
442C2×C4—544no—
443C2×C4—541no—
444C2×C4C2×D3×C3⋊D311yescharacteristic
445C8—32no—
446C8—92no—
447C2×C3—28no—
448C2×C3—24no—
449C2×C3—416no—
450C2×C3—48no—
451C2×C3—42no—
452C2×C3#6393118yescomplemented
453C2×C3#6412412yescomplemented
454C2×C3C2×#381114yes—
455C2×C3C2×D4×C3⋊D311yescharacteristic
456D3—188no—
457D3—182no—
458D3—368no—
459C22—22no—
460C22—42no—
461C22—272no—
462C22—542no—
463C22#8903911yescharacteristic
464C4—62no—
465C4—182no—
466C4—272no—
467C4—542no—
468C4#8903911yescharacteristic
469C4C22×D3×C3⋊D311yescharacteristic
470C3—24no—
471C3C2×#6393114yescomplemented
472C3C2×#6412411yescharacteristic, complemented
473C2—42no—
474C2—542no—
475C2#1172355412yesdirect product factor
476C2C2×#8903911yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC2×#89078C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33C2×#137
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×#89019C2semidirect
3. C2×C3×#3754C2semidirect
4. C2×C32×#214C2semidirect
5. #11723554C2direct (N × H)
6. C2×#88800C2semidirect
7. C2×#88799C2semidirect
8. C2×#88788C2semidirect
23. #89019C22semidirect
24. C3×#3754C22semidirect
26. C32×#214C22semidirect
28. #88800C22semidirect
29. #88799C22semidirect
30. #88788C22semidirect
38. C2×#3754D3semidirect
43. C2×C3×#214D3semidirect
109. #3754C2×D3semidirect
114. C3×#214C2×D3semidirect
141. C2×#214C3⋊D3semidirect
209. C2×C33#137semidirect
252. #214C2×C3⋊D3semidirect
298. C33C2×#137semidirect
363. C2×C32#915semidirect
364. C2×C32#915semidirect
422. C32C2×#915semidirect
423. C32C2×#915semidirect
452. C2×C3#63931semidirect
453. C2×C3#64124semidirect
471. C3C2×#63931semidirect
472. C3C2×#64124semidirect
475. C2#11723554direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890782
C2×#890192
C2×C3×#37542
C2×C32×#2142
#117235542
C2×#888002
C2×#887992
C2×#887882
C2×#641243
C2×#639313