Finite Group Databasebeta

← C2×C3×D32⋊Q8 #13181011

Subgroups — C2×C3×D32⋊Q8 #13181011

Order 1728.

1128 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1128 conjugacy classes of subgroups — too many to draw. Only its 210 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×C3×C4×D32C211yesmaximal, characteristic
2C2×C3×#3797C211yesmaximal, characteristic, complemented
3C2×C3×C4⋊#157C211yesmaximal, characteristic, complemented
4C2×C3×#3773C212yesmaximal, complemented
5C2×C3×D32⋊C4C212yesmaximal
6C3×D32⋊Q8C218yesmaximal, direct product factor
7C2×D32⋊Q8C311yesmaximal, characteristic, direct product factor
8C22×C3×D32C2211yescharacteristic
9C22×C3×#157C2212yes—
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C32×C4×D3—21no—
12C2×C32×Dic6—21no—
13C2×C3×#627—21no—
14C2×C3×#624C2211yescharacteristic
15C2×C3×#624C2212yes—
16C2×C3×D3×Dic3—21no—
17C3×C4×D32—22no—
18C3×C4×D32C2214yes—
19C3×#3797—22no—
20C3×#3797C2214yescomplemented
21C3×C4⋊#157C2214yescomplemented
22C3×#3773C2218yescomplemented
23C3×D32⋊C4C2218yes—
24C2×C3×D3×Q8—61no—
25C22×C3×C4×D3—61no—
26C2×C4×D32C2×C311yescharacteristic
27C2×#3797C2×C311yescharacteristic, complemented
28C2×C4⋊#157C2×C311yescharacteristic, complemented
29C2×#3773C2×C312yescomplemented
30C2×D32⋊C4C2×C312yes—
31D32⋊Q8C2×C318yesdirect product factor
32C22×C3×C3⋊D3C2311yescharacteristic
33C22×C32×D3—21no—
34C2×C3×D32—21no—
35C2×C3×D32—22no—
36C2×C3×D32Q814yescomplemented
37C2×C3×D32C2312yes—
38C2×C3×#157—24no—
39C2×C3×#157C2314yes—
40C2×C33×C4D411yescharacteristic
41C2×C3×C3⋊Dic3D411yescharacteristic
42C3×C4×C3⋊D3D412yes—
43C3×C4×C3⋊D3C2312yes—
44C2×C32×Dic3—21no—
45C2×C32×Dic3—22no—
46C32×C4×D3—24no—
47C32×Dic6—24no—
48C3×#627—24no—
49C3×#624—22no—
50C3×#624—21no—
51C3×#624C2314yes—
52C3×#624C2312yes—
53C3×D3×Dic3—24no—
54C2×C3×C22⋊Q8—91nomaximal
55C23×C3×D3—61no—
56C22×D32C22×C311yescharacteristic
57C22×#157C22×C312yes—
58C2×C32×Q8—61no—
59C22×C32×C4—61no—
60C2×C4×C3⋊D3C22×C311yescharacteristic
61C22×C3×Dic3—61no—
62C3×D3×Q8—64no—
63C3×D3×Q8—64no—
64C2×C3×C4×D3—21no—
65C2×C3×C4×D3—21no—
66C2×C3×C4×D3—41no—
67C2×C3×C4×D3—64no—
68C2×C3×C4×D3—64no—
69C2×C3×C4×D3—61no—
70C2×C3×C4×D3—62no—
71C2×C3×C4×D3—61no—
72C2×C3×C4×D3—61no—
73C2×C3×C4×D3—61no—
74C2×C3×Dic6—21no—
75C2×C3×Dic6—21no—
76C2×C3×Dic6—41no—
77C2×C3×Dic6—62no—
78C2×#627—21no—
79C2×#624C22×C312yes—
80C2×#624C22×C311yescharacteristic
81C2×D3×Dic3—21no—
82C4×D32—22no—
83C4×D32C22×C314yes—
84#3797—22no—
85#3797C22×C314yescomplemented
86C4⋊#157C22×C314yescomplemented
87#3773C22×C318yescomplemented
88D32⋊C4C22×C318yes—
89C22×C33C2×D411yescharacteristic
90C2×C3×C3⋊D3C2×D411yescharacteristic
91C2×C3×C3⋊D3C2×Q812yes—
92C2×C3×C3⋊D3#4112yes—
93C2×C3×C3⋊D3C2411yescharacteristic
94C2×C32×D3—24no—
95C2×C32×D3—22no—
96C3×D32—24no—
97C3×D32—22no—
98C3×D32C2×Q814yescomplemented
99C3×#157—24no—
100C33×C4C2×D412yes—
101C3×C3⋊Dic3C2×D412yes—
102C32×Dic3—22no—
103C32×Dic3—24no—
104C22×C3×Q8—91no—
105C23×C3×C4—91no—
106C2×D3×Q8—61no—
107C22×C4×D3—61no—
108C3×C22⋊Q8—98no—
109C2×C3×C4⋊C4—92no—
110C2×C3×C4⋊C4—91no—
111C2×C3×C22⋊C4—92no—
112C23×C32—61no—
113C22×C3⋊D3C23×C311yescharacteristic
114C22×C3×D3—21no—
115C22×C3×D3—21no—
116C22×C3×D3—41no—
117C22×C3×D3—64no—
118C22×C3×D3—62no—
119C22×C3×D3—61no—
120C22×C3×D3—61no—
121C22×C3×D3—64no—
122C22×C3×D3—62no—
123C2×D32—21no—
124C2×D32—22no—
125C2×D32C3×Q814yescomplemented
126C2×D32C23×C312yes—
127C2×#157—24no—
128C2×#157C23×C314yes—
129C32×Q8—64no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—41no—
133C2×C32×C4—41no—
134C2×C32×C4—64no—
135C2×C32×C4—62no—
136C2×C32×C4—61no—
137C2×C32×C4C3×D411yescharacteristic
138C2×C3⋊Dic3C3×D411yescharacteristic
139C4×C3⋊D3C3×D412yes—
140C4×C3⋊D3C23×C312yes—
141C2×C3×Dic3—22no—
142C2×C3×Dic3—21no—
143C2×C3×Dic3—21no—
144C2×C3×Dic3—22no—
145C2×C3×Dic3—42no—
146C2×C3×Dic3—41no—
147C2×C3×Dic3—61no—
148C2×C3×Dic3—61no—
149C2×C3×Dic3—64no—
150C3×C4×D3—24no—
151C3×C4×D3—24no—
152C3×C4×D3—44no—
153C3×C4×D3—62no—
154C3×C4×D3—64no—
155C3×C4×D3—62no—
156C3×C4×D3—62no—
157C3×C4×D3—62no—
158C3×C4×D3—62no—
159C3×C4×D3—62no—
160C3×C4×D3—64no—
161C3×C4×D3—64no—
162C3×Dic6—24no—
163C3×Dic6—24no—
164C3×Dic6—44no—
165C3×Dic6—68no—
166#627—24no—
167#624—22no—
168#624—21no—
169#624C23×C314yes—
170#624C23×C312yes—
171D3×Dic3—24no—
172C2×C22⋊Q8—91noSylow
173C2×C33C22⋊Q812yescomplemented
174C2×C33C22×D411yescharacteristic
175C3×C3⋊D3C22⋊Q812yescomplemented
176C3×C3⋊D3C22×Q811yescharacteristic
177C3×C3⋊D3C2×#4111yescharacteristic
178C32×D3—24no—
179C24×C3—91no—
180C23×D3—61no—
181C2×C3×Q8—61no—
182C2×C3×Q8—61no—
183C2×C3×Q8—94no—
184C2×C3×Q8—121no—
185C2×C3×Q8—181no—
186C2×C3×Q8—182no—
187C22×C3×C4—61no—
188C22×C3×C4—61no—
189C22×C3×C4—92no—
190C22×C3×C4—91no—
191C22×C3×C4—94no—
192C22×C3×C4—91no—
193C22×C3×C4—92no—
194C22×C3×C4—121no—
195C22×C3×C4—182no—
196C22×C3×C4—181no—
197C22×Dic3—61no—
198D3×Q8—64no—
199D3×Q8—64no—
200C2×C4×D3—21no—
201C2×C4×D3—64no—
202C2×C4×D3—61no—
203C2×C4×D3—62no—
204C2×C4×D3—61no—
205C2×C4×D3—61no—
206C2×C4×D3—64no—
207C2×C4×D3—61no—
208C2×Dic6—21no—
209C2×Dic6—62no—
210C3×C4⋊C4—98no—
211C3×C4⋊C4—94no—
212C3×C22⋊C4—98no—
213C22×C32—21no—
214C22×C32—21no—
215C22×C32—41no—
216C22×C32—41no—
217C22×C32—62no—
218C22×C32—64no—
219C22×C32C2×C3×D411yescharacteristic
220C2×C3⋊D3C2×C3×D411yescharacteristic
221C2×C3⋊D3C2×C3×Q812yes—
222C2×C3⋊D3C3×#4112yes—
223C2×C3⋊D3C24×C311yescharacteristic
224C2×C3×D3—22no—
225C2×C3×D3—24no—
226C2×C3×D3—24no—
227C2×C3×D3—22no—
228C2×C3×D3—42no—
229C2×C3×D3—44no—
230C2×C3×D3—64no—
231C2×C3×D3—61no—
232C2×C3×D3—61no—
233C2×C3×D3—62no—
234C2×C3×D3—62no—
235C2×C3×D3—62no—
236C2×C3×D3—62no—
237C2×C3×D3—61no—
238C2×C3×D3—61no—
239C2×C3×D3—68no—
240C2×C3×D3—64no—
241D32—24no—
242D32—22no—
243D32C2×C3×Q814yescomplemented
244#157—24no—
245C32×C4—22no—
246C32×C4—22no—
247C32×C4—42no—
248C32×C4—42no—
249C32×C4—64no—
250C32×C4—62no—
251C32×C4C2×C3×D412yes—
252C3⋊Dic3C2×C3×D412yes—
253C3×Dic3—24no—
254C3×Dic3—22no—
255C3×Dic3—22no—
256C3×Dic3—24no—
257C3×Dic3—44no—
258C3×Dic3—42no—
259C3×Dic3—62no—
260C3×Dic3—62no—
261C22×Q8—91no—
262C23×C4—91no—
263C22⋊Q8—98no—
264C2×C4⋊C4—91no—
265C2×C4⋊C4—92no—
266C2×C22⋊C4—92no—
267C33C2×C22⋊Q811yescharacteristic, complemented, Sylow
268C23×C3—61no—
269C23×C3—61no—
270C23×C3—92no—
271C23×C3—94no—
272C23×C3—91no—
273C23×C3—121no—
274C23×C3—181no—
275C23×C3—182no—
276C22×D3—21no—
277C22×D3—64no—
278C22×D3—62no—
279C22×D3—61no—
280C22×D3—61no—
281C22×D3—62no—
282C22×D3—64no—
283C3×Q8—64no—
284C3×Q8—64no—
285C3×Q8—124no—
286C3×Q8—184no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—64no—
292C2×C3×C4—62no—
293C2×C3×C4—61no—
294C2×C3×C4—61no—
295C2×C3×C4—62no—
296C2×C3×C4—64no—
297C2×C3×C4—94no—
298C2×C3×C4—92no—
299C2×C3×C4—91no—
300C2×C3×C4—92no—
301C2×C3×C4—92no—
302C2×C3×C4—94no—
303C2×C3×C4—124no—
304C2×C3×C4—121no—
305C2×C3×C4—122no—
306C2×C3×C4—181no—
307C2×C3×C4—182no—
308C2×C3×C4—184no—
309C2×C3×C4—184no—
310C2×C3×C4GO+(4,𝔽2)11yescharacteristic
311C2×Dic3—22no—
312C2×Dic3—21no—
313C2×Dic3—61no—
314C2×Dic3—61no—
315C2×Dic3—64no—
316C4×D3—24no—
317C4×D3—62no—
318C4×D3—64no—
319C4×D3—62no—
320C4×D3—62no—
321C4×D3—62no—
322C4×D3—62no—
323C4×D3—62no—
324C4×D3—64no—
325C4×D3—64no—
326Dic6—24no—
327Dic6—68no—
328C2×C32—21no—
329C2×C32—21no—
330C2×C32—22no—
331C2×C32—22no—
332C2×C32—42no—
333C2×C32—42no—
334C2×C32—41no—
335C2×C32—41no—
336C2×C32—64no—
337C2×C32C3×C22⋊Q812yescomplemented
338C2×C32C22×C3×D411yescharacteristic
339C3⋊D3C3×C22⋊Q812yescomplemented
340C3⋊D3C22×C3×Q811yescharacteristic
341C3⋊D3C2×C3×#4111yescharacteristic
342C3×D3—24no—
343C3×D3—24no—
344C3×D3—44no—
345C3×D3—62no—
346C3×D3—62no—
347C3×D3—61no—
348C3×D3—61no—
349C3×D3—61no—
350C3×D3—61no—
351C24—91no—
352C2×Q8—61no—
353C2×Q8—94no—
354C2×Q8—181no—
355C2×Q8—182no—
356C22×C4—61no—
357C22×C4—92no—
358C22×C4—91no—
359C22×C4—94no—
360C22×C4—91no—
361C22×C4—92no—
362C22×C4—182no—
363C22×C4—181no—
364C4⋊C4—98no—
365C4⋊C4—94no—
366C22⋊C4—98no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—41no—
370C22×C3—41no—
371C22×C3—62no—
372C22×C3—64no—
373C22×C3—62no—
374C22×C3—64no—
375C22×C3—94no—
376C22×C3—91no—
377C22×C3—91no—
378C22×C3—92no—
379C22×C3—92no—
380C22×C3—122no—
381C22×C3—124no—
382C22×C3—182no—
383C22×C3—184no—
384C22×C3C2×GO+(4,𝔽2)11yescharacteristic
385C2×D3—22no—
386C2×D3—24no—
387C2×D3—64no—
388C2×D3—61no—
389C2×D3—61no—
390C2×D3—61no—
391C2×D3—61no—
392C2×D3—62no—
393C2×D3—62no—
394C2×D3—62no—
395C2×D3—62no—
396C2×D3—68no—
397C2×D3—64no—
398C3×C4—22no—
399C3×C4—22no—
400C3×C4—42no—
401C3×C4—42no—
402C3×C4—64no—
403C3×C4—62no—
404C3×C4—62no—
405C3×C4—64no—
406C3×C4—92no—
407C3×C4—122no—
408C3×C4—124no—
409C3×C4—184no—
410C3×C4C2×GO+(4,𝔽2)12yes—
411Dic3—24no—
412Dic3—22no—
413Dic3—62no—
414Dic3—62no—
415C32—21no—
416C32—21no—
417C32—41no—
418C32—41no—
419C32C2×C3×C22⋊Q811yescharacteristic, complemented
420C23—61no—
421C23—92no—
422C23—94no—
423C23—91no—
424C23—181no—
425C23—182no—
426Q8—64no—
427Q8—184no—
428C2×C4—61no—
429C2×C4—62no—
430C2×C4—64no—
431C2×C4—94no—
432C2×C4—92no—
433C2×C4—91no—
434C2×C4—92no—
435C2×C4—92no—
436C2×C4—94no—
437C2×C4—181no—
438C2×C4—182no—
439C2×C4—184no—
440C2×C4—184no—
441C2×C4C3×GO+(4,𝔽2)11yescharacteristic
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—22no—
445C2×C3—22no—
446C2×C3—41no—
447C2×C3—41no—
448C2×C3—42no—
449C2×C3—42no—
450C2×C3—64no—
451C2×C3—64no—
452C2×C3—91no—
453C2×C3—91no—
454C2×C3—92no—
455C2×C3—124no—
456C2×C3D32⋊Q812yesdirect product factor
457C2×C3C22×GO+(4,𝔽2)11yescharacteristic
458D3—24no—
459D3—62no—
460D3—61no—
461D3—61no—
462D3—62no—
463D3—61no—
464D3—61no—
465C22—62no—
466C22—64no—
467C22—94no—
468C22—91no—
469C22—91no—
470C22—92no—
471C22—92no—
472C22—182no—
473C22—184no—
474C22C2×C3×GO+(4,𝔽2)11yescharacteristic
475C4—62no—
476C4—64no—
477C4—92no—
478C4—184no—
479C4C2×C3×GO+(4,𝔽2)12yes—
480C3—21no—
481C3—21no—
482C3—41no—
483C3—41no—
484C3C2×D32⋊Q811yescharacteristic, direct product factor
485C2—64no—
486C2—91no—
487C2—91no—
488C2—92no—
489C2C3×D32⋊Q812yesdirect product factor
490C2C22×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊D3C24×C3
FrattiniC2C22×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×C3×D32C22
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C2×C3×C22⋊Q8
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3797C2semidirect
3. C2×C3×C4⋊#157C2semidirect
4. C2×C3×#3773C2semidirect
6. C3×D32⋊Q8C2direct (N × H)
7. C2×D32⋊Q8C3direct (N × H)
20. C3×#3797C22semidirect
21. C3×C4⋊#157C22semidirect
22. C3×#3773C22semidirect
27. C2×#3797C2×C3semidirect
28. C2×C4⋊#157C2×C3semidirect
29. C2×#3773C2×C3semidirect
31. D32⋊Q8C2×C3direct (N × H)
36. C2×C3×D32Q8semidirect
85. #3797C22×C3semidirect
86. C4⋊#157C22×C3semidirect
87. #3773C22×C3semidirect
98. C3×D32C2×Q8semidirect
125. C2×D32C3×Q8semidirect
173. C2×C33C22⋊Q8semidirect
175. C3×C3⋊D3C22⋊Q8semidirect
243. D32C2×C3×Q8semidirect
267. C33C2×C22⋊Q8semidirect
337. C2×C32C3×C22⋊Q8semidirect
339. C3⋊D3C3×C22⋊Q8semidirect
419. C32C2×C3×C22⋊Q8semidirect
456. C2×C3D32⋊Q8direct (N × H)
484. C3C2×D32⋊Q8direct (N × H)
489. C2C3×D32⋊Q8direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×D322
C2×C3×#37972
C2×C3×C4⋊#1572
C2×C3×#37732
C2×C3×D32⋊C42
C3×D32⋊Q82
C2×D32⋊Q83
C2×C3×C22⋊Q89