Finite Group Databasebeta

← C2×#11722793 #13167565

Subgroups — C2×#11722793 #13167565

Order 1728.

920 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 920 conjugacy classes of subgroups — too many to draw. Only its 111 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×#6295C211yesmaximal, characteristic
2C22×C3×#622C211yesmaximal, characteristic, complemented
3C2×#88815C211yesmaximal, characteristic, complemented
4#11722793C214yesmaximal, direct product factor
5C2×#63739—31nomaximal
6C23×C3×C3⋊D3—21no—
7C22×C3×C3⋊Dic3—21no—
8C22×C3×C3⋊Dic3C2211yescharacteristic
9C2×#6295—21no—
10C2×#6295—22no—
11C2×#6295C414yescomplemented
12C2×#6295C2212yes—
13C2×C3×#622—22no—
14C2×C3×#622C2212yescomplemented
15#88815—21no—
16#88815C2212yescomplemented
17C22×#626—61no—
18C22×#625—31no—
19C22×#622D311yescharacteristic, complemented
20C2×#3724—31no—
21C2×Dic3⋊Dic3—61no—
22#63739—34no—
23C23×C33D411yescharacteristic
24C22×C3×C3⋊D3—24no—
25C22×C3×C3⋊D3—22no—
26C22×C3×C3⋊D3—42no—
27C22×C3×C3⋊D3—42no—
28C2×C3×C3⋊Dic3—22no—
29C2×C3×C3⋊Dic3—42no—
30C2×C3×C3⋊Dic3C2×C412yes—
31C2×C3×C3⋊Dic3D411yescharacteristic
32C2×C3×C3⋊Dic3D412yes—
33C2×C3×C3⋊Dic3C2311yescharacteristic
34#6295—24no—
35#6295—22no—
36#6295C2×C414yescomplemented
37C3×#622—22no—
38C2×#821—91nomaximal
39C23×C3⋊D3—21no—
40C23×C3×D3—61no—
41C23×C3×D3—61no—
42C23×C3×D3—121no—
43C22×C3⋊Dic3—21no—
44C22×C3⋊Dic3C2×D311yescharacteristic
45C22×C3×Dic3—61no—
46C22×C3×Dic3—61no—
47C22×C3×Dic3—61no—
48C22×C3×Dic3—61no—
49C22×C3×Dic3—121no—
50C2×#626—64no—
51C2×#626—62no—
52C2×#626—62no—
53C2×#626—64no—
54C2×#625—34no—
55C2×#625—32no—
56C2×#625—61no—
57C2×#625—62no—
58C2×#622—22no—
59C2×#622C2×D312yescomplemented
60#3724—32no—
61#3724—61no—
62Dic3⋊Dic3—62no—
63Dic3⋊Dic3—62no—
64C22×C33C22⋊C412yes—
65C22×C33D812yes—
66C22×C33SD1612yes—
67C22×C33C2×D411yescharacteristic
68C2×C3×C3⋊D3—24no—
69C2×C3×C3⋊D3—48no—
70C2×C3×C3⋊D3—42no—
71C2×C3×C3⋊D3—42no—
72C3×C3⋊Dic3—42no—
73C3×C3⋊Dic3C22⋊C412yes—
74C3×C3⋊Dic3C22×C411yescharacteristic
75C3×C3⋊Dic3C2×D411yescharacteristic
76C22×#67—181no—
77C22×D12—91no—
78C22×C3×C8—91no—
79C2×C4⋊Dic3—91no—
80C2×Dic3⋊C4—181no—
81#821—94no—
82C23×C32—21no—
83C23×C32—21no—
84C23×C32—41no—
85C23×C32—41no—
86C23×C32D1211yescharacteristic
87C22×C3⋊D3—22no—
88C22×C3⋊D3—24no—
89C22×C3⋊D3—42no—
90C22×C3⋊D3—42no—
91C22×C3×D3—62no—
92C22×C3×D3—64no—
93C22×C3×D3—64no—
94C22×C3×D3—62no—
95C22×C3×D3—124no—
96C22×C3×D3—122no—
97C22×C3×D3—122no—
98C22×C3×D3—122no—
99C22×C3×D3—124no—
100C22×C3×D3—124no—
101C22×C3×D3—122no—
102C22×C3×D3—122no—
103C2×C3⋊Dic3—22no—
104C2×C3⋊Dic3—42no—
105C2×C3⋊Dic3C4×D312yes—
106C2×C3⋊Dic3D1211yescharacteristic
107C2×C3⋊Dic3#6712yes—
108C2×C3⋊Dic3C22×D311yescharacteristic
109C2×C3×Dic3—61no—
110C2×C3×Dic3—61no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—61no—
114C2×C3×Dic3—62no—
115C2×C3×Dic3—62no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3—62no—
118C2×C3×Dic3—62no—
119C2×C3×Dic3—122no—
120C2×C3×Dic3—122no—
121C2×C3×Dic3—124no—
122C2×C3×Dic3—122no—
123#626—64no—
124#626—68no—
125#626—64no—
126#625—34no—
127#625—64no—
128#625—62no—
129#622—22no—
130C2×#102—271noSylow
131C2×C33#10214yescomplemented
132C2×C33C2×C22⋊C411yescharacteristic
133C2×C33C2×D811yescharacteristic
134C2×C33C2×SD1611yescharacteristic
135C3×C3⋊D3—44no—
136C24×C3—181no—
137C23×D3—61no—
138C23×D3—61no—
139C23×D3—121no—
140C22×C3×C4—91no—
141C22×C3×C4—181no—
142C2×#67—182no—
143C2×#67—184no—
144C2×#67—182no—
145C2×#67—184no—
146C22×Dic3—61no—
147C22×Dic3—61no—
148C22×Dic3—61no—
149C22×Dic3—61no—
150C22×Dic3—121no—
151C2×D12—92no—
152C2×D12—94no—
153C2×D12—181no—
154C2×D12—182no—
155C2×C3×C8—92no—
156C2×C3×C8—182no—
157C4⋊Dic3—92no—
158C4⋊Dic3—181no—
159Dic3⋊C4—182no—
160Dic3⋊C4—182no—
161C22×C32—21no—
162C22×C32—21no—
163C22×C32—22no—
164C22×C32—22no—
165C22×C32—22no—
166C22×C32—22no—
167C22×C32—22no—
168C22×C32—22no—
169C22×C32—42no—
170C22×C32—42no—
171C22×C32—42no—
172C22×C32—42no—
173C22×C32—42no—
174C22×C32—42no—
175C22×C32—41no—
176C22×C32—41no—
177C22×C32#20412yes—
178C22×C32D2412yes—
179C22×C32(C2×D3)⋊C412yes—
180C22×C32C2×D1211yescharacteristic
181C2×C3⋊D3—24no—
182C2×C3⋊D3—42no—
183C2×C3⋊D3—42no—
184C2×C3⋊D3—48no—
185C2×C3×D3—64no—
186C2×C3×D3—64no—
187C2×C3×D3—124no—
188C2×C3×D3—122no—
189C2×C3×D3—122no—
190C2×C3×D3—128no—
191C2×C3×D3—1216no—
192C2×C3×D3—124no—
193C2×C3×D3—124no—
194C2×C3×D3—128no—
195C2×C3×D3—122no—
196C2×C3×D3—122no—
197C3⋊Dic3—42no—
198C3⋊Dic3(C2×D3)⋊C412yes—
199C3⋊Dic3C2×C4×D311yescharacteristic
200C3⋊Dic3C2×#6711yescharacteristic
201C3×Dic3—62no—
202C3×Dic3—62no—
203C3×Dic3—61no—
204C3×Dic3—61no—
205C3×Dic3—61no—
206C3×Dic3—61no—
207C3×Dic3—122no—
208C3×Dic3—124no—
209C3×Dic3—122no—
210C22×D4—271no—
211C22×C8—91no—
212C2×C4⋊C4—271no—
213#102—274no—
214C33C2×#10211yescharacteristic, complemented, Sylow
215C23×C3—21no—
216C23×C3—21no—
217C23×C3—41no—
218C23×C3—41no—
219C23×C3—182no—
220C23×C3—184no—
221C23×C3—362no—
222C23×C3—362no—
223C23×C3GO+(4,𝔽2)11yescharacteristic
224C22×D3—62no—
225C22×D3—64no—
226C22×D3—62no—
227C22×D3—64no—
228C22×D3—122no—
229C22×D3—124no—
230C22×D3—122no—
231C22×D3—122no—
232C22×D3—124no—
233C22×D3—124no—
234C22×D3—122no—
235C22×D3—122no—
236C2×C3×C4—91no—
237C2×C3×C4—91no—
238C2×C3×C4—92no—
239C2×C3×C4—92no—
240C2×C3×C4—182no—
241C2×C3×C4—362no—
242#67—184no—
243#67—188no—
244#67—184no—
245C2×Dic3—62no—
246C2×Dic3—62no—
247C2×Dic3—61no—
248C2×Dic3—61no—
249C2×Dic3—62no—
250C2×Dic3—62no—
251C2×Dic3—61no—
252C2×Dic3—61no—
253C2×Dic3—62no—
254C2×Dic3—62no—
255C2×Dic3—122no—
256C2×Dic3—122no—
257C2×Dic3—124no—
258C2×Dic3—122no—
259D12—94no—
260D12—182no—
261D12—184no—
262C3×C8—182no—
263C2×C32—21no—
264C2×C32—21no—
265C2×C32—21no—
266C2×C32—21no—
267C2×C32—21no—
268C2×C32—21no—
269C2×C32—24no—
270C2×C32—24no—
271C2×C32—44no—
272C2×C32—44no—
273C2×C32—41no—
274C2×C32—41no—
275C2×C32—41no—
276C2×C32—41no—
277C2×C32—41no—
278C2×C32—41no—
279C2×C32#82114yescomplemented
280C2×C32C2×#20411yescharacteristic
281C2×C32C2×D2411yescharacteristic
282C2×C32C2×(C2×D3)⋊C411yescharacteristic
283C3⋊D3—44no—
284C3×D3—124no—
285C3×D3—128no—
286C3×D3—124no—
287C24—181no—
288C2×D4—272no—
289C2×D4—274no—
290C2×D4—541no—
291C2×D4—542no—
292C22×C4—91no—
293C22×C4—181no—
294C2×C8—92no—
295C2×C8—182no—
296C4⋊C4—272no—
297C4⋊C4—541no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—22no—
301C22×C3—22no—
302C22×C3—22no—
303C22×C3—22no—
304C22×C3—22no—
305C22×C3—22no—
306C22×C3—41no—
307C22×C3—41no—
308C22×C3—42no—
309C22×C3—42no—
310C22×C3—42no—
311C22×C3—42no—
312C22×C3—42no—
313C22×C3—42no—
314C22×C3—184no—
315C22×C3—368no—
316C22×C3—362no—
317C22×C3—362no—
318C22×C3D32⋊C412yes—
319C22×C3#377412yes—
320C22×C3#377512yes—
321C22×C3C2×GO+(4,𝔽2)11yescharacteristic
322C2×D3—64no—
323C2×D3—64no—
324C2×D3—124no—
325C2×D3—128no—
326C2×D3—122no—
327C2×D3—122no—
328C2×D3—1216no—
329C2×D3—124no—
330C2×D3—124no—
331C2×D3—122no—
332C2×D3—122no—
333C2×D3—128no—
334C3×C4—91no—
335C3×C4—91no—
336C3×C4—92no—
337C3×C4—362no—
338Dic3—62no—
339Dic3—61no—
340Dic3—61no—
341Dic3—62no—
342Dic3—61no—
343Dic3—61no—
344Dic3—122no—
345Dic3—124no—
346Dic3—122no—
347C32—21no—
348C32—21no—
349C32—41no—
350C32—41no—
351C32C2×#82111yescharacteristic, complemented
352C23—182no—
353C23—184no—
354C23—362no—
355C23—362no—
356C23#157⋊D311yescharacteristic
357D4—274no—
358D4—542no—
359D4—544no—
360C2×C4—91no—
361C2×C4—91no—
362C2×C4—92no—
363C2×C4—92no—
364C2×C4—182no—
365C2×C4—362no—
366C8—182no—
367C2×C3—24no—
368C2×C3—24no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3—21no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—44no—
376C2×C3—44no—
377C2×C3—41no—
378C2×C3—41no—
379C2×C3—41no—
380C2×C3—41no—
381C2×C3—41no—
382C2×C3—41no—
383C2×C3—364no—
384C2×C3#6373914yescomplemented
385C2×C3C2×D32⋊C411yescharacteristic
386C2×C3C2×#377411yescharacteristic
387C2×C3C2×#377511yescharacteristic
388D3—124no—
389D3—128no—
390D3—124no—
391C22—184no—
392C22—368no—
393C22—362no—
394C22—362no—
395C22#8893912yes—
396C22#8894112yes—
397C22#8894212yes—
398C22C2×#157⋊D311yescharacteristic
399C4—91no—
400C4—91no—
401C4—92no—
402C4—362no—
403C3—21no—
404C3—21no—
405C3—41no—
406C3—41no—
407C3C2×#6373911yescharacteristic, complemented
408C2—364no—
409C2#1172279314yesdirect product factor
410C2C2×#8893911yescharacteristic
411C2C2×#8894111yescharacteristic
412C2C2×#8894211yescharacteristic
subgroupG/H
center, hypercenterC23#157⋊D3
commutatorC3×C3⋊Dic3C22×C4
FrattiniC22C2×#157⋊D3
Fitting, socleC23×C33D4
prime radicalC22×#6295C2
cosocleC2×C3×C3⋊Dic3C23
hypocenterC33C2×#102
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C22×C3×#622C2semidirect
3. C2×#88815C2semidirect
4. #11722793C2direct (N × H)
11. C2×#6295C4semidirect
14. C2×C3×#622C22semidirect
16. #88815C22semidirect
19. C22×#622D3semidirect
36. #6295C2×C4semidirect
59. C2×#622C2×D3semidirect
131. C2×C33#102semidirect
214. C33C2×#102semidirect
279. C2×C32#821semidirect
351. C32C2×#821semidirect
384. C2×C3#63739semidirect
407. C3C2×#63739semidirect
409. C2#11722793direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62952
C22×C3×#6222
C2×#888152
#117227932
C2×#637393
C2×#8219