Finite Group Databasebeta

← #13155414

Subgroups — #13155414

Order 1728.

878 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 878 conjugacy classes of subgroups — too many to draw. Only its 128 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⋊D3×Dic6C211yesmaximal, characteristic
2C2×#88809C211yesmaximal, characteristic
3C3×#63849C211yesmaximal, characteristic
4#11722041C212yesmaximal, complemented
5C3×#63565C212yesmaximal, complemented
6#10591562—31nomaximal
7#10590600—32nomaximal
8C2×#6308C2211yescharacteristic
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C32×Dic6C2211yescharacteristic
11C2×#6293—21no—
12C2×Dic3×C3⋊D3—21no—
13C3⋊D3×Dic6—22no—
14C3⋊D3×Dic6C414yescomplemented
15C2×C3×C3⋊(C3⋊C8)C2212yes—
16C32×C4⋊Dic3C2212yes—
17#88809—22no—
18C3×#3722—21no—
19C3×Dic32—21no—
20C2×Q8×C3⋊D3—31no—
21C2×D3×Dic6—32no—
22C2×D3×Dic6—61no—
23C3×#892—32no—
24#63849D311yescharacteristic
25C2×#3712—31no—
26C2×#3711—32no—
27#63637—32no—
28C3×#835—32no—
29C3×#810—32no—
30#63567—32no—
31#63567—32no—
32#63567—62no—
33#63565D312yescomplemented
34C22×C3×C3⋊D3D411yescharacteristic
35C2×C33×C4C2311yescharacteristic
36C2×C3⋊(C3⋊Dic3)—21no—
37#6308—21no—
38#6308C2×C412yes—
39C2×C3×C3⋊Dic3D411yescharacteristic
40C3×C4×C3⋊D3C2×C412yes—
41C3×C4×C3⋊D3D412yes—
42C2×C32×Dic3—21no—
43C2×C32×Dic3—22no—
44C32×Dic6—21no—
45C32×Dic6C2×C412yes—
46#6293—24no—
47Dic3×C3⋊D3—24no—
48C3×C3⋊(C3⋊C8)—22no—
49#5004—91no—
50#4802—92no—
51C2×C32×Q8—31no—
52Q8×C3⋊D3—34no—
53Q8×C3⋊D3—62no—
54C2×C4×C3⋊D3—61no—
55C2×C4×C3⋊D3C2×D311yescharacteristic
56C2×#634—32no—
57C2×#634—31no—
58C2×#634—61no—
59C2×#634—61no—
60C2×#634—62no—
61C2×#634—121no—
62C2×C3×C4×D3—32no—
63C2×C3×C4×D3—61no—
64C2×C3×Dic6—21no—
65C2×C3×Dic6C2×D312yes—
66C2×#627—62no—
67C2×#627—62no—
68C2×D3×Dic3—62no—
69C2×D3×Dic3—62no—
70D3×Dic6—38no—
71D3×Dic6—64no—
72D3×Dic6—64no—
73D3×Dic6—64no—
74C32×C4⋊C4—32no—
75C2×C3⋊(C3⋊C8)C2×D312yes—
76C3×(C2×D3)⋊C4—62no—
77C3×C4⋊Dic3—22no—
78C3×C4⋊Dic3C2×D312yes—
79C3×C4⋊Dic3C2×D312yes—
80C3×C4×Dic3—62no—
81C2×C3×C3⋊C8—32no—
82C2×C3×C3⋊C8—32no—
83C2×C3×C3⋊C8—62no—
84#3722—21no—
85Dic32—21no—
86#3712—62no—
87#3711—64no—
88C22×C33C2×D411yescharacteristic
89C2×C3×C3⋊D3—22no—
90C2×C3×C3⋊D3C22⋊C412yes—
91C33×C4C22×C411yescharacteristic
92C33×C4C2×D411yescharacteristic
93C3⋊(C3⋊Dic3)—22no—
94C3×C3⋊Dic3C22⋊C412yes—
95C32×Dic3—22no—
96C32×Dic3—42no—
97C2×D3×Q8—92no—
98C2×D3×Q8—181no—
99C22×Dic6—91no—
100C3×#117—91no—
101C2×#208—91no—
102C2×ΓL(1,𝔽25)—92no—
103#892—32no—
104C3×#103—92no—
105#835—32no—
106#816—92no—
107#810—32no—
108#810—92no—
109#810—92no—
110#810—182no—
111C22×C3⋊D3#6711yescharacteristic
112C22×C3×D3—32no—
113C22×C3×D3—61no—
114C32×Q8—32no—
115C32×Q8—61no—
116C2×C32×C4—22no—
117C2×C32×C4—21no—
118C2×C32×C4—41no—
119C2×C32×C4—61no—
120C2×C32×C4—62no—
121C2×C32×C4C22×D311yescharacteristic
122C2×C32×C4C22×D312yes—
123C2×C3⋊Dic3—61no—
124C2×C3⋊Dic3—62no—
125C2×C3⋊Dic3—62no—
126C2×C3⋊Dic3—122no—
127C2×C3⋊Dic3—122no—
128C2×C3⋊Dic3#6711yescharacteristic
129C4×C3⋊D3—64no—
130C4×C3⋊D3C4×D312yes—
131C4×C3⋊D3D1212yes—
132#634—34no—
133#634—32no—
134#634—62no—
135#634—64no—
136#634—61no—
137#634—64no—
138#634—62no—
139#634—62no—
140#634—122no—
141#634—122no—
142#634—122no—
143C2×C3×Dic3—22no—
144C2×C3×Dic3—22no—
145C2×C3×Dic3—22no—
146C2×C3×Dic3—22no—
147C2×C3×Dic3—32no—
148C2×C3×Dic3—42no—
149C2×C3×Dic3—61no—
150C3×C4×D3—34no—
151C3×C4×D3—34no—
152C3×C4×D3—62no—
153C3×C4×D3—62no—
154C3×Dic6—22no—
155C3×Dic6—22no—
156C3×Dic6—22no—
157C3×Dic6C4×D314yes—
158#627—68no—
159#627—68no—
160D3×Dic3—68no—
161D3×Dic3—68no—
162C3⋊(C3⋊C8)—22no—
163C3×C3⋊C8—62no—
164C3×C3⋊C8—62no—
165C3×C3⋊C8—122no—
166#419—271noSylow
167C2×C33C2×C22⋊C411yescharacteristic
168C2×C33#13712yes—
169C3×C3⋊D3—22no—
170C2×C3×Q8—32no—
171C2×C3×Q8—61no—
172C22×C3×C4—91no—
173C22×Dic3—181no—
174D3×Q8—98no—
175D3×Q8—184no—
176D3×Q8—184no—
177D3×Q8—184no—
178C2×C4×D3—32no—
179C2×C4×D3—61no—
180C2×C4×D3—182no—
181C2×C4×D3—182no—
182C2×Dic6—91no—
183C2×Dic6—92no—
184C2×Dic6—94no—
185C2×Dic6—181no—
186C2×Dic6—182no—
187C2×Dic6—182no—
188C2×Dic6—181no—
189C2×Dic6—182no—
190C2×Dic6—182no—
191C2×Dic6—361no—
192C2×Dic6D3211yescharacteristic
193C2×C3×C8—92no—
194C3×C4⋊C4—32no—
195C3×C4⋊C4—32no—
196C3×C4⋊C4—62no—
197C3×C22⋊C4—181no—
198C3×C42—181no—
199(C2×D3)⋊C4—62no—
200C4⋊Dic3D3212yes—
201C4×Dic3—62no—
202#208—182no—
203C2×C3⋊C8—32no—
204C2×C3⋊C8—32no—
205C2×C3⋊C8—62no—
206ΓL(1,𝔽25)—184no—
207C22×C32—22no—
208C22×C32—21no—
209C22×C32—41no—
210C22×C32D3×D412yes—
211C22×C32C2×#6711yescharacteristic
212C2×C3⋊D3—22no—
213C2×C3⋊D3(C2×D3)⋊C412yes—
214C2×C3×D3—34no—
215C2×C3×D3—62no—
216C2×C3×D3—64no—
217C2×C3×D3—62no—
218C2×C3×D3—62no—
219C32×C4—22no—
220C32×C4—22no—
221C32×C4—21no—
222C32×C4—21no—
223C32×C4—41no—
224C32×C4—41no—
225C32×C4—62no—
226C32×C4—122no—
227C32×C4C2×C4×D311yescharacteristic
228C32×C4C2×C4×D312yes—
229C32×C4C2×D1211yescharacteristic
230C32×C4D3×D412yes—
231C3⋊Dic3—62no—
232C3⋊Dic3—64no—
233C3⋊Dic3—64no—
234C3⋊Dic3—124no—
235C3⋊Dic3—124no—
236C3⋊Dic3(C2×D3)⋊C412yes—
237C3×Dic3—24no—
238C3×Dic3—24no—
239C3×Dic3—34no—
240C3×Dic3—42no—
241C3×Dic3—42no—
242C3×Dic3—62no—
243C3×Dic3—82no—
244C22×Q8—271no—
245C2×M16—271no—
246#117—91no—
247#103—92no—
248#103—272no—
249C33#41911yescharacteristic, complemented, Sylow
250C23×C3—91no—
251C22×D3—32no—
252C22×D3—61no—
253C3×Q8—34no—
254C3×Q8—62no—
255C3×Q8—62no—
256C3×Q8—62no—
257C2×C3×C4—22no—
258C2×C3×C4—21no—
259C2×C3×C4—41no—
260C2×C3×C4—62no—
261C2×C3×C4—62no—
262C2×C3×C4—62no—
263C2×C3×C4—62no—
264C2×C3×C4—91no—
265C2×C3×C4—92no—
266C2×C3×C4—92no—
267C2×C3×C4—122no—
268C2×C3×C4C2×D3212yes—
269C2×C3×C4C2×D3211yescharacteristic
270C2×Dic3—21no—
271C2×Dic3—22no—
272C2×Dic3—32no—
273C2×Dic3—61no—
274C2×Dic3—184no—
275C2×Dic3—182no—
276C2×Dic3—182no—
277C2×Dic3—181no—
278C2×Dic3—362no—
279C2×Dic3—362no—
280C4×D3—34no—
281C4×D3—34no—
282C4×D3—62no—
283C4×D3—62no—
284C4×D3—188no—
285C4×D3—188no—
286Dic6—21no—
287Dic6—94no—
288Dic6—92no—
289Dic6—182no—
290Dic6—184no—
291Dic6—184no—
292Dic6—188no—
293Dic6—188no—
294Dic6—182no—
295Dic6—182no—
296Dic6—181no—
297Dic6—362no—
298Dic6—362no—
299Dic6—362no—
300Dic6#62412yes—
301C3×C8—182no—
302C3⋊C8—62no—
303C3⋊C8—62no—
304C3⋊C8—122no—
305C2×C32—22no—
306C2×C32—22no—
307C2×C32—22no—
308C2×C32—22no—
309C2×C32—21no—
310C2×C32—41no—
311C2×C32—42no—
312C2×C32D3×C22⋊C412yes—
313C2×C32#91512yes—
314C2×C32#91812yes—
315C2×C32C2×(C2×D3)⋊C411yescharacteristic
316C2×C32#95112yes—
317C3⋊D3—22no—
318C3×D3—64no—
319C3×D3—64no—
320C2×Q8—31no—
321C2×Q8—274no—
322C2×Q8—271no—
323C2×Q8—541no—
324C2×Q8—542no—
325C22×C4—91no—
326C22×C4—541no—
327M16—542no—
328C2×C8—92no—
329C4⋊C4—32no—
330C22⋊C4—181no—
331C42—181no—
332C22×C3—21no—
333C22×C3—22no—
334C22×C3—41no—
335C22×C3—92no—
336C22×C3—182no—
337C22×C3D3×#6712yes—
338C22×C3#381111yescharacteristic
339C2×D3—34no—
340C2×D3—62no—
341C2×D3—64no—
342C2×D3—62no—
343C2×D3—62no—
344C3×C4—21no—
345C3×C4—22no—
346C3×C4—22no—
347C3×C4—21no—
348C3×C4—41no—
349C3×C4—41no—
350C3×C4—64no—
351C3×C4—64no—
352C3×C4—92no—
353C3×C4—122no—
354C3×C4—122no—
355C3×C4—242no—
356C3×C4C4×D3212yes—
357C3×C4D3×D1212yes—
358C3×C4C2×#62411yescharacteristic
359C3×C4#381111yescharacteristic
360Dic3—22no—
361Dic3—34no—
362Dic3—42no—
363Dic3—62no—
364Dic3—184no—
365Dic3—184no—
366Dic3—182no—
367Dic3—364no—
368Dic3—364no—
369C32—21no—
370C32—22no—
371C32—41no—
372C32#480212yescomplemented
373C32#500411yescharacteristic, complemented
374C23—91no—
375Q8—32no—
376Q8—61no—
377Q8—272no—
378Q8—544no—
379Q8—541no—
380C2×C4—61no—
381C2×C4—62no—
382C2×C4—91no—
383C2×C4—92no—
384C2×C4—92no—
385C2×C4—544no—
386C2×C4—541no—
387C2×C4D3311yescharacteristic
388C8—182no—
389C2×C3—22no—
390C2×C3—22no—
391C2×C3—22no—
392C2×C3—21no—
393C2×C3—22no—
394C2×C3—42no—
395C2×C3—41no—
396C2×C3—182no—
397C2×C3D3×(C2×D3)⋊C412yes—
398C2×C3#6392912yes—
399C2×C3#6394312yes—
400C2×C3#6395112yes—
401C2×C3#6397411yescharacteristic
402D3—64no—
403D3—64no—
404C22—92no—
405C22—182no—
406C22#8895211yescharacteristic
407C4—62no—
408C4—92no—
409C4—122no—
410C4—542no—
411C4D3×#62411yescharacteristic
412C4#8895511yescharacteristic
413C3—21no—
414C3—22no—
415C3—41no—
416C3#1059060012yescomplemented
417C3#1059156211yescharacteristic, complemented
418C2—182no—
419C2#1172288812yes—
420C2#1172290411yescharacteristic
subgroupG/H
centerC22#88952
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypocenterC33#419
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722041C2semidirect
5. C3×#63565C2semidirect
14. C3⋊D3×Dic6C4semidirect
33. #63565D3semidirect
249. C33#419semidirect
372. C32#4802semidirect
373. C32#5004semidirect
416. C3#10590600semidirect
417. C3#10591562semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×Dic62
C2×#888092
C3×#638492
#117220412
C3×#635652
#105915623
#105906003