Finite Group Databasebeta

← #13155528

Subgroups — #13155528

Order 1728.

702 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 118 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
1C4×#6289C211yesmaximal, characteristic
2#11722078C211yesmaximal, characteristic, complemented
3#11722024C212yesmaximal
4C3×#63572C211yesmaximal, characteristic, complemented
5C3×#63554C212yesmaximal, complemented
6#10590665—32nomaximal
7#10590299—31nomaximal
8C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
9C2×#6289C411yescharacteristic
10C2×#6289C411yescharacteristic
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C2×C3×C3⋊(C3⋊C8)C2212yes—
13C32×C4×Dic3C2211yescharacteristic
14C2×C32×C3⋊C8C2212yes—
15Dic3×C3⋊Dic3C411yescharacteristic
16Dic3×C3⋊Dic3C411yescharacteristic
17C42×C3⋊D3—31no—
18C4×#624—32no—
19C4×#624—61no—
20#63647—32no—
21C3×#815—32no—
22C3×Dic3⋊C8—32no—
23C3×#803—32no—
24#63572—31no—
25#63572D311yescharacteristic, complemented
26Dic3⋊(C3⋊C8)—32no—
27#63557—32no—
28#63557—32no—
29#63557—62no—
30#63554D312yescomplemented
31C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
32C2×C33×C4C2311yescharacteristic
33C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
34C4×C3⋊(C3⋊D3)—21no—
35C4×C3⋊(C3⋊D3)—21no—
36C2×C3×C3⋊Dic3C2×C411yescharacteristic
37C2×C3×C3⋊Dic3C2×C411yescharacteristic
38C2×C32×Dic3C2×C411yescharacteristic
39C2×C32×Dic3C2×C411yescharacteristic
40#6289—22no—
41#6289—22no—
42C3×C3⋊(C3⋊C8)—22no—
43C32×C3⋊C8—22no—
44#4835—91no—
45#4705—92no—
46C2×C4×C3⋊D3—31no—
47C2×C4×C3⋊D3—32no—
48C2×C4×C3⋊D3—31no—
49C2×C4×C3⋊D3—31no—
50C2×C4×C3⋊D3—61no—
51C2×C4×C3⋊D3—62no—
52C2×C4×C3⋊D3—121no—
53C2×#624—32no—
54C2×#624—32no—
55C2×#624—61no—
56C2×#624—61no—
57C2×C32×C8—32no—
58C32×C42—31no—
59C4×C3⋊Dic3—31no—
60C4×C3⋊Dic3—31no—
61C4×C3⋊Dic3C2×D311yescharacteristic
62C2×C3⋊(C3⋊C8)C2×D312yes—
63C3×C4×Dic3—21no—
64C3×C4×Dic3—32no—
65C3×C4×Dic3—61no—
66C3×C4×Dic3C2×D312yes—
67C2×C3×C3⋊C8—22no—
68C2×C3×C3⋊C8—32no—
69C2×C3×C3⋊C8—32no—
70C2×C3×C3⋊C8—62no—
71C2×C3×C3⋊C8C2×D312yes—
72C2×C3×C3⋊C8C2×D312yes—
73Dic32—32no—
74Dic32—32no—
75Dic32—61no—
76Dic32—61no—
77C22×C33C22×C411yescharacteristic
78C2×C3⋊(C3⋊D3)—21no—
79C2×C3⋊(C3⋊D3)—21no—
80C2×C3⋊(C3⋊D3)M1612yes—
81C33×C4#4111yescharacteristic
82C33×C4#4111yescharacteristic
83C3⋊(C3⋊Dic3)—21no—
84C3×C3⋊Dic3—21no—
85C3×C3⋊Dic3M1612yes—
86C32×Dic3—21no—
87C32×Dic3—21no—
88C42×D3—92no—
89C42×D3—91no—
90C42×D3—181no—
91C3×C4⋊C8—91no—
92C3×#97—92no—
93(C2×D3)⋊C8—92no—
94(C2×D3)⋊C8—92no—
95(C2×D3)⋊C8—92no—
96(C2×D3)⋊C8—182no—
97#815—32no—
98Dic3⋊C8—32no—
99Dic3⋊C8—92no—
100C4⋊(C3⋊C8)—91no—
101#803—32no—
102C22×C3⋊D3—32no—
103C22×C3⋊D3—31no—
104C22×C3⋊D3—61no—
105C22×C3⋊D3—62no—
106C22×C3⋊D3—121no—
107C2×C32×C4—22no—
108C2×C32×C4—21no—
109C2×C32×C4—31no—
110C2×C32×C4—31no—
111C2×C32×C4—41no—
112C2×C32×C4C22×D312yes—
113C2×C32×C4C22×D311yescharacteristic
114C2×C3⋊Dic3—32no—
115C2×C3⋊Dic3—31no—
116C2×C3⋊Dic3—61no—
117C2×C3⋊Dic3—62no—
118C2×C3⋊Dic3—121no—
119C2×C3⋊Dic3C4×D311yescharacteristic
120C2×C3⋊Dic3C4×D311yescharacteristic
121C4×C3⋊D3—62no—
122C4×C3⋊D3—61no—
123C4×C3⋊D3—61no—
124C4×C3⋊D3—62no—
125C4×C3⋊D3—62no—
126C4×C3⋊D3—62no—
127C4×C3⋊D3—62no—
128C4×C3⋊D3—62no—
129C4×C3⋊D3—122no—
130C4×C3⋊D3—122no—
131C4×C3⋊D3—122no—
132C4×C3⋊D3—122no—
133C2×C3×Dic3—21no—
134C2×C3×Dic3—21no—
135C2×C3×Dic3—32no—
136C2×C3×Dic3—32no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3—61no—
139C2×C3×Dic3C4×D312yes—
140C2×C3×Dic3C4×D312yes—
141#624—64no—
142#624—64no—
143#624—64no—
144#624—64no—
145C32×C8—62no—
146C3⋊(C3⋊C8)—22no—
147C3×C3⋊C8—22no—
148C3×C3⋊C8—22no—
149C3×C3⋊C8—42no—
150C3×C3⋊C8—62no—
151C3×C3⋊C8—62no—
152C3×C3⋊C8—122no—
153#432—271noSylow
154C2×C33#11711yescharacteristic
155C2×C33C2×M1611yescharacteristic
156C2×C33C2×M1611yescharacteristic
157C3⋊(C3⋊D3)—22no—
158C2×C4×D3—92no—
159C2×C4×D3—92no—
160C2×C4×D3—91no—
161C2×C4×D3—92no—
162C2×C4×D3—91no—
163C2×C4×D3—91no—
164C2×C4×D3—181no—
165C2×C4×D3—181no—
166C2×C4×D3—181no—
167C2×C4×D3—182no—
168C2×C4×D3—361no—
169C2×C3×C8—32no—
170C2×C3×C8—32no—
171C2×C3×C8—62no—
172C2×C3×C8—92no—
173C3×C42—32no—
174C3×C42—61no—
175C3×C42—91no—
176C4×Dic3—32no—
177C4×Dic3—61no—
178C4×Dic3—92no—
179C4×Dic3—92no—
180C4×Dic3—91no—
181C4×Dic3—91no—
182C4×Dic3—181no—
183C4×Dic3—181no—
184C4×Dic3D3211yescharacteristic
185C2×C3⋊C8—32no—
186C2×C3⋊C8—32no—
187C2×C3⋊C8—62no—
188C2×C3⋊C8D3212yes—
189C22×C32—22no—
190C22×C32—21no—
191C22×C32—41no—
192C22×C32C2×C4×D311yescharacteristic
193C22×C32C2×C4×D312yes—
194C2×C3⋊D3—34no—
195C2×C3⋊D3—32no—
196C2×C3⋊D3—62no—
197C2×C3⋊D3—64no—
198C2×C3⋊D3—61no—
199C2×C3⋊D3—61no—
200C2×C3⋊D3—62no—
201C2×C3⋊D3—62no—
202C2×C3⋊D3—62no—
203C2×C3⋊D3—62no—
204C2×C3⋊D3—122no—
205C2×C3⋊D3—122no—
206C2×C3⋊D3—122no—
207C2×C3⋊D3—122no—
208C2×C3⋊D3—122no—
209C32×C4—21no—
210C32×C4—21no—
211C32×C4—22no—
212C32×C4—22no—
213C32×C4—41no—
214C32×C4—41no—
215C32×C4—61no—
216C32×C4—61no—
217C32×C4#23512yes—
218C32×C4#23512yes—
219C32×C4#23711yescharacteristic
220C32×C4#23911yescharacteristic
221C3⋊Dic3—21no—
222C3⋊Dic3—61no—
223C3⋊Dic3—62no—
224C3⋊Dic3—62no—
225C3⋊Dic3—122no—
226C3⋊Dic3—122no—
227C3⋊Dic3ΓL(1,𝔽25)12yes—
228C3×Dic3—22no—
229C3×Dic3—22no—
230C3×Dic3—22no—
231C3×Dic3—22no—
232C3×Dic3—34no—
233C3×Dic3—62no—
234C3×Dic3—62no—
235C3×Dic3—62no—
236C2×C42—271no—
237C4⋊C8—91no—
238C4⋊C8—271no—
239C22⋊C8—272no—
240#97—92no—
241C33#43211yescharacteristic, complemented, Sylow
242C22×D3—92no—
243C22×D3—91no—
244C22×D3—181no—
245C22×D3—182no—
246C22×D3—361no—
247C2×C3×C4—22no—
248C2×C3×C4—21no—
249C2×C3×C4—32no—
250C2×C3×C4—32no—
251C2×C3×C4—41no—
252C2×C3×C4—61no—
253C2×C3×C4—61no—
254C2×C3×C4—91no—
255C2×C3×C4—91no—
256C2×C3×C4C2×D3211yescharacteristic
257C2×C3×C4C2×D3212yes—
258C2×Dic3—32no—
259C2×Dic3—32no—
260C2×Dic3—61no—
261C2×Dic3—61no—
262C2×Dic3—92no—
263C2×Dic3—91no—
264C2×Dic3—181no—
265C2×Dic3—182no—
266C2×Dic3—361no—
267C2×Dic3#62411yescharacteristic
268C2×Dic3#62411yescharacteristic
269C4×D3—182no—
270C4×D3—182no—
271C4×D3—184no—
272C4×D3—184no—
273C4×D3—184no—
274C4×D3—184no—
275C4×D3—182no—
276C4×D3—182no—
277C4×D3—181no—
278C4×D3—181no—
279C4×D3—182no—
280C4×D3—182no—
281C4×D3—362no—
282C4×D3—362no—
283C4×D3—362no—
284C4×D3—362no—
285C3×C8—62no—
286C3×C8—62no—
287C3×C8—122no—
288C3×C8—182no—
289C3⋊C8—22no—
290C3⋊C8—62no—
291C3⋊C8—62no—
292C3⋊C8—122no—
293C2×C32—22no—
294C2×C32—22no—
295C2×C32—21no—
296C2×C32—21no—
297C2×C32—22no—
298C2×C32—21no—
299C2×C32—41no—
300C2×C32—41no—
301C2×C32—41no—
302C2×C32#87212yes—
303C2×C32#89211yescharacteristic
304C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
305C2×C32D3×M1612yes—
306C2×C32D3×M1611yescharacteristic
307C2×C32D3×M1612yes—
308C3⋊D3—62no—
309C3⋊D3—64no—
310C3⋊D3—64no—
311C3⋊D3—124no—
312C3⋊D3—124no—
313C22×C4—271no—
314C22×C4—271no—
315C22×C4—271no—
316C2×C8—32no—
317C2×C8—92no—
318C42—31no—
319C42—91no—
320C42—271no—
321C42—271no—
322C22×C3—21no—
323C22×C3—22no—
324C22×C3—41no—
325C22×C3C4×D3212yes—
326C22×C3C2×#62411yescharacteristic
327C2×D3—94no—
328C2×D3—92no—
329C2×D3—182no—
330C2×D3—184no—
331C2×D3—182no—
332C2×D3—182no—
333C2×D3—182no—
334C2×D3—181no—
335C2×D3—182no—
336C2×D3—181no—
337C2×D3—362no—
338C2×D3—362no—
339C2×D3—362no—
340C2×D3—362no—
341C2×D3—362no—
342C3×C4—22no—
343C3×C4—21no—
344C3×C4—22no—
345C3×C4—21no—
346C3×C4—41no—
347C3×C4—41no—
348C3×C4—62no—
349C3×C4—62no—
350C3×C4—62no—
351C3×C4—62no—
352C3×C4—92no—
353C3×C4—181no—
354C3×C4#379811yescharacteristic
355C3×C4#379811yescharacteristic
356C3×C4#379912yes—
357C3×C4#380412yes—
358Dic3—21no—
359Dic3—21no—
360Dic3—34no—
361Dic3—62no—
362Dic3—62no—
363Dic3—62no—
364Dic3—182no—
365Dic3—182no—
366Dic3—181no—
367Dic3—362no—
368Dic3—362no—
369C32—21no—
370C32—22no—
371C32—41no—
372C32#470512yescomplemented
373C32#483511yescharacteristic, complemented
374C23—271no—
375C2×C4—31no—
376C2×C4—31no—
377C2×C4—91no—
378C2×C4—91no—
379C2×C4—271no—
380C2×C4—542no—
381C2×C4—542no—
382C2×C4—541no—
383C2×C4—541no—
384C2×C4D3311yescharacteristic
385C8—62no—
386C8—182no—
387C2×C3—21no—
388C2×C3—22no—
389C2×C3—21no—
390C2×C3—22no—
391C2×C3—21no—
392C2×C3—22no—
393C2×C3—41no—
394C2×C3—41no—
395C2×C3—41no—
396C2×C3D3×ΓL(1,𝔽25)12yes—
397C2×C3#6379112yes—
398C2×C3#6381811yescharacteristic
399C2×C3#6381811yescharacteristic
400C2×C3#6383612yes—
401C2×C3#6387411yescharacteristic
402D3—184no—
403D3—182no—
404D3—184no—
405D3—364no—
406D3—364no—
407C22—272no—
408C22—541no—
409C22—541no—
410C22D3×#62411yescharacteristic
411C4—61no—
412C4—61no—
413C4—92no—
414C4—181no—
415C4—541no—
416C4#8895911yescharacteristic
417C4#8896511yescharacteristic
418C3—21no—
419C3—22no—
420C3—41no—
421C3#1059029911yescharacteristic, complemented
422C3#1059066512yescomplemented
423C2—542no—
424C2#1172281811yescharacteristic
425C2#1172281911yescharacteristic
426C2#1172291811yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D33
commutator, socleC22×C33C22×C4
Fitting, cosocleC2×C33×C4C23
prime radicalC22×C3⋊(C3⋊D3)C2×C4
hypocenterC33#432
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11722078C2semidirect
4. C3×#63572C2semidirect
5. C3×#63554C2semidirect
25. #63572D3semidirect
30. #63554D3semidirect
241. C33#432semidirect
372. C32#4705semidirect
373. C32#4835semidirect
421. C3#10590299semidirect
422. C3#10590665semidirect

Maximal subgroups

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

subgroupindex
C4×#62892
#117220782
#117220242
C3×#635722
C3×#635542
#105906653
#105902993