Finite Group Databasebeta

← #13167530

Subgroups — #13167530

Order 1728.

848 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 848 conjugacy classes of subgroups — too many to draw. Only its 99 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×#157⋊D3C211yesmaximal, characteristic
2C2×C3×C4×#157C211yesmaximal, characteristic, complemented
3C2×#88940C211yesmaximal, characteristic, complemented
4C2×#88939C212yesmaximal, complemented
5#11723700C212yesmaximal, complemented
6#10593454—31nomaximal
7C22×#1113C2212yes—
8C22×C3×#157C2211yescharacteristic
9C22×C3×#157C2211yescharacteristic
10C2×#157⋊D3—22no—
11C2×#157⋊D3—22no—
12C2×#157⋊D3C414yescomplemented
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×#6294C2212yes—
15C3×C4×#157—22no—
16#88940—22no—
17#88939—24no—
18C3×#3752—22no—
19#88814—22no—
20C22×GO+(4,𝔽2)—31no—
21C2×C4×#157D311yescharacteristic, complemented
22C2×#3773—31no—
23C2×D32⋊C4—32no—
24#63921—32no—
25D3×(C2×D3)⋊C4—62no—
26C22×C3×C3⋊D3—22no—
27C22×C3×C3⋊D3C2311yescharacteristic
28C2×#1113—22no—
29C2×#1113—22no—
30C2×#1113—22no—
31C2×#1113—22no—
32C2×#1113—22no—
33C2×#1113C2×C414yes—
34C2×C3×#157—21no—
35C2×C3×#157—21no—
36C2×C3×#157C2×C412yes—
37C2×C3×#157D412yes—
38C2×C3×#157D412yes—
39C2×C3×#157D412yescomplemented
40#157⋊D3—28no—
41C2×C33×C4D411yescharacteristic, complemented
42C2×C3×C3⋊Dic3—22no—
43C2×C3×C3⋊Dic3D411yescharacteristic, complemented
44C3×C4×C3⋊D3—21no—
45C3×C4×C3⋊D3—21no—
46#6294—22no—
47#6294—22no—
48#4939—91nomaximal
49C22×D32—32no—
50C22×D32—62no—
51C22×#157C2×D311yescharacteristic
52C22×#157C2×D311yescharacteristic
53C2×GO+(4,𝔽2)—34no—
54C2×GO+(4,𝔽2)—62no—
55C2×GO+(4,𝔽2)—62no—
56C2×C4×C3⋊D3C2×D311yescharacteristic
57C2×C3×C4×D3—62no—
58C2×#624—32no—
59C2×D3×Dic3—62no—
60C4×#157—22no—
61#3773—62no—
62D32⋊C4—64no—
63#3752—22no—
64C3×(C2×D3)⋊C4—62no—
65C3×(C2×D3)⋊C4—62no—
66C3×(C2×D3)⋊C4—122no—
67#3722—62no—
68(C2×D3)⋊Dic3—62no—
69(C2×D3)⋊Dic3—62no—
70C22×C33C2×D411yescharacteristic
71C2×C3×C3⋊D3—24no—
72C2×C3×C3⋊D3—42no—
73C2×C3×C3⋊D3—42no—
74C2×C3×C3⋊D3C22×C411yescharacteristic
75C2×C3×C3⋊D3C2×D411yescharacteristic
76C2×C3×C3⋊D3C2×D411yescharacteristic
77C2×C3×C3⋊D3C2×D411yescharacteristic
78C2×C3×C3⋊D3#4111yescharacteristic
79C2×C3×C3⋊D3#4111yescharacteristic
80#1113—24no—
81#1113—24no—
82#1113—42no—
83#1113—42no—
84C3×#157—22no—
85C3×#157C22⋊C414yescomplemented
86C33×C4—21no—
87C3×C3⋊Dic3—21no—
88C3×C3⋊Dic3—42no—
89C22×D12—91no—
90C2×C3×C42—91no—
91C2×(C2×D3)⋊C4—92no—
92C2×(C2×D3)⋊C4—92no—
93C2×C4⋊Dic3—91no—
94D3×C22⋊C4—182no—
95C22×C3⋊D3—22no—
96C22×C3⋊D3C22×D311yescharacteristic
97C22×C3×D3—62no—
98C22×C3×D3—62no—
99C22×C3×D3—62no—
100C22×C3×D3—122no—
101C2×D32—34no—
102C2×D32—64no—
103C2×D32—64no—
104C2×D32—62no—
105C2×D32—62no—
106C2×D32—62no—
107C2×D32—62no—
108C2×D32—62no—
109C2×D32—122no—
110C2×D32—122no—
111C2×D32—122no—
112C2×D32—122no—
113C2×#157—21no—
114C2×#157—21no—
115C2×#157C4×D312yes—
116C2×#157D1212yes—
117C2×#157D1212yescomplemented
118C2×#157#6712yes—
119GO+(4,𝔽2)—68no—
120C2×C32×C4—22no—
121C2×C32×C4—42no—
122C2×C32×C4D1211yescharacteristic, complemented
123C2×C3⋊Dic3—22no—
124C2×C3⋊Dic3D1211yescharacteristic, complemented
125C4×C3⋊D3—21no—
126C4×C3⋊D3—21no—
127C2×C3×Dic3—62no—
128C2×C3×Dic3—62no—
129C2×C3×Dic3—62no—
130C2×C3×Dic3—122no—
131C3×C4×D3—122no—
132C3×C4×D3—122no—
133#624—62no—
134#624—62no—
135D3×Dic3—122no—
136D3×Dic3—122no—
137#390—271noSylow
138C2×C33C4×D411yescharacteristic
139C2×C33#12111yescharacteristic
140C2×C33#12711yescharacteristic
141C3×C3⋊D3—44no—
142C3×C3⋊D3C2×C22⋊C411yescharacteristic
143C3×C3⋊D3C4×D411yescharacteristic
144C3×C3⋊D3#12111yescharacteristic
145C3×C3⋊D3#12411yescharacteristic
146C23×D3—92no—
147C23×D3—182no—
148C22×C3×C4—91no—
149C22×C3×C4—91no—
150C22×C3×C4—91no—
151C22×Dic3—92no—
152C2×D12—94no—
153C2×D12—182no—
154C2×D12—182no—
155C2×C4×D3—62no—
156C2×C4×D3—182no—
157C3×C22⋊C4—182no—
158C3×C42—182no—
159#217—182no—
160(C2×D3)⋊C4—62no—
161(C2×D3)⋊C4—62no—
162(C2×D3)⋊C4—122no—
163(C2×D3)⋊C4—182no—
164(C2×D3)⋊C4—182no—
165(C2×D3)⋊C4—184no—
166C4⋊Dic3—182no—
167C22×C32—22no—
168C22×C32—42no—
169C22×C32C2×D1211yescharacteristic
170C2×C3⋊D3—24no—
171C2×C3⋊D3—42no—
172C2×C3⋊D3—42no—
173C2×C3⋊D3C2×C4×D311yescharacteristic
174C2×C3⋊D3C2×D1211yescharacteristic
175C2×C3⋊D3C2×D1211yescharacteristic
176C2×C3⋊D3#23511yescharacteristic
177C2×C3⋊D3#23511yescharacteristic
178C2×C3⋊D3C2×#6711yescharacteristic
179C2×C3×D3—62no—
180C2×C3×D3—62no—
181C2×C3×D3—62no—
182C2×C3×D3—62no—
183C2×C3×D3—62no—
184C2×C3×D3—62no—
185C2×C3×D3—64no—
186C2×C3×D3—64no—
187C2×C3×D3—124no—
188C2×C3×D3—122no—
189C2×C3×D3—122no—
190C2×C3×D3—122no—
191C2×C3×D3—122no—
192C2×C3×D3—242no—
193C2×C3×D3—242no—
194D32—64no—
195D32—64no—
196D32—124no—
197D32—124no—
198D32—124no—
199D32—124no—
200D32—122no—
201D32—122no—
202#157—22no—
203#157(C2×D3)⋊C414yescomplemented
204C32×C4—21no—
205C32×C4—42no—
206C32×C4—82no—
207C3⋊Dic3—21no—
208C3⋊Dic3—42no—
209C3×Dic3—122no—
210C3×Dic3—122no—
211C3×Dic3—122no—
212C3×Dic3—242no—
213C22×D4—271no—
214C2×C4⋊C4—271no—
215C2×C22⋊C4—272no—
216C2×C22⋊C4—272no—
217C2×C42—91no—
218C33#39011yescharacteristic, complemented, Sylow
219C23×C3—91no—
220C23×C3—182no—
221C22×D3—62no—
222C22×D3—62no—
223C22×D3—62no—
224C22×D3—94no—
225C22×D3—122no—
226C22×D3—184no—
227C22×D3—182no—
228C22×D3—184no—
229C22×D3—182no—
230C22×D3—182no—
231C22×D3—182no—
232C22×D3—182no—
233C22×D3—362no—
234C22×D3—362no—
235C22×D3—362no—
236C22×D3—362no—
237C2×C3×C4—22no—
238C2×C3×C4—42no—
239C2×C3×C4—92no—
240C2×C3×C4—92no—
241C2×C3×C4—92no—
242C2×C3×C4—92no—
243C2×C3×C4—91no—
244C2×C3×C4—181no—
245C2×C3×C4—181no—
246C2×C3×C4—181no—
247C2×C3×C4—181no—
248C2×C3×C4—182no—
249C2×C3×C4GO+(4,𝔽2)11yescharacteristic, complemented
250C2×Dic3—62no—
251C2×Dic3—62no—
252C2×Dic3—62no—
253C2×Dic3—122no—
254C2×Dic3—182no—
255C2×Dic3—182no—
256D12—188no—
257C4×D3—122no—
258C4×D3—122no—
259C4×D3—362no—
260C4×D3—362no—
261C2×C32—22no—
262C2×C32—22no—
263C2×C32—22no—
264C2×C32—42no—
265C2×C32—42no—
266C2×C32—42no—
267C2×C32C4×D1211yescharacteristic
268C2×C32#87411yescharacteristic
269C2×C32#93011yescharacteristic
270C3⋊D3—44no—
271C3⋊D3C4×D1211yescharacteristic
272C3⋊D3#87511yescharacteristic
273C3⋊D3C2×(C2×D3)⋊C411yescharacteristic
274C3⋊D3#93011yescharacteristic
275C3×D3—62no—
276C3×D3—62no—
277C3×D3—62no—
278C3×D3—62no—
279C3×D3—124no—
280C3×D3—124no—
281C3×D3—244no—
282C24—272no—
283C2×D4—274no—
284C2×D4—542no—
285C2×D4—542no—
286C22×C4—91no—
287C22×C4—91no—
288C22×C4—91no—
289C22×C4—272no—
290C4⋊C4—542no—
291C22⋊C4—182no—
292C22⋊C4—542no—
293C22⋊C4—544no—
294C42—182no—
295C22×C3—22no—
296C22×C3—42no—
297C22×C3—91no—
298C22×C3—91no—
299C22×C3—91no—
300C22×C3—91no—
301C22×C3—91no—
302C22×C3—91no—
303C22×C3—184no—
304C22×C3—362no—
305C22×C3—362no—
306C22×C3C2×GO+(4,𝔽2)11yescharacteristic
307C2×D3—62no—
308C2×D3—62no—
309C2×D3—62no—
310C2×D3—62no—
311C2×D3—62no—
312C2×D3—62no—
313C2×D3—64no—
314C2×D3—64no—
315C2×D3—124no—
316C2×D3—122no—
317C2×D3—122no—
318C2×D3—122no—
319C2×D3—122no—
320C2×D3—184no—
321C2×D3—184no—
322C2×D3—242no—
323C2×D3—242no—
324C2×D3—364no—
325C2×D3—364no—
326C2×D3—364no—
327C2×D3—364no—
328C2×D3—362no—
329C2×D3—362no—
330C3×C4—21no—
331C3×C4—42no—
332C3×C4—82no—
333C3×C4—94no—
334C3×C4—182no—
335C3×C4—181no—
336C3×C4—362no—
337Dic3—122no—
338Dic3—122no—
339Dic3—122no—
340Dic3—242no—
341C32—22no—
342C32—42no—
343C32#493911yescharacteristic, complemented
344C23—91no—
345C23—182no—
346C23—274no—
347C23—542no—
348C23—542no—
349C23—542no—
350C23—542no—
351C23—542no—
352D4—548no—
353C2×C4—92no—
354C2×C4—92no—
355C2×C4—92no—
356C2×C4—92no—
357C2×C4—91no—
358C2×C4—181no—
359C2×C4—181no—
360C2×C4—181no—
361C2×C4—181no—
362C2×C4—182no—
363C2×C4—542no—
364C2×C4—542no—
365C2×C4#157⋊D311yescharacteristic, complemented
366C2×C3—22no—
367C2×C3—22no—
368C2×C3—22no—
369C2×C3—42no—
370C2×C3—42no—
371C2×C3—42no—
372C2×C3—91no—
373C2×C3—91no—
374C2×C3—91no—
375C2×C3—91no—
376C2×C3—364no—
377C2×C3C4×GO+(4,𝔽2)11yescharacteristic
378C2×C3#6423111yescharacteristic
379C2×C3#6424211yescharacteristic
380D3—62no—
381D3—62no—
382D3—62no—
383D3—62no—
384D3—124no—
385D3—124no—
386D3—244no—
387C22—91no—
388C22—91no—
389C22—91no—
390C22—91no—
391C22—91no—
392C22—91no—
393C22—184no—
394C22—362no—
395C22—362no—
396C22—544no—
397C22—544no—
398C22—1082no—
399C22—1082no—
400C22C2×#157⋊D311yescharacteristic
401C4—21no—
402C4—94no—
403C4—182no—
404C4—181no—
405C4—362no—
406C3—22no—
407C3—42no—
408C3#1059345411yescharacteristic, complemented
409C2—91no—
410C2—91no—
411C2—91no—
412C2—91no—
413C2—364no—
414C2C4×#157⋊D311yescharacteristic
415C2#1172434411yescharacteristic
416C2#1172435511yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutatorC2×C3×C3⋊D3C22×C4
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×#157⋊D3C2
cosocleC22×C3×C3⋊D3C23
hypercenterC2×C4#157⋊D3
hypocenterC33#390
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4×#157C2semidirect
3. C2×#88940C2semidirect
4. C2×#88939C2semidirect
5. #11723700C2semidirect
12. C2×#157⋊D3C4semidirect
21. C2×C4×#157D3semidirect
39. C2×C3×#157D4semidirect
41. C2×C33×C4D4semidirect
43. C2×C3×C3⋊Dic3D4semidirect
85. C3×#157C22⋊C4semidirect
117. C2×#157D12semidirect
122. C2×C32×C4D12semidirect
124. C2×C3⋊Dic3D12semidirect
203. #157(C2×D3)⋊C4semidirect
218. C33#390semidirect
249. C2×C3×C4GO+(4,𝔽2)semidirect
343. C32#4939semidirect
365. C2×C4#157⋊D3semidirect
408. C3#10593454semidirect

Maximal subgroups

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

subgroupindex
C22×#157⋊D32
C2×C3×C4×#1572
C2×#889402
C2×#889392
#117237002
#105934543
#49399