Finite Group Databasebeta

← #13166469

Subgroups — #13166469

Order 1728.

752 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 72 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×#89018C211yesmaximal, characteristic
2C2×#88926C211yesmaximal, characteristic, complemented
3C2×#88924C211yesmaximal, characteristic, complemented
4#11723103C211yesmaximal, characteristic, complemented
5C3×#63778C211yesmaximal, characteristic, complemented
6#11722726C211yesmaximal, characteristic, complemented
7#11722726C211yesmaximal, characteristic, complemented
8#10593834—31nomaximal
9C2×C4⋊(C3⋊(C3⋊D3))C411yescharacteristic
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C32×Dic6C411yescharacteristic
12C2×#6292C411yescharacteristic
13C2×#6292C411yescharacteristic
14C2×#6289C2211yescharacteristic
15C2×#6289C2211yescharacteristic
16#89018—22no—
17#89018—22no—
18C2×C3⋊#622C2211yescharacteristic
19C2×C3⋊#622C2211yescharacteristic
20C2×C3×#622C2211yescharacteristic
21C2×C3×#622C2211yescharacteristic
22#88926—22no—
23#88924—22no—
24C2×#3832—31no—
25C2×#3799—62no—
26C2×#3788—31no—
27C2×#3787—31no—
28#63778—31no—
29#63778—31no—
30#63778—31no—
31#63778D311yescharacteristic, complemented
32C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
33C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
34C22×C3×C3⋊D3C2×C411yescharacteristic
35C2×C33×C4C2×C411yescharacteristic
36C4⋊(C3⋊(C3⋊D3))—22no—
37C2×C3×C3⋊Dic3C2311yescharacteristic
38C3×C4×C3⋊D3—21no—
39C3×C4×C3⋊D3—21no—
40C2×C32×Dic3C2×C411yescharacteristic
41C2×C32×Dic3C2×C411yescharacteristic
42C32×Dic6—22no—
43#6292—22no—
44#6292—22no—
45#6289—21no—
46#6289—21no—
47#6289D412yes—
48#6289D412yes—
49C3⋊#622—21no—
50C3⋊#622—21no—
51C3×#622—21no—
52C3×#622—21no—
53#4727—91nomaximal
54C2×C32×Q8—31no—
55#3832—62no—
56#3832—62no—
57C2×C4⋊(C3⋊D3)—31no—
58C2×C4⋊(C3⋊D3)—31no—
59C2×C4⋊(C3⋊D3)—31no—
60C2×C4⋊(C3⋊D3)—62no—
61C2×C4⋊(C3⋊D3)—122no—
62C2×C4×C3⋊D3—31no—
63C2×C4×C3⋊D3—31no—
64C2×C4×C3⋊D3C2×D311yescharacteristic
65C2×C3×C4×D3—62no—
66C2×C3×Dic6—22no—
67C2×#626—62no—
68C2×#626—62no—
69C2×#624—62no—
70C2×#624—62no—
71#3799—68no—
72#3799—68no—
73C2×#622—31no—
74C2×#622—31no—
75C2×#622C2×D311yescharacteristic
76C2×#622C2×D311yescharacteristic
77#3788—62no—
78#3787—62no—
79C22×C33C22×C411yescharacteristic
80C2×C3⋊(C3⋊D3)—21no—
81C2×C3⋊(C3⋊D3)—21no—
82C2×C3⋊(C3⋊D3)—21no—
83C2×C3⋊(C3⋊D3)—21no—
84C2×C3⋊(C3⋊D3)—21no—
85C2×C3⋊(C3⋊D3)C22⋊C412yes—
86C2×C3×C3⋊D3—21no—
87C2×C3×C3⋊D3—21no—
88C2×C3×C3⋊D3—21no—
89C33×C4—21no—
90C3×C3⋊Dic3C2×D411yescharacteristic
91C3×C3⋊Dic3C2×D411yescharacteristic
92C32×Dic3—21no—
93C32×Dic3C22⋊C412yes—
94C2×#239—182no—
95C2×#235—91no—
96C2×ΓL(1,𝔽25)—91no—
97C2×D3×C8—91no—
98C3×C22⋊C8—91no—
99#830—91no—
100(C2×D3)⋊C8—91no—
101(C2×D3)⋊C8—91no—
102C22×C3⋊D3—31no—
103C22×C3⋊D3—31no—
104C22×C3⋊D3—62no—
105C22×C3⋊D3—62no—
106C22×C3⋊D3—122no—
107C22×C3⋊D3—122no—
108C22×C3⋊D3C4×D311yescharacteristic
109C22×C3×D3—62no—
110C32×Q8—62no—
111C2×C32×C4—22no—
112C2×C32×C4—31no—
113C2×C32×C4—31no—
114C2×C32×C4—42no—
115C2×C32×C4C4×D311yescharacteristic
116C2×C3⋊Dic3C22×D311yescharacteristic
117C4⋊(C3⋊D3)—62no—
118C4⋊(C3⋊D3)—62no—
119C4⋊(C3⋊D3)—62no—
120C4⋊(C3⋊D3)—68no—
121C4⋊(C3⋊D3)—128no—
122C4×C3⋊D3—21no—
123C4×C3⋊D3—21no—
124C4×C3⋊D3—32no—
125C4×C3⋊D3—32no—
126C4×C3⋊D3—61no—
127C4×C3⋊D3—61no—
128C2×C3×Dic3—22no—
129C2×C3×Dic3—22no—
130C2×C3×Dic3—62no—
131C3×C4×D3—64no—
132C3×C4×D3—64no—
133C3×Dic6—28no—
134#626—68no—
135#626—68no—
136#624—64no—
137#624—64no—
138#624—64no—
139#624—64no—
140#622—21no—
141#622—21no—
142#622—61no—
143#622—61no—
144#408—271noSylow
145C2×C33C2×C22⋊C411yescharacteristic
146C2×C33#13111yescharacteristic
147C2×C33#13111yescharacteristic
148C3⋊(C3⋊D3)—22no—
149C3⋊(C3⋊D3)—41no—
150C3×C3⋊D3—41no—
151C2×C3×Q8—62no—
152C22×C3×C4—91no—
153C2×#67—91no—
154C2×#67—91no—
155#239—188no—
156#239—188no—
157#235—182no—
158#235—182no—
159C2×D12—91no—
160C2×D12—182no—
161C2×D12—182no—
162C2×D12—182no—
163C2×D12—362no—
164C2×C4×D3—62no—
165C2×C4×D3—91no—
166C2×C4×D3—91no—
167C2×C4×D3—182no—
168C2×C4×D3—182no—
169C2×Dic6#15711yescharacteristic
170C2×C3×C8—91no—
171C2×C3×C8—91no—
172C2×C3⋊C8—91no—
173C2×C3⋊C8—91no—
174ΓL(1,𝔽25)—182no—
175D3×C8—182no—
176C22×C32—22no—
177C22×C32—42no—
178C22×C32C2×C4×D311yescharacteristic
179C2×C3⋊D3—21no—
180C2×C3⋊D3—21no—
181C2×C3⋊D3—21no—
182C2×C3⋊D3—32no—
183C2×C3⋊D3—64no—
184C2×C3⋊D3—61no—
185C2×C3⋊D3—64no—
186C2×C3⋊D3—61no—
187C2×C3⋊D3—61no—
188C2×C3⋊D3—61no—
189C2×C3⋊D3—61no—
190C2×C3⋊D3—124no—
191C2×C3⋊D3—124no—
192C2×C3⋊D3—122no—
193C2×C3⋊D3—122no—
194C2×C3⋊D3—122no—
195C2×C3⋊D3—122no—
196C2×C3⋊D3—242no—
197C2×C3⋊D3—242no—
198C2×C3⋊D3—242no—
199C2×C3⋊D3—242no—
200C2×C3×D3—64no—
201C2×C3×D3—122no—
202C2×C3×D3—122no—
203C32×C4—21no—
204C32×C4—24no—
205C32×C4—32no—
206C32×C4—44no—
207C32×C4—61no—
208C3⋊Dic3D3×D411yescharacteristic
209C3⋊Dic3D3×D411yescharacteristic
210C3×Dic3—24no—
211C3×Dic3—24no—
212C3×Dic3—62no—
213C3×Dic3—62no—
214C2×#41—271no—
215C2×M16—271no—
216C22×C8—271no—
217C22⋊C8—91no—
218C22⋊C8—271no—
219C22⋊C8—271no—
220C22⋊C8—271no—
221C33#40811yescharacteristic, complemented, Sylow
222C23×C3—91no—
223C22×D3—62no—
224C22×D3—91no—
225C22×D3—91no—
226C22×D3—182no—
227C22×D3—182no—
228C22×D3—362no—
229C22×D3—362no—
230C3×Q8—68no—
231C2×C3×C4—22no—
232C2×C3×C4—42no—
233C2×C3×C4—62no—
234C2×C3×C4—62no—
235C2×C3×C4—91no—
236C2×C3×C4—181no—
237C2×C3×C4—181no—
238C2×C3×C4C2×#15711yescharacteristic
239#67—182no—
240#67—182no—
241C2×Dic3—62no—
242C2×Dic3C2×#15711yescharacteristic
243C2×Dic3C2×#15711yescharacteristic
244D12—182no—
245D12—188no—
246D12—188no—
247D12—188no—
248D12—368no—
249C4×D3—64no—
250C4×D3—64no—
251C4×D3—92no—
252C4×D3—92no—
253C4×D3—184no—
254C4×D3—184no—
255C4×D3—181no—
256C4×D3—181no—
257C4×D3—184no—
258C4×D3—184no—
259Dic6—22no—
260C3×C8—181no—
261C3×C8—181no—
262C3⋊C8—181no—
263C3⋊C8—181no—
264C2×C32—22no—
265C2×C32—22no—
266C2×C32—22no—
267C2×C32—42no—
268C2×C32—42no—
269C2×C32—42no—
270C2×C32D3×C22⋊C411yescharacteristic
271C2×C32#90111yescharacteristic
272C2×C32#90711yescharacteristic
273C3⋊D3—41no—
274C3⋊D3—62no—
275C3⋊D3—124no—
276C3⋊D3—121no—
277C3⋊D3—124no—
278C3⋊D3—244no—
279C3⋊D3—244no—
280C3×D3—124no—
281#41—542no—
282#41—542no—
283C2×Q8—31no—
284C2×D4—271no—
285C2×D4—271no—
286C2×D4—271no—
287C22×C4—91no—
288C22×C4—271no—
289C22×C4—271no—
290M16—542no—
291C2×C8—91no—
292C2×C8—91no—
293C2×C8—271no—
294C2×C8—271no—
295C2×C8—542no—
296C22×C3—22no—
297C22×C3—42no—
298C22×C3—181no—
299C22×C3—181no—
300C22×C3—181no—
301C22×C3C22×#15711yescharacteristic
302C2×D3—64no—
303C2×D3—92no—
304C2×D3—122no—
305C2×D3—122no—
306C2×D3—184no—
307C2×D3—181no—
308C2×D3—181no—
309C2×D3—184no—
310C2×D3—181no—
311C2×D3—181no—
312C2×D3—181no—
313C2×D3—364no—
314C2×D3—362no—
315C2×D3—362no—
316C2×D3—362no—
317C2×D3—362no—
318C2×D3—364no—
319C2×D3—722no—
320C2×D3—722no—
321C2×D3—722no—
322C2×D3—722no—
323C3×C4—24no—
324C3×C4—21no—
325C3×C4—44no—
326C3×C4—64no—
327C3×C4—64no—
328C3×C4—91no—
329C3×C4—91no—
330Dic3—21no—
331Dic3—62no—
332Dic3—62no—
333Dic3C22⋊#15712yes—
334C32—22no—
335C32—42no—
336C32#472711yescharacteristic, complemented
337C23—91no—
338C23—271no—
339C23—271no—
340Q8—62no—
341D4—542no—
342D4—542no—
343D4—542no—
344C2×C4—31no—
345C2×C4—31no—
346C2×C4—91no—
347C2×C4—181no—
348C2×C4—181no—
349C2×C4—272no—
350C2×C4—272no—
351C2×C4—541no—
352C2×C4—541no—
353C2×C4D3×#15711yescharacteristic
354C8—181no—
355C8—181no—
356C8—541no—
357C8—541no—
358C2×C3—22no—
359C2×C3—22no—
360C2×C3—22no—
361C2×C3—42no—
362C2×C3—42no—
363C2×C3—42no—
364C2×C3—361no—
365C2×C3#6428911yescharacteristic
366C2×C3#6428911yescharacteristic
367C2×C3C2×C22⋊#15711yescharacteristic
368D3—124no—
369D3—182no—
370D3—364no—
371D3—364no—
372D3—361no—
373D3—724no—
374D3—724no—
375C22—181no—
376C22—181no—
377C22—181no—
378C22—272no—
379C22—541no—
380C22—541no—
381C22—541no—
382C22—541no—
383C22—541no—
384C22C2×D3×#15711yescharacteristic
385C4—21no—
386C4—32no—
387C4—61no—
388C4—91no—
389C4—91no—
390C3—22no—
391C3—42no—
392C3#1059383411yescharacteristic, complemented
393C2—361no—
394C2—542no—
395C2—1081no—
396C2#1172425211yescharacteristic
397C2#1172425311yescharacteristic
398C2D3×C22⋊#15711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socleC22×C33C22×C4
FittingC2×C33×C4C2×C4
prime radicalC2×#89018C2
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D3×#157
hypocenterC33#408
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88926C2semidirect
3. C2×#88924C2semidirect
4. #11723103C2semidirect
5. C3×#63778C2semidirect
6. #11722726C2semidirect
7. #11722726C2semidirect
31. #63778D3semidirect
221. C33#408semidirect
336. C32#4727semidirect
392. C3#10593834semidirect

Maximal subgroups

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

subgroupindex
C2×#890182
C2×#889262
C2×#889242
#117231032
C3×#637782
#117227262
#105938343
#47279