Finite Group Databasebeta

← D4×#88907 #72253056

Subgroups — D4×#88907 #72253056

Order 3456.

10888 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 10888 conjugacy classes of subgroups — too many to draw. Only its 150 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×#88907C212yesmaximal, complemented
2C4×#88907C211yesmaximal, characteristic, complemented
3D4×C22⋊(C22⋊C9)C313yesmaximal, complemented
4C24×C32×D4C311yesmaximal, characteristic
5C2×#88907—22no—
6C2×#88907C2211yescharacteristic
7D4×#1094—45nomaximal
8C26×C32C2×C312yes—
9C23×C32×D4—35no—
10C23×C32×D4—310no—
11C23×C32×D4—35no—
12C24×C32×C4C2×C311yescharacteristic
13C22×C22⋊(C22⋊C9)C2×C316yescomplemented
14C4×C22⋊(C22⋊C9)C2×C313yescomplemented
15#88907D411yescharacteristic, direct product factor
16C22×#1094—410no—
17C4×#1094—45no—
18C24×C3×D4—31no—
19C24×C3×D4C3211yescharacteristic
20C25×C32—22no—
21C25×C32—310no—
22C25×C32—310no—
23C25×C32—610no—
24C25×C32C22×C311yescharacteristic
25C22×C32×D4—360no—
26C22×C32×D4—330no—
27C22×C32×D4—360no—
28C22×C32×D4—310no—
29C22×C32×D4—310no—
30C22×C32×D4—35no—
31C22×C32×D4—310no—
32C22×C32×D4A415yes—
33C23×C32×C4—35no—
34C23×C32×C4—35no—
35C2×C22⋊(C22⋊C9)—26no—
36C2×C22⋊(C22⋊C9)C22×C313yes—
37D4×C22⋊C9—415no—
38C2×#1094—45no—
39C2×#1094—810no—
40D4×3−1+2—161no—
41C26×C3—32no—
42C26×C3C2×C3212yes—
43C23×C3×D4—35no—
44C23×C3×D4—310no—
45C23×C3×D4—315no—
46C23×C3×D4—315no—
47C23×C3×D4—330no—
48C23×C3×D4—35no—
49C24×C3×C4—31no—
50C24×C3×C4C2×C3211yescharacteristic
51C24×C32—35no—
52C24×C32—35no—
53C24×C32—360no—
54C24×C32—320no—
55C24×C32—310no—
56C24×C32—610no—
57C24×C32—610no—
58C24×C32—660no—
59C24×C32—660no—
60C24×C32—610no—
61C24×C32—610no—
62C24×C32C3×D411yescharacteristic
63C24×C32C2×A4110yes—
64C2×C32×D4—3120no—
65C2×C32×D4—360no—
66C2×C32×D4—360no—
67C2×C32×D4—330no—
68C2×C32×D4—330no—
69C2×C32×D4—35no—
70C2×C32×D4—35no—
71C2×C32×D4—310no—
72C22×C32×C4—330no—
73C22×C32×C4—310no—
74C22×C32×C4—35no—
75C22×C32×C4C2×A415yes—
76C22⋊(C22⋊C9)C3×D413yescomplemented
77C22×C22⋊C9—430no—
78C4×C22⋊C9—415no—
79C24×D43−1+211yescharacteristic, complemented, Sylow
80C22×3−1+2—162no—
81#1094—45no—
82C4×3−1+2—161no—
83C25×C3—22no—
84C25×C3—330no—
85C25×C3—330no—
86C25×C3—310no—
87C25×C3—310no—
88C25×C3—31no—
89C25×C3—62no—
90C25×C3—610no—
91C25×C3—630no—
92C25×C3C22×C3211yescharacteristic
93C22×C3×D4—360no—
94C22×C3×D4—3180no—
95C22×C3×D4—3180no—
96C22×C3×D4—390no—
97C22×C3×D4—330no—
98C22×C3×D4—330no—
99C22×C3×D4—360no—
100C22×C3×D4—330no—
101C22×C3×D4—310no—
102C22×C3×D4—310no—
103C22×C3×D4—330no—
104C22×C3×D4—315no—
105C22×C3×D4—310no—
106C22×C3×D4—35no—
107C22×C3×D4—35no—
108C22×C3×D4C3×A415yes—
109C23×C3×C4—315no—
110C23×C3×C4—315no—
111C23×C3×C4—35no—
112C23×C3×C4—35no—
113C23×C32—210no—
114C23×C32—310no—
115C23×C32—330no—
116C23×C32—35no—
117C23×C32—35no—
118C23×C32—360no—
119C23×C32—310no—
120C23×C32—310no—
121C23×C32—610no—
122C23×C32—630no—
123C23×C32—620no—
124C23×C32—660no—
125C23×C32—660no—
126C23×C32—660no—
127C23×C32—6120no—
128C23×C32—610no—
129C23×C32C22×A415yes—
130C32×D4—360no—
131C32×D4—310no—
132C32×D4—310no—
133C32×D4—35no—
134C32×D4C22⋊A411yescharacteristic
135C2×C32×C4—330no—
136C2×C32×C4—35no—
137C2×C32×C4—35no—
138C2×C22⋊C9—415no—
139C2×C22⋊C9—830no—
140D4×C9—163no—
141C26C2×3−1+212yescomplemented
142C23×D4—35no—
143C23×D4—35no—
144C23×D4—310no—
145C24×C4C2×3−1+211yescharacteristic, complemented
146C2×3−1+2—161no—
147C2×3−1+2—322no—
148C24×C3—35no—
149C24×C3—315no—
150C24×C3—315no—
151C24×C3—35no—
152C24×C3—3180no—
153C24×C3—360no—
154C24×C3—360no—
155C24×C3—320no—
156C24×C3—330no—
157C24×C3—310no—
158C24×C3—31no—
159C24×C3—310no—
160C24×C3—610no—
161C24×C3—610no—
162C24×C3—610no—
163C24×C3—610no—
164C24×C3—630no—
165C24×C3—630no—
166C24×C3—630no—
167C24×C3—630no—
168C24×C3—660no—
169C24×C3—6180no—
170C24×C3—6180no—
171C24×C3—660no—
172C24×C3C32×D411yescharacteristic
173C24×C3C2×C3×A4110yes—
174C2×C3×D4—3360no—
175C2×C3×D4—3120no—
176C2×C3×D4—3180no—
177C2×C3×D4—3180no—
178C2×C3×D4—390no—
179C2×C3×D4—390no—
180C2×C3×D4—360no—
181C2×C3×D4—360no—
182C2×C3×D4—330no—
183C2×C3×D4—330no—
184C2×C3×D4—315no—
185C2×C3×D4—315no—
186C2×C3×D4—330no—
187C2×C3×D4—35no—
188C2×C3×D4—35no—
189C2×C3×D4—310no—
190C22×C3×C4—390no—
191C22×C3×C4—330no—
192C22×C3×C4—330no—
193C22×C3×C4—310no—
194C22×C3×C4—315no—
195C22×C3×C4—35no—
196C22×C3×C4—35no—
197C22×C3×C4C2×C3×A415yes—
198C22×C32—35no—
199C22×C32—330no—
200C22×C32—310no—
201C22×C32—35no—
202C22×C32—310no—
203C22×C32—610no—
204C22×C32—610no—
205C22×C32—610no—
206C22×C32—610no—
207C22×C32—660no—
208C22×C32—660no—
209C22×C32D4×A415yes—
210C22×C32C2×C22⋊A412yes—
211C32×C4—35no—
212C32×C4C2×C22⋊A411yescharacteristic
213C22×C9—166no—
214C22⋊C9—415no—
215C4×C9—163no—
216C25—22no—
217C25—310no—
218C25—310no—
219C25—610no—
220C25C22×3−1+211yescharacteristic
221C22×D4—360no—
222C22×D4—360no—
223C22×D4—330no—
224C22×D4—310no—
225C22×D4—310no—
226C22×D4—310no—
227C22×D4—35no—
228C22×D4#109415yescomplemented
229C23×C4—35no—
230C23×C4—35no—
2313−1+2—161noSylow
232C23×C3—210no—
233C23×C3—330no—
234C23×C3—310no—
235C23×C3—330no—
236C23×C3—390no—
237C23×C3—315no—
238C23×C3—315no—
239C23×C3—35no—
240C23×C3—35no—
241C23×C3—3180no—
242C23×C3—360no—
243C23×C3—330no—
244C23×C3—330no—
245C23×C3—310no—
246C23×C3—310no—
247C23×C3—35no—
248C23×C3—610no—
249C23×C3—610no—
250C23×C3—630no—
251C23×C3—630no—
252C23×C3—620no—
253C23×C3—660no—
254C23×C3—660no—
255C23×C3—660no—
256C23×C3—6120no—
257C23×C3—690no—
258C23×C3—660no—
259C23×C3—6180no—
260C23×C3—6180no—
261C23×C3—6180no—
262C23×C3—6360no—
263C23×C3—630no—
264C23×C3—610no—
265C23×C3C22×C3×A415yes—
266C3×D4—3180no—
267C3×D4—330no—
268C3×D4—360no—
269C3×D4—310no—
270C3×D4—330no—
271C3×D4—315no—
272C3×D4—310no—
273C3×D4—35no—
274C3×D4—31no—
275C3×D4C3×C22⋊A411yescharacteristic
276C2×C3×C4—390no—
277C2×C3×C4—330no—
278C2×C3×C4—315no—
279C2×C3×C4—315no—
280C2×C3×C4—35no—
281C2×C3×C4—35no—
282C2×C32—22no—
283C2×C32—35no—
284C2×C32—35no—
285C2×C32—610no—
286C2×C32C22×C22⋊A411yescharacteristic
287C2×C9—163no—
288C2×C9—326no—
289C24—35no—
290C24—35no—
291C24—360no—
292C24—320no—
293C24—310no—
294C24—610no—
295C24—610no—
296C24—610no—
297C24—610no—
298C24—660no—
299C24—660no—
300C24D4×3−1+211yescharacteristic, complemented
301C24C2×#1094110yescomplemented
302C2×D4—3120no—
303C2×D4—360no—
304C2×D4—360no—
305C2×D4—330no—
306C2×D4—330no—
307C2×D4—35no—
308C2×D4—35no—
309C2×D4—310no—
310C22×C4—330no—
311C22×C4—310no—
312C22×C4—35no—
313C22×C4C2×#109415yescomplemented
314C22×C3—35no—
315C22×C3—315no—
316C22×C3—315no—
317C22×C3—330no—
318C22×C3—390no—
319C22×C3—310no—
320C22×C3—330no—
321C22×C3—35no—
322C22×C3—330no—
323C22×C3—310no—
324C22×C3—32no—
325C22×C3—35no—
326C22×C3—610no—
327C22×C3—610no—
328C22×C3—610no—
329C22×C3—610no—
330C22×C3—630no—
331C22×C3—630no—
332C22×C3—630no—
333C22×C3—630no—
334C22×C3—6180no—
335C22×C3—6180no—
336C22×C3—660no—
337C22×C3—660no—
338C22×C3C3×D4×A415yes—
339C22×C3C2×C3×C22⋊A412yes—
340C3×C4—315no—
341C3×C4—35no—
342C3×C4—31no—
343C3×C4C2×C3×C22⋊A411yescharacteristic
344C32D4×C22⋊A411yescharacteristic
345C9—163no—
346C23—210no—
347C23—310no—
348C23—330no—
349C23—35no—
350C23—35no—
351C23—360no—
352C23—310no—
353C23—310no—
354C23—610no—
355C23—630no—
356C23—620no—
357C23—660no—
358C23—660no—
359C23—660no—
360C23—6120no—
361C23—610no—
362C23C22×#109415yes—
363D4—360no—
364D4—310no—
365D4—310no—
366D4—35no—
367D4#8890711yescharacteristic, direct product factor
368C2×C4—330no—
369C2×C4—35no—
370C2×C4—35no—
371C2×C3—22no—
372C2×C3—315no—
373C2×C3—315no—
374C2×C3—35no—
375C2×C3—35no—
376C2×C3—31no—
377C2×C3—62no—
378C2×C3—630no—
379C2×C3—610no—
380C2×C3C22×C3×C22⋊A411yescharacteristic
381C22—35no—
382C22—310no—
383C22—330no—
384C22—35no—
385C22—310no—
386C22—610no—
387C22—610no—
388C22—610no—
389C22—610no—
390C22—660no—
391C22—660no—
392C22D4×#109415yescomplemented
393C22C2×#8890712yescomplemented
394C4—35no—
395C4C2×#8890711yescharacteristic, complemented
396C3—31no—
397C3C3×D4×C22⋊A411yescharacteristic
398C2—22no—
399C2—35no—
400C2—35no—
401C2—610no—
402C2C22×#8890711yescharacteristic
subgroupG/H
center, FrattiniC2×C3C22×C3×C22⋊A4
commutator, socle, cosocleC25×C3C22×C32
Fitting, prime radicalC24×C32×D4C3
hypercenterC32×D4C22⋊A4
hypocenterC24D4×3−1+2
LCSG ▷ C25×C3 ▷ C24
UCSC1 ◁ C2×C3 ◁ C32×D4

Complements

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

NH ≅ G/Ntype
1. C22×#88907C2semidirect
2. C4×#88907C2semidirect
3. D4×C22⋊(C22⋊C9)C3semidirect
13. C22×C22⋊(C22⋊C9)C2×C3semidirect
14. C4×C22⋊(C22⋊C9)C2×C3semidirect
15. #88907D4direct (N × H)
76. C22⋊(C22⋊C9)C3×D4semidirect
79. C24×D43−1+2semidirect
141. C26C2×3−1+2semidirect
145. C24×C4C2×3−1+2semidirect
228. C22×D4#1094semidirect
300. C24D4×3−1+2semidirect
301. C24C2×#1094semidirect
313. C22×C4C2×#1094semidirect
367. D4#88907direct (N × H)
392. C22D4×#1094semidirect
393. C22C2×#88907semidirect
395. C4C2×#88907semidirect

Maximal subgroups

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

subgroupindex
C22×#889072
C4×#889072
D4×C22⋊(C22⋊C9)3
C24×C32×D43
D4×#10944